Real Numbers and Properties

Natural Numbers……

Known as “Counting” Numbers

Example: 1, 2, 3, 4, 5,…….

Whole Numbers……

You add the number 0 to the natural numbers.

Example: 0, 1, 2, 3, 4, 5…….

Integers……

Integers are made up of whole numbers and their opposites.

Example: …-4,-3,-2,-1,0,1,2,3,4….

Rational Numbers…… The set of rational numbers is

made up of all of the following a. Natural Numbers b. Whole Numbers c. Integers d. Plus every repeating and terminating decimal.

Examples of Rational Numbers…… A. ½ = 0.5 (Terminating Decimal)

B. 1.23232323 (Repeating Decimal)

C. 0.256256256 (Repeating Decimal)

D. 2.735 (Terminating Decimal)

Irrational Numbers….

Consists of Non-Terminating and Non-Repeating Decimals.

Example: 0.9482137507264

Real Numbers (ℝ)

Irrational NumbersRational Numbers (ℚ)

Decimal form either terminates or repeats

Decimal form is non-terminating and non-repeating

Integers (ℤ)

Whole Numbers

Natural Numbers (ℕ)

1, 2, 3, …

0, 1, 2, 3, …

…-3, -2, -1, 0, 1, 2, 3, …

All rational and irrational numbers

The Number Line…… A number line consists of positive

numbers (right of 0) and negative numbers (left of 0).

A real life example of a number line is a temperature thermometer.

Negative Positive0

For example…..

-5 would represent 5 degrees below zero.

+4 would represent 4 degrees above zero.

Make the Comparison…… 7 degrees below 0

is (warmer/colder) than 4 degrees above 0.

7 degrees below 0 is a (lower/higher) temperature than 4 degrees above 0.

Colder

Lower

Coordinates on a Graph….

Find the best estimate of the point. a. -2 b. 2 c. -1.75 d. -1.5

0 1 212

Answer: -1.75

Sets and Subsets…… A set is a group of numbers. Example: Set A = {1,2,3,4,5}

A subset is a group of numbers in which every member is in another set.

Example: Set B = {1,2,3}

So, B is a subset of A.

Which of the following would represent a subset of integers?

States Sales Tax Rate Amount of Gas in a Car Number of Students in

Class A Dinner Receipt

Strategy: Eliminate those that are NOT integers.

7.5% - NO 6.5 Gallons – NO

12 – YES $10.31 - NO

You Try…Which of the following would represent a subset of integers?

Costs of a TV

# of miles on the odometer of a car

A person’s weight

Number of residents in South Carolina

No

No

No

Yes

Inequalities…..

We use inequalities to compare numbers.

The following are inequalities:

Examples…….

“4 is less than 7” -

“9 is greater than or equal to 5” -

74

59

You Try……Insert the appropriate inequality sign.

1. -5 -2

2. -7 2

3. 4 -12

1. <

2. <

3. >

Least to Greatest…… This means to arrange numbers in

the order from the smallest to the largest.

HINT: If there are fractions it might be easier to convert to decimals first.

Which Number is Smaller?

9

2

7

3 or

smalleris7

3

22222.09

2

42857.07

3

Which Number is Larger?.......

32.013

5 or

.arg32.0,

32.032.0

3846.013

5

erlisSo

You Try…Compare -0.67 > -0.68

-0.8 > -0.86

3

2 68.0

5

4

7

6

Which Set is Ordered from Least to Greatest?

1. {-3/2, -3, 0, 2/3}

2. {-3, -3/2, 0, 2/3}

3. {0, 2/3, -3/2, -3}

4. {0, -3/2, -3, 2/3}

1. {-3/2, -3, 0, 2/3}

2. {-3, -3/2, 0, 2/3}

3. {0, 2/3, -3/2, -3}

4. {0, -3/2, -3, 2/3

What kinds of numbers are used to represent numbers below zero?

Answer:

NEGATIVE Numbers

Make -8 -4 a true statement.

Answer:

<

Quick Review

0-400 -200 200 400

1) Coordinate of A:

a) -250 b) -300 c) -325 d) -500

2) Coordinate of B:

a) -210 b) -350 c) -100 d) -50

3) Coordinate of C:

a) 350 b) 425 c) 325 d) 275

Quick Review4) Use , : -8 5

5) Which is smaller? or

6) Write from smallest to largest:-3, -3.8, -5, 5.6, -5.6

7

5

8

3

Number Properties

Commutative Property-Changes Order For Addition

A+B = B+A

Ex. 2+3 = 5 3+2 = 5 2+3=3+2

For Multiplication

AB = BA

Ex. 4(8) = 328(4) = 32

4(8) = 8(4)

THIS IS NOT TRUE FOR SUBTRACTION OR DIVISION!

Associative Property-Changes Grouping

For AdditionA + (B + C) = (A + B) + C

Ex.5 + (2 + 4) (5 + 2) + 4 = 5 + 6 = 7 + 4 = 11 = 11

5 + (2 + 4) = (5 + 2) + 4

For MultiplicationA(BC) = (AB)C

Ex.2 (3 5) (2 3) 5 = 2(15) = (6)5 = 30 = 30

2 x (3 x 5) = (2 x 3) x 5

This is not true for subtraction or division!

Which Property?

1) 3x 4 = 4 3x 2) 6y + (7 + 3z) = (6y +7) +3z3) (5x + 7) + 8y = 5x + (7 + 8y)4) (3x)(2x + 5) = (2x + 5)(3x)5) 10x + 4y = 4y + 10x6) (2x 5)(10y) = (2x)(5 10y)

Distributive Property

A (B + C) = AB + AC

(B + C) A = BA + CA

A (B – C) = AB – AC

(B – C) A = BA – CAEx. -3 (4 – 2x)

Strategy: Think -3 (4 – 2x) means -3 (4 + -2x)

= -3(4) + (-3)(-2x)

= -12 + 6x

TRY THESE:

A) 4 (6 +2a) B) -7 (-3m – 5)

Which Property?

1) -3x(y + 2) + 4y = -3x(y) – 3x(2) + 4y

2) -3y + 4x(y + 2) = -3y + 4xy + 4x(2)

3) 6x + (3y + 1) = (3y +1) + 6x

What is an example of the commutative prop. of addition?

A) 3 + 5m = 3 + (1 + 4)mB) 3 + 5m = 5m + 3C) 3 + 5m = 5 + 3mD) 3 + 5m = 3m + 5

A) 7 + 4m = (7 +4)mB) (5 + 2) + 4m = 7 + 4mC) 7 + 4m = 4 + 7mD) 7 + 4m = 4m + 7

Homework