a store front display in nyc showing price tags with decimals. the use of decimals in the real world
TRANSCRIPT
A store front display in NYC showing price tags with decimals
The Use of Decimals in the Real World
Place Values in a Decimal and the Expanded Form of a Decimal
Given a decimal
58147
place value is 1
100001
is valueplace
1000
1 is valueplace
1001
is valueplace
101
is valueplace
Place Values in a Decimal and the Expanded Form of a Decimal
Given a decimal
58147We can therefore rewrite it in an expanded form
which can then be converted to a mixed number
100007
10004
1001
108
5
100008147
5
How a decimal is read
A decimal is read as if they were written in fraction form except that the decimal point is read ldquoandrdquo We donrsquot use the word ldquoandrdquo in any other places
Example 1204657 is read as
ldquoOne thousand two hundred four and six hundred fifty seven thousandthsrdquo
ie
This method works only for short decimals and when there are many digits after the decimal point such as 271828 the scientists and engineers will call it
ldquoTwo point seven one eight two eightrdquo
6571204
1000
How is a decimal read
A problem for discussion
Is there any chance of confusion when a student reads the decimal 120438 as
ldquoOne thousand and two hundred and four and thirty eight hundredthsrdquo
ResponseIn daily uses it seldomly causes a problem but it doesnrsquot mean that confusion will never happen either For instance consider the number
36800041If you read this as
ldquoThirty six thousand and eight hundred and forty one thousandthsrdquo
then the person who heard this may interpret it as ldquo3600841rdquo
Remarks1 The decimal notation is not unique throughout the world even up to this dayFor example the British uses 3middot1416 (with the dot higher) while the French and German use 31416
2 When we change a fraction into a decimal the representation is not always terminating
and we will see later that even terminating decimals have non-terminating representations such as
025 = 02499999 hellip
333031
eg
dot comma Momayyez unknown
Comparing Decimals
Research shows that most students believe that 0287 is bigger than 035
becausea) 0287 has more digits than 035
b) 0287 is read as ldquotwo hundred eighty seven thousandthsrdquo which sounds larger than ldquothirty five hundredthsrdquo particularly when they are not familiar with the fact than ldquoone thousandthrdquo is really smaller than ldquoone hundredthrdquo
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Place Values in a Decimal and the Expanded Form of a Decimal
Given a decimal
58147
place value is 1
100001
is valueplace
1000
1 is valueplace
1001
is valueplace
101
is valueplace
Place Values in a Decimal and the Expanded Form of a Decimal
Given a decimal
58147We can therefore rewrite it in an expanded form
which can then be converted to a mixed number
100007
10004
1001
108
5
100008147
5
How a decimal is read
A decimal is read as if they were written in fraction form except that the decimal point is read ldquoandrdquo We donrsquot use the word ldquoandrdquo in any other places
Example 1204657 is read as
ldquoOne thousand two hundred four and six hundred fifty seven thousandthsrdquo
ie
This method works only for short decimals and when there are many digits after the decimal point such as 271828 the scientists and engineers will call it
ldquoTwo point seven one eight two eightrdquo
6571204
1000
How is a decimal read
A problem for discussion
Is there any chance of confusion when a student reads the decimal 120438 as
ldquoOne thousand and two hundred and four and thirty eight hundredthsrdquo
ResponseIn daily uses it seldomly causes a problem but it doesnrsquot mean that confusion will never happen either For instance consider the number
36800041If you read this as
ldquoThirty six thousand and eight hundred and forty one thousandthsrdquo
then the person who heard this may interpret it as ldquo3600841rdquo
Remarks1 The decimal notation is not unique throughout the world even up to this dayFor example the British uses 3middot1416 (with the dot higher) while the French and German use 31416
2 When we change a fraction into a decimal the representation is not always terminating
and we will see later that even terminating decimals have non-terminating representations such as
025 = 02499999 hellip
333031
eg
dot comma Momayyez unknown
Comparing Decimals
Research shows that most students believe that 0287 is bigger than 035
becausea) 0287 has more digits than 035
b) 0287 is read as ldquotwo hundred eighty seven thousandthsrdquo which sounds larger than ldquothirty five hundredthsrdquo particularly when they are not familiar with the fact than ldquoone thousandthrdquo is really smaller than ldquoone hundredthrdquo
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Place Values in a Decimal and the Expanded Form of a Decimal
Given a decimal
58147We can therefore rewrite it in an expanded form
which can then be converted to a mixed number
100007
10004
1001
108
5
100008147
5
How a decimal is read
A decimal is read as if they were written in fraction form except that the decimal point is read ldquoandrdquo We donrsquot use the word ldquoandrdquo in any other places
Example 1204657 is read as
ldquoOne thousand two hundred four and six hundred fifty seven thousandthsrdquo
ie
This method works only for short decimals and when there are many digits after the decimal point such as 271828 the scientists and engineers will call it
ldquoTwo point seven one eight two eightrdquo
6571204
1000
How is a decimal read
A problem for discussion
Is there any chance of confusion when a student reads the decimal 120438 as
ldquoOne thousand and two hundred and four and thirty eight hundredthsrdquo
ResponseIn daily uses it seldomly causes a problem but it doesnrsquot mean that confusion will never happen either For instance consider the number
36800041If you read this as
ldquoThirty six thousand and eight hundred and forty one thousandthsrdquo
then the person who heard this may interpret it as ldquo3600841rdquo
Remarks1 The decimal notation is not unique throughout the world even up to this dayFor example the British uses 3middot1416 (with the dot higher) while the French and German use 31416
2 When we change a fraction into a decimal the representation is not always terminating
and we will see later that even terminating decimals have non-terminating representations such as
025 = 02499999 hellip
333031
eg
dot comma Momayyez unknown
Comparing Decimals
Research shows that most students believe that 0287 is bigger than 035
becausea) 0287 has more digits than 035
b) 0287 is read as ldquotwo hundred eighty seven thousandthsrdquo which sounds larger than ldquothirty five hundredthsrdquo particularly when they are not familiar with the fact than ldquoone thousandthrdquo is really smaller than ldquoone hundredthrdquo
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
How a decimal is read
A decimal is read as if they were written in fraction form except that the decimal point is read ldquoandrdquo We donrsquot use the word ldquoandrdquo in any other places
Example 1204657 is read as
ldquoOne thousand two hundred four and six hundred fifty seven thousandthsrdquo
ie
This method works only for short decimals and when there are many digits after the decimal point such as 271828 the scientists and engineers will call it
ldquoTwo point seven one eight two eightrdquo
6571204
1000
How is a decimal read
A problem for discussion
Is there any chance of confusion when a student reads the decimal 120438 as
ldquoOne thousand and two hundred and four and thirty eight hundredthsrdquo
ResponseIn daily uses it seldomly causes a problem but it doesnrsquot mean that confusion will never happen either For instance consider the number
36800041If you read this as
ldquoThirty six thousand and eight hundred and forty one thousandthsrdquo
then the person who heard this may interpret it as ldquo3600841rdquo
Remarks1 The decimal notation is not unique throughout the world even up to this dayFor example the British uses 3middot1416 (with the dot higher) while the French and German use 31416
2 When we change a fraction into a decimal the representation is not always terminating
and we will see later that even terminating decimals have non-terminating representations such as
025 = 02499999 hellip
333031
eg
dot comma Momayyez unknown
Comparing Decimals
Research shows that most students believe that 0287 is bigger than 035
becausea) 0287 has more digits than 035
b) 0287 is read as ldquotwo hundred eighty seven thousandthsrdquo which sounds larger than ldquothirty five hundredthsrdquo particularly when they are not familiar with the fact than ldquoone thousandthrdquo is really smaller than ldquoone hundredthrdquo
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
How is a decimal read
A problem for discussion
Is there any chance of confusion when a student reads the decimal 120438 as
ldquoOne thousand and two hundred and four and thirty eight hundredthsrdquo
ResponseIn daily uses it seldomly causes a problem but it doesnrsquot mean that confusion will never happen either For instance consider the number
36800041If you read this as
ldquoThirty six thousand and eight hundred and forty one thousandthsrdquo
then the person who heard this may interpret it as ldquo3600841rdquo
Remarks1 The decimal notation is not unique throughout the world even up to this dayFor example the British uses 3middot1416 (with the dot higher) while the French and German use 31416
2 When we change a fraction into a decimal the representation is not always terminating
and we will see later that even terminating decimals have non-terminating representations such as
025 = 02499999 hellip
333031
eg
dot comma Momayyez unknown
Comparing Decimals
Research shows that most students believe that 0287 is bigger than 035
becausea) 0287 has more digits than 035
b) 0287 is read as ldquotwo hundred eighty seven thousandthsrdquo which sounds larger than ldquothirty five hundredthsrdquo particularly when they are not familiar with the fact than ldquoone thousandthrdquo is really smaller than ldquoone hundredthrdquo
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Remarks1 The decimal notation is not unique throughout the world even up to this dayFor example the British uses 3middot1416 (with the dot higher) while the French and German use 31416
2 When we change a fraction into a decimal the representation is not always terminating
and we will see later that even terminating decimals have non-terminating representations such as
025 = 02499999 hellip
333031
eg
dot comma Momayyez unknown
Comparing Decimals
Research shows that most students believe that 0287 is bigger than 035
becausea) 0287 has more digits than 035
b) 0287 is read as ldquotwo hundred eighty seven thousandthsrdquo which sounds larger than ldquothirty five hundredthsrdquo particularly when they are not familiar with the fact than ldquoone thousandthrdquo is really smaller than ldquoone hundredthrdquo
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
dot comma Momayyez unknown
Comparing Decimals
Research shows that most students believe that 0287 is bigger than 035
becausea) 0287 has more digits than 035
b) 0287 is read as ldquotwo hundred eighty seven thousandthsrdquo which sounds larger than ldquothirty five hundredthsrdquo particularly when they are not familiar with the fact than ldquoone thousandthrdquo is really smaller than ldquoone hundredthrdquo
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Comparing Decimals
Research shows that most students believe that 0287 is bigger than 035
becausea) 0287 has more digits than 035
b) 0287 is read as ldquotwo hundred eighty seven thousandthsrdquo which sounds larger than ldquothirty five hundredthsrdquo particularly when they are not familiar with the fact than ldquoone thousandthrdquo is really smaller than ldquoone hundredthrdquo
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103 = 3 7 6 1 5
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Multiplying or Dividing Decimals by Powers of 10
Multiplying a decimal by 10n is the same as moving the decimal point to the right n places adding place holders if necessary
eg 37615 times 103
Dividing a decimal by 10n is the same as moving the decimal point to the left n places adding place holders if necessary
eg 74328 times 105 = 74328000
eg 37615 divide 103 = 00037615
eg 74328 divide 102 = 74328
= 37615
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up (click to see the animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
6724
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Next we compare the whole number portions ndash whichever has the larger whole number portion is the bigger decimal
In this case they are both equal so we have to compare the digits in the first column on the right of the decimal point
672189
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
6724
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
(Please click to see animation)
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6724672189
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
In this column the two digits are also equal so we have to keep moving to the right until we can find a column that has two different digits
Ordering Decimals
Given two decimals how do we determine quickly which one is larger
The method is rather easy to learn from just a few examples
Example
Which one is larger 6724 or 672189
SolutionWe first put one of them above the other such that the decimal points are lined up
6721896724
Now we see that the upper digit in the highlighted column is larger there for the corresponding number (ie the upper one) is larger than the lower one
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Addition and Subtraction of Decimals These two operations are respectively similar to the addition and subtraction of whole numbers except that the numbers are aligned by the decimal points rather than the last digits (counting from the left)
Example
3416 + 23096
Incorrect
3416 + 23096
Correctand we have to treat any empty space as a 0
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
There are two ways to carry out this operation(I) Converting the decimals to fractions
Example
065 times 2417
Multiplication of Decimals
= 157105
1000417
210065
10002417
10065
1000100241765
000100
157105
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
(II) Ignore the decimal points first and multiply the two numbers as if they were whole numbers In the end we insert the decimal point back to the answer in the proper position
Example
065 times 2417 can be first treated as 65 times 2417 = 157105
the decimal point is then re-inserted to the product such that
ldquothe number of decimal places in the answer is equal to the sum of the number of decimal places in the multiplicandsrdquo
In this particular case 065 has two decimal places and 2417 has three decimal places Hence their product should have 5 decimal places and this means that
065 times 2417 = 157105
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Division of Decimals
6 5 5 2 2 1
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 1 Move the decimal point in the divisor to the right until it becomes a whole number (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend (click)
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
6 5 5 2 2 1 Step 1 Move the decimal point in the divisor to the right until it becomes a whole number
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers (click)
Division of Decimals
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1
5
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3
6
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Step 2 Move the decimal point in the dividend to the right by the same amount
Step 3 Put a decimal point above that one in the
dividend
Division of Decimals
The process of long division is similar to that of dividing whole numbers with some modifications
Example 2556 divide 12
6 5 5 2 2 1
Step 4 Divide as if we are dividing whole numbers
2
2 4-
1 5
1
1 2-
3 6
3
3 6-
0
Therefore 2556 divide 12 = 213
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Converting Fractions to Decimals
10001
)74000(1000
17
400070004000
74
Exploration
Find the first 3 digits in the decimal expansion of 47
We first consider the following
From long division we have 4000 divide 7 = 571 r3 hence
It is not hard to see that we can actually insert the decimal point first in the quotient and then continue the long division as long as we want (see next page)
57101000571
10001
57174
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Converting Fractions to Decimals
ConclusionThe decimal expansion of a fraction ab can be obtained by long division
53
49 50
1
0571428 0000000047
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Converting Fractions to Decimals
FactThe decimal expansion of any fraction ab is either terminating or repeating
TheoremIf the fraction ab is in its reduced form then its decimal expansion is terminating if and only if b is one of the following forms
(1) a product of 2rsquos only(2) a product of 5rsquos only(3) a product of 2rsquos and 5rsquos only
Examples is not terminating113 is terminating
62517
is terminating17 52
76407
192021
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Converting Fractions to Decimals
TheoremIf the fraction ab is in its reduced form and
b = 2m5n
then the decimal expansion of ab is terminating with number of decimal places exactly equal to
maxm n
Now we know what kind of fractions will have terminating decimal expansions but can we predict how many decimal places there will be in the expansion
Example The decimal expansion of
1352
13
40
13 will have exactly 3 decimal places
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Converting Fractions to Decimals
One more question If we know that a certain fraction has repeating decimal expansion can we predict its cycle length
Unfortunately there is no formula to calculate the precise cycle length All we know is an upper bound and a small (not too helpful) property
TheoremIf p is a prime number other than 2 and 5 then the cycle length of 1p is at most (p ndash 1) and the cycle length must divide (p ndash 1)
ExampleThe cycle length of 131 is at most 30 and it must divide 30 In fact the cycle length of 131 is 15
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Converting Fractions to Decimals
More examples
prime number p cycle length of 1p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
There is no obvious pattern on the cycle length and a large denominator can have a small cycle length
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Converting Fractions to Decimals
More facts (optional)
1 If p is a prime other than 2 or 5 then the cycle length of 1(p2) is at mostp(p ndash 1) and the cycle length must divide p(p ndash 1)
2 If p and q are different primes other than 2 and 5 then the cycle length of 1pq will be at most (p ndash 1)(q ndash 1) and divides (p ndash 1)(q ndash 1)
Example cycle length of 17 is 6 cycle length of 149 is 42 (= 7times6)
ExampleCycle length of 1(7times11) is less than 6times10 = 60 and must divide 60 It turns out that the cycle length of 177 is only 6
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Converting Decimals to Fractions
From the previous theorem we see that only repeating or terminating decimals can be converted to a fraction
Procedures(1) terminating decimal eg
035742 = 35742 100000
The number of 0rsquos in the denominator is equal to the number of decimal places
(2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Converting Decimals to Fractions
Procedures2) repeating decimals of type I eg
02222 middotmiddotmiddot = 2 9
047474747 middotmiddotmiddot = 47 99
0528528528 middotmiddotmiddot = 528 999
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-
900519
90057576
900576570
9006571057
9006
900957
9006
10057
Converting Decimals to Fractions
Procedures
3) repeating decimals of type II eg
00626262 middotmiddotmiddot = 62 990
000626262 middotmiddotmiddot = 62 9900
0000344934493449 middotmiddotmiddot = 3449 9999000
4) repeating decimals of type III eg
0576666 middotmiddotmiddot = 057 + 0006666 middotmiddotmiddot
- PowerPoint Presentation
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Addition and Subtraction of Decimals
- Multiplication of Decimals
- Slide 32
- Division of Decimals
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Converting Fractions to Decimals
- Slide 46
- Slide 47
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Converting Decimals to Fractions
- Slide 53
- Slide 54
- Slide 55
-