Real Numbers Slide 1.7-6

Properties of Real Numbers 1.7 Properties of Real Numbers Use the commutative properties. Use the associative properties. Use the identity properties. Use the inverse properties. Use the distributive properties. 2 3 4 5

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

Use the commutative properties. Objective 1 Use the commutative properties. Slide 1.7-4

Commutative Properties 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. Commutative Properties Addition Multiplication Slide 1.7-5

Using the Commutative Properties EXAMPLE 1 Using the Commutative Properties Use a commutative property to complete each statement. Solution: Slide 1.7-6

Use the associative properties. Objective 2 Use the associative properties. Slide 1.7-7

Associative Properties Use the associative properties. 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 1.7-8

Using the Associative Properties EXAMPLE 2 Using the Associative Properties Use an associative property to complete each statement. Solution: Slide 1.7-9

Distinguishing Between Properties EXAMPLE 3 Distinguishing Between Properties Is an example of the associative property or the commutative property? Solution: Commutative Slide 1.7-10

Using the Commutative and Associative Properties EXAMPLE 4 Using the Commutative and Associative Properties Find the sum. Solution: Slide 1.7-11

Use the identity properties. Objective 3 Use the identity properties. Slide 1.7-12

Identity Properties Addition Multiplication 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. Likewise, the identity of a real number is left unchanged when identity properties are applied. Identity Properties 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. Slide 1.7-13

Using the Identity Properties EXAMPLE 5 Using the Identity Properties Use an identity property to complete each statement. Solution: Slide 1.7-14

Using the Identity Property to Simplify Expressions EXAMPLE 6 Using the Identity Property to Simplify Expressions Simplify. Solution: Slide 1.7-15

Use the inverse properties. Objective 4 Use the inverse properties. Slide 1.7-16

Inverse Properties Addition Multiplication Use 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. Inverse Properties Addition Multiplication Slide 1.7-17

Using the Inverse Properties EXAMPLE 7 Using the Inverse Properties Use an inverse property to complete each statement. Solution: Slide 1.7-18

Using Properties to Simplify an Expression EXAMPLE 8 Using Properties to Simplify an Expression Simplify. Solution: Slide 1.7-19

Use the distributive property. Objective 5 Use the distributive property. Slide 1.7-20

Using the Distributive Property EXAMPLE 9 Using the Distributive Property Use the distributive property to rewrite each expression. Solution: Slide 1.7-23

Use the distributive properties. 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 1.7-21

Distributive Property Use the distributive properties. (cont’d) 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. Distributive Property 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 1.7-22

Solution: EXAMPLE 10 Write the expression without parentheses. Using the Distributive Property to Remove (Clear) Parentheses Write the expression without parentheses. Solution: Slide 1.7-24