Chapter 1 Section 7

Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. 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 © 2012, 2008, 2004 Pearson Education, Inc. Properties of Real Numbers 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. There are 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. Slide 1.7-3

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 1 Use the commutative properties. Slide 1.7-4

Copyright © 2012, 2008, 2004 Pearson Education, Inc. 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. Commutative Properties Addition Multiplication Slide Use the commutative properties.

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Use a commutative property to complete each statement. Slide EXAMPLE 1 Using the Commutative Properties

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the associative properties. Objective 2 Slide 1.7-7

Copyright © 2012, 2008, 2004 Pearson Education, Inc. When we associate one object with another, we think of those objects as being grouped together. We can group the first two together or the last two together and get the same answer. Associative Properties Addition Multiplication Slide Use the associative properties.

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use an associative property to complete each statement. Solution: Slide EXAMPLE 2 Using the Associative Properties

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Commutative Is an example of the associative property or the commutative property? Slide EXAMPLE 3 Distinguishing Between Properties

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Find the sum. Slide EXAMPLE 4 Using the Commutative and Associative Properties

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the identity properties. Objective 3 Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. If a child wears a costume on Halloween, the child’s appearance is changed, but his or her identity is unchanged. Likewise, the identity of a real number is left unchanged when identity properties are applied. 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. Identity Properties Addition Multiplication Slide Use the identity properties.

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Use an identity property to complete each statement. Slide EXAMPLE 5 Using the Identity Properties

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Simplify. Slide EXAMPLE 6 Using the Identity Property to Simplify Expressions

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the inverse properties. Objective 4 Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. 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. Inverse Properties Addition Multiplication Slide Use the inverse properties.

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Use an inverse property to complete each statement. Slide EXAMPLE 7 Using the Inverse Properties

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Simplify. Solution: Slide EXAMPLE 8 Using Properties to Simplify an Expression

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the distributive property. Objective 5 Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the distributive property to rewrite each expression. Solution: Slide EXAMPLE 9 Using the Distributive Property

Copyright © 2012, 2008, 2004 Pearson Education, Inc. 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, Slide Use the distributive properties.

Copyright © 2012, 2008, 2004 Pearson Education, Inc. 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. The distributive property is also valid for multiplication over subtraction. and The distributive property can be extended to more than two numbers. The distributive property can be used “in reverse.” Slide Use the distributive properties. (cont’d) Distributive Property

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Write the expression without parentheses. Slide EXAMPLE 10 Using the Distributive Property to Remove (Clear) Parentheses