decimals as an extension of percents © math as a second language all rights reserved next #8 taking...

Decimals as an

Extension of

Percents© Math As A Second Language All Rights Reserved

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#8

Taking the Fearout of Math

0.25

25%

In our discussion of percents we pointed out that people, for whatever

reason, were most comfortable with scales that went from 0 to a power of 10. We

happened to focus on percents (that is, per hundred). It would have been just as logical to choose a scale that went from 0 to 10 (perhaps calling it a “perdec”) or a scale that went from 0 to 1,000 (perhaps

calling it “permill”), etc.

© Math As A Second Language All Rights Reserved

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As an illustration, let’s suppose that there was a quiz with 7 equally weighted

questions and you got the correct answer to 6 of the problems but omitted answering the seventh question. Then the fractional part of the test that you did correctly was 6/7. If the perfect score was 10, then your exact score would be given by…

6/7 of 10 = 6/7 × 10 = 60/7

= 60 ÷ 7

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To begin to capture the flavor of how

decimals are related to our present

discussion, let’s carry out the division using

the traditional algorithm…

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7 60

8

– 56

4

4/7

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next In terms of our corn bread model, this

tells us that if a perfect score is represented by 10 (the corn bread is presliced into

10 equally-sized pieces), 6/7 is worth more than 8 pieces but less than 9 pieces

(in fact, it is exactly 84/7 pieces).

Less than 6/7

Greater than 6/7

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next Next suppose that instead of a perfect score being 10, it is 100. In this case, we

would obtain…

60/7 of 100 = 60/7× 100 = 600/7 = 600 ÷ 7

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Performing the division, we would find

that…

7 600

8

– 5640

5/75

– 355

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next This tells us that if a perfect score is represented by 100 (the corn bread is

presliced into 100 equally-sized pieces), 6/7 is worth more than 85 pieces but less than 86 pieces (in fact it is exactly 855/7 pieces).

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Less than 6/7

Greater than 6/7

The observation we wish to make now is that part of the above division

problem was done previously when we assumed that the perfect score was 10.


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7 600

8

– 5640

R 55

– 355

The highlighted portion shows what

we could have started with when

we did the computation.

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In a similar way if the perfect score was represented by 1,000, our division

problem would look like…


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This shows us that 6 on a scale of 7 is greater than 857 on

a scale of 1,000but less than 858. It is exactly 8571/7

per 1,000.

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7 6000

8

– 5640

R 157

– 3550

– 491

We can now appreciate what the role of a decimal

point might mean here.In terms of decimals, the first

digit to the right of the decimal point modifies “per 10”, the

second digit to the right of the decimal point modifies

“per 100” etc. In other words, in decimal form, 6/7 can be

represented by…


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7 6.000

0.8

– 5640

57

– 3550

– 491

The fact that the digit furthest to the right (7)

occurs in the third column to the right of the decimal point tells us that 857 is modifying

“thousandths”.


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7 6.000

0.8

– 5640

57

– 3550

– 491

And if we want to see how many parts per 10,000 is

represented by 6/7, wecould annex another 0 in the previous computation and simply continue to thenext step as shown by the

highlighted numerals.


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7 6.000

0.8

– 5640

57

– 3550

– 491

0

0

1

– 73

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Switching to “short division” to save space, we can continue the division

process to obtain…


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7 6. 0 40 50 10 30 20 60 0

0. 8 5 7 1 4 2 …

…and from the above we can see a problem that arises when we want to express the

common fraction as an equivalent decimal fraction. Namely, when we started the

division the dividend consisted of a 6 followed by nothing but 0’s.

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However, as indicated by the numeral (6) above, after we obtained the quotient

0.857142, the dividend was again a 6 followed by nothing except 0’s.


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7 6. 0 40 50 10 30 20 60 0

0. 8 5 7 1 4 2 …

In other words, when 6/7 is represented as a decimal, the cycle of digits “857142”

will repeat endlessly.

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The fact that the decimal representation never comes to an end means that we have to “invent” a notation that takes the place of us having to write the decimal “endlessly”.


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Notes

What we do is place a bar overthe repeating cycle.

For example, if we write 0.6173, it means 0.617361736173...endlessly.

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In summary, only the digits that are under the bar repeat endlessly. We will discuss

endless representations in greater detail in a later presentation.


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Notes

The fact that a decimal representation is endless doesn’t worry us in the

“real world”.

Even the very best measuring devices have a limited number of decimal place accuracy.


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Notes

For example, 0.6173 represents a rate of 6,173 per 10,000 while 0.6174 represents a rate of 6,174 per 10,000. So having to choose between 0.6173 and 0.6174 as an

estimate for 0.6173 involves an error of lessthan 1 part per 10,000.

While 0.6173 represents an endlessly repeating decimal, both 0.6173

and 0.6174 do end.


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Notes

Because they end, we can look at the digit furthest to the right of the decimal

point to determine that 0.6173 represents 6,173 ten thousandths, and 0.6174 represents 6,174 ten thousandths.


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Enrichment Note #1

In general our intuitive ideas are based on things that we have experienced, and in mathematics no matter how

many steps there are in a process, it is still a finite number.

An Obsolete Hybrid Decimal Notation


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Note #1

So, for example, while a number such as 50,000,0000,000,000,000,000 is a very

large number, it is still less than50,000,0000,000,000,000,001. In essence there is a noticeable difference between

being “very huge” and being infinite.

Thus, the idea of a decimal having endless many digits is far more subtle

than a decimal that has 50,000,0000,000,000,000,000 digits.

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Enrichment Note #1

So it should not come as a big surprise that in the “old” days people felt

uncomfortable with expressing a non-ending decimal by putting a bar over

the endlessly repeating cycle. Specifically, they coped with this

notion by using a hybrid notation that lasted well into the 18th century.


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Note #1

Rather than represent 6/7 as 0.8 R4, they would write it as 0.84/7 . The fraction was

not a separate entry, but rather it was attached to the digit immediately to its

left. In other words, 0.84/7 was used as anabbreviation for 84/7 tenths.1

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1 1While the notation might have been confusing, it allowed people to avoid having to come to grips with endlessness. More specifically…

0.84/7 = 8 4/7 per 10 = 60/7 per 10 = 60/7 ÷10 = 60/7 × 1/10 = 60/70 = 6/7


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Enrichment Note #2

Earlier in our discussion we talked about eventually having to round off

endless decimals when we are making real world measurements.

This introduces an error but it does convert the endless decimal into an

“almost equivalent common fraction”.

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Enrichment Note #2

For example, if we look at theendless decimal 0.857142 and round it off

to 0.857, we obtain almost equivalent common fraction 857/1,000. The error in this

approximation is given by…

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0.857142 – 0.857 = 0.857142 – 0.857 = 0.000142857

…and in most real-life applications this error is less than the limitations that are

present in our measuring devices.

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next The corn bread model makes it easier for

students to visualize this ratherabstract concept.

For example, in the case where our measuring instrument can only give us guaranteed

accuracy to the nearest tenth, it is equivalent to the case in which our corn bread is

presliced into 10 equally sized indivisible pieces. In this case, 0.857142 is more than 8 of these 10 pieces but less than 9 of these

10 pieces. Hence, the maximum error in this approximation is less than 1 piece of these

10 pieces.

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In a similar way, if our measuring instrument can only give us guaranteed accuracy to the

nearest hundredth, it is equivalent to the case in which our corn bread is presliced into 100 equally sized indivisible pieces. In this case, 0.857142 is more than 85 of these 100 pieces but less than 86 of these 100 pieces. Hence, the maximum error in this approximation is

less than 1 piece of these 100 pieces.

C o r n B r e a dehT


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In a similar way, if our measuring instrument can only give us guaranteed accuracy to the

nearest thousandth, it is equivalent to the case in which our corn bread is presliced into 1,000 equally sized indivisible pieces. In this case,

0.857142 is more than 857 of these 1,000 pieces but less than 858 of these

1,000 pieces. Hence, the maximum error in this approximation is less than 1 piece of

these 1,000 pieces.



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Notice that in each case, the maximum error is less than 1 piece, but as the power of 10 increases, the size of the

indivisible pieces get smaller and smaller and eventually

the “thickness” of each piece will become less than the size of the error in our

measuring instrument.



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As technology continues to improve, the error in our measuring instrument will most likely continue to improve. However, any line we draw must have some thickness (otherwise the line would be invisible). Therefore, the

error will never completely disappear, and the error in our measurement will always be 1 of

the indivisible pieces. However, eventually thethickness of each of the pieces will be less than the accuracy of our measuring device, and therefore, for all practical purposes, the

error will become completely negligible.



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Enrichment Note #3

Our previous two enrichment notes (hopefully) explain quite clearly that when it comes to making measurements in the “real world” there is no need to “invent”

any numbers beyond the rational numbers (that is, common fractions).


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Enrichment Note #3

Yet in the study of pure mathematics much is made of endless decimals and irrational numbers (which will be discussed in detail

later in our discussion of decimals). Examples such as this show that

mathematics belongs as much to the field of liberal arts, as well as to the fields of

science and engineering.


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Enrichment Note #3

We feel it is important for you to try to help your students see mathematics in this

light. In short, there are people who study mathematics for the same reasons that

people study poetry and other liberal arts; that is, not to build better bombs but rather

to better engage the human mind.

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This completes our present discussion. In

our next presentation we shall discuss how we

compare decimalsby size. In particular we

show the rationale by which we know that

0.6173 is greater than 0.6173 but less than

0.6174.


Comparing Decimals

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