Chapter 1 Section 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Chapter Chapter 11Section Section 77

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Properties of Real Numbers

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44

33

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55

1.71.71.71.7Use the commutative properties.Use the associative properties.Use the identity properties.Use the inverse properties.Use the distributive properties.


Properties of Real Numbers

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If you were asked to fine the sum 3 + 89 + 97, you might mentally add 3 + 97 to get 100 and then add 100 + 89 to get 189.

While the rules for the order of operations say to add from left to right, we may change the order of the terms and group them in any way we choose without affecting the sum.

These are examples of shortcuts that we use in everyday mathematics. Such shortcuts are justified by the basic properties of addition and multiplication, discussed in this section.

In these properties, a, b, and c represent real numbers.


Objective 11

Use the commutative properties.

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Use the commutative properties.

The word commute means to go back and forth. Many people commute to work or to school. If you travel from home to work and follow the same route from work to home, you travel the same distance each time.

Addition

Multiplication

a b b a

ab ba

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

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EXAMPLE 1

Solution:

Using the Commutative Properties

2 2 _____x

5 ____x x

x

5

Use a commutative property to complete each statement.

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Objective 22

Use the associative properties.

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Use the associative properties.

When we associate one object with another, we think of those objects as being grouped together.

Addition

Multiplication

( ) ( )a b c a b c

( ) ( )ab c a bc

The associative properties say that when we add or multiply three numbers, we can group the first two together or the last two together and get the same answer.

The various properties are often represented by acronyms. CPA can represent the commutative property of addition, APM can represent the associative property of multiplication, and so on.

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Use an associative property to complete each statement.

EXAMPLE 2

Solution:

Using the Associative Properties

5 (2 8) ________

10 ( 8) ( 3) ________

(5 2) 8

10 ( 8) ( 3)

By the associative properties of addition and multiplication, the sum or product of three numbers will be the same no matter how the numbers are “associated” in groups. So parentheses can be left out in many problems.

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EXAMPLE 3

Solution:Commutative

Distinguishing between the Associative and Commutative Properties

Is an example of the associative property or the commutative property?

(2 4)6 (4 2)6

Note that with the commutative properties, the number sequence changes on opposite sides of the equal sign. With the associative properties, the number sequence is the same on either side.

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EXAMPLE 4

Solution:

Using the Commutative and Associative Properties

Find the sum.

43 26 17 24 6 (43 17) (26 24) 6

60 50 6

116

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Objective 33

Use the identity properties.

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If a child wears a costume on Halloween, the child’s appearance is changed, but his or her identity is unchanged.

Use the identity properties.

The number 0 leaves the identity, or value, of any real number unchanged by addition. So 0 is called the identity element for addition, or the additive identity.

Since multiplication by 1 leaves any real number unchanged, 1 is the identity element for multiplication, or the multiplicative identity.

and Addition

and Multiplication

0a a 0 a a

1a a 1 a a

The identity of a real number is left unchanged when identity properties are applied. The identity properties say:

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EXAMPLE 5

Solution:

Using the Identity Properties

Complete each statement so that it is an example of an identity property.

5 ___ 5

1 1___

3 3

0

1

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EXAMPLE 6Using Identity Properties to Simplify Expressions

Solution:

Simplify.

36

48

3 5

8 24

6 6

6 8

3 2

4 2

3

4

13 5

8 24

3 3 5

8 3 24

9 5

24 24

4

24

6

4

4

1

6

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Objective 44

Use the inverse properties.

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Us the inverse properties.

Each day before you go to work or school, you probably put on your shoes before you leave. Before you go to sleep at night, you probably take them off, and this leads to the same situation that existed before you put them on. These operations from everyday life are examples of inverse operations.

The inverse properties of addition and multiplication lead to the additive and multiplicative identities, respectively.

and Addition

and Multiplication

( ) 0a a 0a a 1

1aa

1

1 0)a aa

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EXAMPLE 7EXAMPLE 7

Solution:

Complete each statement so that it is an example of an inverse property.

___ 6 0

1___ 1

9

6

9

Using the Inverse Properties

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EXAMPLE 8

Simplify the expression.

Solution:

Using Properties to Simplify an Expression

1 13

2 2y

1 13

2 2y

3y

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Objective 55

Use the distributive properties.

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Use the distributive property.The everyday meaning of the word distribute is “to give out

from one to several.”

Look at the value of the following expressions:

, which equals , or 26

, which equals , or 26.

Since both expressions equal 26, .

2(5 8) 2(13)2(5) 2(8) 10 16

2(5 8) 2(5) 2(8) With this property, a product can be changed to a sum or

difference. This idea is illustrated by the divided rectangle below.

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The distributive property can be used “in reverse.” For example, we can write .

The distributive property can be extended to more than two numbers.

Use the distributive property. (cont’d)

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

and( )a b c ab ac ( )b c a ba ca

The distributive property is also valid for multiplication over subtraction.

and( )a b c ab ac ( )b c a ba ca

( )a b c d ab ac ad

( )ac bc a b c

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EXAMPLE 9 Using the Distributive Property

Use the distributive property to rewrite each expression.

4(3 7)

6( )x y z

3 3a b

4 3 4 7 12 28 40

6 ( 6 ) ( 6 )x y z 6 6 6x y z

3( )a b

Solution:

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Solution:

EXAMPLE 10Using the Distributive Property to Remove Parentheses

Write the expression without parentheses.

( 5 8)y 5 8y

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Chapter 1 Section 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley