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HAWKES LEARNING SYSTEMS

Students Matter. Success Counts.

Copyright © 2013 by Hawkes Learning

Systems/Quant Systems, Inc.

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Section 7.6

Properties of Real Numbers






Objectives

o Apply the properties of real numbers.o Identify properties that justify given statements.






Applying the Properties of Real Numbers

Properties of Addition and MultiplicationIn this table, a, b, and c are real numbers. For Addition Name of Property For

Multiplication

3+ = ++ 6 = 6 + 3

a b b a Commutativeproperty 9 4

=4 = 9 ab ba

2 + 5 + 4 = 2 +

+ +

5

+ =

+

+

4

a b c a b c 6 2 7

=

= 6 2 7

a bc ab cAssociativeproperty







Properties of Addition and Multiplication (cont.)Addition Name of Property Multiplication

20 + 0+ =

= 0+ =+ 20 = 20

0 0a a a

2 1 = 1 2 == =

2

1 1a a aIdentity

10 + 1

+ = 00 = 0

a aInverse

1 = 13

=

3

for 01 1 aaa







Properties of Addition and Multiplication (cont.)

Zero-Factor Law

Distributive Property of Multiplication over Addition

= = 0 0 0a a 5 0 = 0 5 = 0

+ = +a b c ab ac 3 + 5 = 3 + 3 5x x







NotesThe number 0 is called the additive identity because when 0 is added to a number the result is the same number. Likewise, the number 1 is called the multiplicative identity because when a number is multiplied by 1 the result is the same number. Also, the additive inverse of a number is its opposite and the multiplicative inverse of a number is its reciprocal.






Example 1: Properties of Addition and Multiplication

State the name of each property being illustrated. Solutioncommutative property of addition

Solutionassociative property of addition

Solutionmultiplicative identity

7 + 13 = 13 + 7 a.

8 + 9 + 1 = 8 + 9 + 1b.

25 1 = 25 c.






Example 1: Properties of Addition and Multiplication (cont.)

Solutiondistributive property

Solutionzero-factor law

3 + = 3 + 3x y x yd.

0 14 = 0e.






Example 2: Properties of Addition and Multiplication

In each of the following equations, state the property illustrated and show that the statement is true for the value given for the variable by substituting the value in the equation and evaluating. a. x + 14 = 14 + x given that x = −4

Solution

and+ 14 = 14 +1 104 0 4 =

The commutative property of addition is illustrated.






Example 2: Properties of Addition and Multiplication (cont.)

b. given that x = 5Solution

c. given that y = −2Solution

3 6 = 3 6x x

and 3 6 = 18 5 = 35 56 = 39 00 93 = 0

12 + 3 = 12 + 36y y

12 + 3 = 12 1 = 12 + 36 =an12 122 =2d2 4 + 36

The associative property of multiplication is illustrated.

The distributive property is illustrated.






Practice Problems

Determine the property being illustrated.

1. 2.

3. 2 + 7 = 7 + 2 4. 2(y + 5) = 2y + 10

2 5 2 = 2 5 2 15 0 = 0 15 = 0






Practice Problem Answers

1. associative property of multiplication2. zero-factor law3. commutative property of addition4. distributive property

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