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Numbers and Operations
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Families of numbers
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Complex numbers
Real Numbers Imaginary Numbers
|Rational Numbers Irrational Numbers
|Integers
|Whole Numbers
|Natural Numbers
The Numbrella
Can be expressed as a fraction Can’t be expressed as a fraction
All “non-decimal” values
All positive integers and zero
All positive integers
i—or bi
a+biHas a real and an imaginary component
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Counting Numbers◦ 1, 2, 3, 4, 5, …
Natural Numbers
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Counting Numbers & Zero◦ 0, 1, 2, 3, 4, 5, …
Whole Numbers
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Positive and Negative Numbers and Zero◦ …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …
Integers
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Can be expressed as the ratio of 2 integers
Rational Numbers
n
m
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Cannot be expressed as the ratio of 2 integers◦ Non-terminating, non-repeating integers◦ Π
Irrational Numbers
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The approximate value of √7:√4 = 2 √9 = 3 so √7 is approx.
2.6
Determine the approximate value of the point:
1 2 3 4 5 6 7 8
The point is about 3.4
Examples:
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Scientific Notation
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1-9 are significant 0’s between digits are significant 0’s at the end suggest rounding and are
not significant Leading 0’s are not significant 0’s at the end of a decimal indicate the
level of precision Every digit in scientific notation is
significant
Significant Digits Rules
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1024 4 Significant Digits
1000 1 Significant Digit
.0005 1 Significant Digit
ALWAYS HAVE ONE SIGNIFICANT DIGIT IN FRONT OF THE DECIMAL FOR SCIENTIFIC NOTATION
Examples
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Expand: 2.15 x 10-3 2.15 x 103
a negative exponent tells you to move the decimal to the left
.00215 2150
Write in scientific notation: 3,145,062 2,230,000 .000345
move the decimal so that there is only one digit in front and count the number of spaces you have moved—moving left is positive here and right is negative.
3.145062 x 106 2.23 x 106 3.45 x 10-4
Examples
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Simplify: do the math on the numeric portion as you normally would, use the rules of exponents on the powers of ten, place in standard scientific notation to finish (one digit before the decimal)
(2.75 x 102)(4 x 103)11 x 105
1.1 x 106
Examples
5 x 106 . 10 x 108
.5 x 10-2
5 x 10-1
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Percent
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Convert 20% to a decimal 20/100= .2
Convert .45 to a percentage .45 * 100= 45%
Convert ¾ to a percentage ¾= .75 .75 * 100=75%
Percentages
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What is 7 percent of 50?◦ .07 * 50 = 3.5
A CD that normally costs $15 is on sale for 20% off. What will you pay◦ Option 1
.2 * 15 = 3 15-3= 12◦ Option 2
If it is 20% off you will pay 80% .8 * 15 = 12
Examples:
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Order of Operations
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PEMDASARAN THESIS
XPONENTS
MULT
&
DIV
ADD
&
SUB
From left to right
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30 ÷ 10 • (20 – 15)2
30 ÷ 10 • 52
30 ÷ 10 • 25
75
Examples:
Parenthesis Exponents
then mult and divFrom left to right
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Absolute Value
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Formal definition
0 when x-
0 when x ||
x
xx
Absolute value is the distance from the origin and distance is always positive.
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|6| |-7| |-9-3| 6 7 |-12|
12
Examples
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GCF and LCM
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GCF—greatest common factor What is the largest number that divides all the given
numbers evenly20 35 60 24
5 4 5 7 6 10 3 8
2 2 2 3 2 5 2 4
2 2
22* 5 5*7 22*3*5 23*3WHAT DO THEY SHARE?
5 22* 3=12
Examples
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LCM—least common multiple What is the smallest number that the given number go
into evenly20 35 60 24
5 4 5 7 6 10 3 8
2 2 2 3 2 5 2 4
2 2
22* 5 5*7 22*3*5 23*3WHAT IS THE LAGEST VALUE SHOWN IN EACH?
22*5*7=140 23*3*5=120
Examples
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Using Proportions
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If Sue charges a flat rate each hour to babysit. If she ears $44 for 8 hours. What will she earn for 5 hours?
PRIMARY RULE:◦ If you put the $ amount in the numerator on one
side put the same value in the numerator on the other side. Etc.
cross mult. 220 = 8x27.5= x
Sue will earn $27.50 for 5 hours.
What is a proportion and how can you solve a problem with it?
58
44 x
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Distance and Work Problems
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Distance problems
rtD
TimeRateDistance
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Example It took the Smith’s 5 hours to go 275 miles.
What was their average rate of speed?
D=rt275 = r(5)55 = r
They went about 55 mph
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Use the reciprocal of the time for the rate of work
W for 1st
person =hours worked * rate of workW for 2
nd person =hours worked * rate of work
Total job always =1
1 = W for 1st
person + W for 2nd person
Work problems
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John and Sam decide to build a bird house. John and build the bird house in 5 hours working alone. Sam can do it in 8 hours alone. How long will it take if they work together?
It will take them 3.08 hours to make the bird house.
Example:
851
xx
xx 5840 x1340 x08.3
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EstimationWhat are the critical terms for estimation?
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The “detail” associated with a measurement
Precision
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Calculations with two different levels of precision can only be accurate to the least precise measure.
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How correct a measurement is
The smaller the unit of measure the more accurate your measurement
Accuracy
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The amount of difference between your measurement and the true value
Error
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Jim bought 3 pounds of nails for $16.25. Which amount is closest to the price per pound?
Round off and check above and below15/3 = 5 and 18/3 = 6
A reasonable values would be between $5 and $6 but closer to $5
Examples:
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Conversions
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1 inch = 2.54 cm
12 inches = 1 foot
3 feet = 1 yard
5280 feet = 1 mile
How many inches are in 1 yard?◦ 1 yard = 3 feet 1 foot = 12 inches
3x12 =36 inches
Length Conversions
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3 Teaspoons = 1 Tablespoon 2 Tablespoons = 1 ounce 8 ounces = 1 cup 2 cups = 1 pint 2 pints = 1 quart 4 quarts = 1 gallon
Fluid Conversions
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16 ounces = 1 pound
2.2 pounds = 1 kilogram
2000 pounds = 1 ton
Weight Conversions
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milli- centi-
-meter = distance -gram = weight -liter = fluid
kilo-