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

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 Slide If you were asked to fine the sum , you might mentally add to get 100 and then add 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 1 Objective 1 Use the commutative properties. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 The commutative properties say that if two numbers are added or multiplied in any order, the result is the same. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Solution: Using the Commutative Properties Use a commutative property to complete each statement. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Use the associative properties. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Use the associative properties. When we associate one object with another, we think of those objects as being grouped together. Addition Multiplication 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. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Use an associative property to complete each statement. EXAMPLE 2 Solution: Using the Associative Properties 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. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution: Commutative Distinguishing between the Associative and Commutative Properties Is an example of the associative property or the commutative property? 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. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Using the Commutative and Associative Properties Find the sum. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Use the identity properties. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. andAddition andMultiplication The identity of a real number is left unchanged when identity properties are applied. The identity properties say: Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solution: Using the Identity Properties Complete each statement so that it is an example of an identity property. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Using Identity Properties to Simplify Expressions Solution: Simplify. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Use the inverse properties. Slide

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

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Solution: Complete each statement so that it is an example of an inverse property. Using the Inverse Properties Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Simplify the expression. Solution: Using Properties to Simplify an Expression Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5 Objective 5 Use the distributive properties. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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,. With this property, a product can be changed to a sum or difference. This idea is illustrated by the divided rectangle below. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 The distributive property is also valid for multiplication over subtraction. and Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Using the Distributive Property Use the distributive property to rewrite each expression. Solution: Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution: EXAMPLE 10 Using the Distributive Property to Remove Parentheses Write the expression without parentheses. Slide