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Chapter 1 Section 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1.7 1 2 3 4 5 Properties of Real Numbers Use the commutative properties. Use the associative properties. Use the identity properties. Use the inverse properties. Use the distributive properties. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Properties of Real Numbers 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 3 Objective 1 Use the commutative properties. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 4 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. The commutative properties say that if two numbers are added or multiplied in any order, the result is the same. ab ba ab ba Addition Multiplication Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 5 EXAMPLE 1 Using the Commutative Properties Use a commutative property to complete each statement. x 2 2 _____ 5x x ____ Solution: x 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 6 Objective 2 Use the associative properties. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 7 Use the associative properties. When we associate one object with another, we think of those objects as being grouped together. 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. (a b) c a (b c) (ab)c a(bc) Addition Multiplication 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 8 EXAMPLE 2 Using the Associative Properties Use an associative property to complete each statement. Solution: 5 (2 8) ________ (5 2) 8 10 (8) (3) ________ 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 9 EXAMPLE 3 Distinguishing between the Associative and Commutative Properties Is (2 4)6 (4 2)6 an example of the associative property or the commutative property? Solution: Commutative 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 10 EXAMPLE 4 Using the Commutative and Associative Properties Find the sum. Solution: 43 26 17 24 6 (43 17) (26 24) 6 60 50 6 116 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 11 Objective 3 Use the identity properties. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 12 Use the identity properties. If a child wears a costume on Halloween, the child’s appearance is changed, but his or her identity is unchanged. The identity of a real number is left unchanged when identity properties are applied. The identity properties say: a0 a a 1 a and 0 a a and 1 a a Addition Multiplication 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 13 EXAMPLE 5 Using the Identity Properties Complete each statement so that it is an example of an identity property. Solution: 5 ___ 5 0 1 1 ___ 3 3 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 14 EXAMPLE 6 Using Identity Properties to Simplify Expressions Simplify. 36 48 Solution: 3 2 66 42 6 8 3 4 3 5 3 3 5 9 5 3 5 1 24 24 8 24 8 3 24 8 24 4 4 1 24 46 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 15 Objective 4 Use the inverse properties. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 16 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. a (a) 0 and 1 a 1 and a a a 0 1 a 1 a 0) a Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Addition Multiplication Slide 1.7- 17 EXAMPLE 7 Using the Inverse Properties Complete each statement so that it is an example of an inverse property. ___ 6 0 1 ___ 1 9 Solution: 6 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 18 EXAMPLE 8 Using Properties to Simplify an Expression Simplify the expression. 1 1 3y 2 2 Solution: 1 1 3y 2 2 3y Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 19 Objective 5 Use the distributive properties. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 20 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: 2(5 8) , which equals 2(13), or 26 2(5) 2(8) , which equals 10 16, or 26. Since both expressions equal 26, 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 21 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. a(b c) ab ac and (b c)a ba ca The distributive property is also valid for multiplication over subtraction. a(b c) ab ac and (b c)a ba ca The distributive property can be extended to more than two a(b c d ) ab ac ad numbers. The distributive property can be used “in reverse.” For example, we can write ac bc (a b)c . Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 22 EXAMPLE 9 Using the Distributive Property Use the distributive property to rewrite each expression. Solution: 4(3 7) 4 3 4 7 12 28 40 6( x y z ) 6 x (6 y) (6 z) 6 x 6 y 6 z 3a 3b 3(a b) Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 23 EXAMPLE 10 Using the Distributive Property to Remove Parentheses Write the expression without parentheses. (5 y 8) Solution: 5y 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.7- 24