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