Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 1.


The Real Number System

Chapter 1


1.7

Properties of Real Numbers


Objectives

1. Use the commutative properties.

2. Use the associative properties.

3. Use the identity properties.

4. Use the inverse properties.

5. Use the distributive property.

1.7 Properties of Real Numbers



The commutative properties say that if two numbers are added or multiplied in any order, they give the same result.

Addition

Multiplication

Using the Commutative Properties

a b b a

a b b a


Example 1

Use a commutative property to complete each statement.

(a) –5 + 7 = 7 + ___

1.7 Properties of Real NumbersUsing the Commutative Properties

(–5)

(b) –3(8) = 8 (___)–3



The associative properties say that when we add or multiply three numbers, we can group them in any manner and get the same answer.

Addition

Multiplication

Using the Associative Properties

a b c a b c

ab c a bc


Example 2

Use an associative property to complete each statement.

(a) –6 + (–3 + 5) = (–6 + ___) + 5

1.7 Properties of Real NumbersUsing the Associative Properties

(–3)

(b) –3(5 · –2) = (–3 ·___) · –25


1.7 Properties of Real NumbersUsing Commutative and Associative Properties

Example 3

Find each sum or product.

(a) 18 + 23 + 9 + 12 + 27 + 11 = (18 + 12) + (23 + 27) + (9 + 11)

= 30 + 50 + 20

= 100

(b) 75(9 · 4) = (75 · 4)9

= 300 · 9

= 2700



The identity properties say that the sum of 0 and any number equals that number, and the product of 1 and any number equals that number.

Addition

Multiplication

Using the Identity Properties

0 and 0a a a a

1 and 1a a a a


1.7 Properties of Real NumbersUsing the Inverse Properties

Example 4

These statements are examples of identity properties.

(a) 5 0 5

(b) 1 18 18


1.7 Properties of Real NumbersUsing the Identity Properties

Example 5

Simplify each expression.

45(a)

81

9 9

95

99

5 9

19

5

9

5

The product of a number and 1is that number.


The opposite of a, –a, is the additive inverse of a.

The inverse property of addition says that the sum of a number and its additive inverse is 0 (the additive identity).

Addition


0 and 0a a a a


The reciprocal of a, 1/a, is the multiplicative inverse of the nonzero number a.

The inverse property of multiplication says that the product of a number and its multiplicative inverse is 1 (the multiplicative identity).

Multiplication


1 11 and 1 0a aa a a



Example 6

The following statements are examples of inverse properties.

2 7(a) 1

7 2

3 3(b) 0

5 5



The distributive property says that multiplying a number a by a sum of numbers b + c gives the same result as multiplying a by b and a by c and then adding the two products.

a(b + c) = ab + ac

The distributive property is also valid for subtraction.

a(b – c) = ab – ac

Using the Distributive Property


Example 7

Use the distributive property to rewrite each expression.

(a) –6(–2 + 5) = –6(–2) + –6(5)

1.7 Properties of Real NumbersUsing the Distributive Property

= 12 + (–30)

= –18

(b) 2(3x – 7) = 2(3x) + 2(–7)

= 6x + (–14)

= 6x –14


Example 7 (concluded)

Use the distributive property to rewrite each expression.

(c) 3 · 9 + 3 · 6 = Here, we can use the distributive property in reverse.

1.7 Properties of Real NumbersUsing the Distributive Property

= 3(15)

= 45

(d) 5(2r – 3s + 4t) = 5(2r) + 5(–3s) + 5(4t)

= 10r – 15s + 20t

3(9 + 6)

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 1.

Documents

Transcript of Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 1.