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Numbers and Operations
Families of numbers
Counting Numbers◦ 1, 2, 3, 4, 5, …
Natural Numbers
Counting Numbers & Zero◦ 0, 1, 2, 3, 4, 5, …
Whole Numbers
Positive and Negative Numbers and Zero◦ …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …
Integers
Can be expressed as the ratio of 2 integers
Rational Numbers
n
m
Cannot be expressed as the ratio of 2 integers◦ Non-terminating, non-repeating integers◦ Π
Irrational Numbers
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
1. Which value is closest to the square root of 273?
A. 16.5 B. 136.5 C. 546 D. 74,529
2. Determine the approximate value of the point:
1 2 3 4 5 6 7 8
Examples:
3. Given the numbers below, what is the order of these numbers from least to greatest?
3.3
A. 3.3
B. 3.3
C. 3.3
D. 3.3
103.3 3
11
10
10
10
10
3.3
3.3
3.3
3.3
3
11
3
11
3
11
3
11
Scientific Notation
Expand: 2.15 x 10-3
.00215
Write in scientific notation: 1. 3,145,062 2. 2,230,000 3. .000345
Examples
Simplify:
1. (1.5 x 102)(4 x 103) 2.
Examples
30 x 106 . 10 x 108
1. The diameter of a red blood cell, in inches, is 3 x 10-4. This expressionis the same as which of the following numbers?
A. 0.00003 B. 0.0003 C. 0.003 D. 3,000
2.
A. 0.5 x 102
B. 2 x 102
C. 2 x 100.5
D. 0.5 x 10-2
E. 2 x 10-2
2
3
103
106
Percent
Convert 20% to a decimal
Convert .45 to a percentage
Convert ¾ to a percentage
Percentages
What is 7 percent of 50?
A CD that normally costs $15 is on sale for 20% off. What will you pay
Examples:
Order of Operations
PEMDASARAN THESIS
XPONENTS
MULT
&
DIV
ADD
&
SUB
From left to right
30 ÷ 10 • (20 – 15)2
Examples:
GCF and LCM
GCF—greatest common factor What is the largest number that divides all the given
numbers evenly20 35 60 24
WHAT DO THEY SHARE?
Examples
LCM—least common multiple What is the smallest number that the given numbers go
into evenly20 35 60 24
WHAT IS THE LARGEST VALUE SHOWN IN EACH?
Examples
Using Proportions
In the town of Centerville, 2.1 centimeters of snow fell in 3 hours. Snow continued to fall at the same rate. How many centimeters of snow had fallen after 7 hours?
A. 0.3 cmB. 2.8 cmC. 4.9 cmD. 14.7 cm
What is a proportion and how can you solve a problem with it?
Distance Problems
Distance problems
rtD
TimeRateDistance
Example It took the Smith’s 5 hours to go 275 miles.
What was their average rate of speed?
D=rt
EstimationWhat are the critical terms for estimation?
1. Mrs. Ditters and her son went to lunch. Their bill came to $27.29. If a fair tip is between 15 and 20 percent, what would be a fair tip to leave their waiter?
A. $2.00B. $2.72C. $5.00D. $20.00
Examples:
Rule of Exponents
xn
Base: the number to be multiplied by itself Exponent: how many times the base is to
multiplied by itself
Exponent
=> EXPONENT=> BASE
Examples (23)4
(23)(24)
x2 + 2x2
4 8x
3
4
2
2
x2 – 2x2
X0 =1
x-n = 1 . xn
And
Negative Exponent
nnx
x
1
A negative exponent means the items belongs on the other side of the fraction bar
Example: simplify
2
43
z
yx
1. What is the value of the expression 2( ¼ )2.
A. 1/16 B. 1/8 C. 2 2/8 D. 8
Simplifying Square Roots
Be familiar with the following values
Perfect Squares and cubes
Base Squares Cubes
0 0 0
1 2 1
2 4 8
3 9 27
4 16 64
5 25 125
6 36 216
7 49
8 64
9 81
10 100
Simplifying Square rootsuse a factor tree and remove groups of 2
(if you are working with cube roots, remove in groups of 3, etc.)
Square Roots
75
The same number must be under the radical in order to add or subtract
Addition and Subtraction
33
Radicals must have the same index in order to multiply or divide
Multiplication or Division
33 22 4