Copyright © 2011 Pearson Education, Inc.

Properties of Real Numbers 1.4 Properties of Real Numbers Use the distributive property. Use the inverse properties. Use the identity properties. Use the commutative and associative properties. Use the multiplication property of 0. 1 2 3 4 5 Copyright © 2011 Pearson Education, Inc.

Copyright © 2011 Pearson Education, Inc. Objective 1 Use the distributive property. Copyright © 2011 Pearson Education, Inc. Slide 1.1- 3

Copyright © 2011 Pearson Education, Inc. The Distributive Property For any real numbers, a, b, and c, a(b + c) = ab + ac and (b + c)a = ba + ca. The Distributive property can be extended to more than two numbers. a(b + c + d) = ab + ac + ad Copyright © 2011 Pearson Education, Inc. Slide 1.1- 4

Copyright © 2011 Pearson Education, Inc. EXAMPLE 1 Use the distributive property to rewrite each expression. a. 4(p – 5) = 4(p – 5) = = 4p + 20 b. 6m + 2m = 6m + 2m = = 4m 4p (4)(5) (6 + 2)m Copyright © 2011 Pearson Education, Inc. Slide 1.1- 5

Copyright © 2011 Pearson Education, Inc. continued c. 2r + 3s Because there is no common number or variable here, we cannot use the distributive property to rewrite the expression. d. 5(4p – 2q + r) = 5(4p – 2q + r) = = 20p – 10q + 5r 5(4p) 5(2q) + 5r Copyright © 2011 Pearson Education, Inc. Slide 1.1- 6

Copyright © 2011 Pearson Education, Inc. Objective 2 Use the inverse properties. Copyright © 2011 Pearson Education, Inc. Slide 1.1- 7

Copyright © 2011 Pearson Education, Inc. Inverse Properties For any real number a, a + a = 0 and a + a = 0 and Copyright © 2011 Pearson Education, Inc. Slide 1.1- 8

Copyright © 2011 Pearson Education, Inc. Objective 3 Use the identity properties. Copyright © 2011 Pearson Education, Inc. Slide 1.1- 9

Copyright © 2011 Pearson Education, Inc. The numbers 0 and 1 each have a special property. Zero is the only number that can be added to any number to get that number. Adding 0 to any number leaves the identity of the number unchanged. For this reason, 0 is called the identity element for addition, or the additive identity. In a similar way, multiplying any number by 1 leaves the identity of the number unchanged, so 1 is the identity element for multiplication, or the multiplicative identity. Copyright © 2011 Pearson Education, Inc.

Copyright © 2011 Pearson Education, Inc. EXAMPLE 2 Simplify each expression. a. x – 3x = 1x – 3x = 1x – 3x Identity property. = (1 – 3)x Distributive property. = 2x Subtract inside parentheses. b. (3 + 4p) = 1(3 + 4p) = 1(3) + (1)(4p) Identity property. = 3 – 4p Multiply. Copyright © 2011 Pearson Education, Inc. Slide 1.1- 11

Copyright © 2011 Pearson Education, Inc. A term is a number or the product of a number and one or more variables raised to powers. The numerical factor in a term is called the numerical coefficient, or just the coefficient. Terms with exactly the same variables raised to exactly the same powers are called like terms. Examples: 5y and 21y 6x2 and 9x2 Some examples of unlike terms are 3m and 16x 7y3 and 3y2 Copyright © 2011 Pearson Education, Inc.

Copyright © 2011 Pearson Education, Inc. Objective 4 Use the commutative and associative properties. Copyright © 2011 Pearson Education, Inc. Slide 1.1- 13

Copyright © 2011 Pearson Education, Inc. Commutative and Associative Properties For any real numbers a, b, and c, a + b = b + a ab = ba and Interchange the order of the two terms or factors. Also, a + (b + c) = (a + b) + c and a(bc) = (ab)c. Shift parentheses among the three terms or factors; the order stays the same. Commutative properties Associative properties Copyright © 2011 Pearson Education, Inc. Slide 1.1- 14

Copyright © 2011 Pearson Education, Inc. EXAMPLE 3 Simplify 12b – 9 + 4b – 7b + 1. 12b – 9 + 4b – 7b + 1 = (12b + 4b) – 9 – 7b + 1 = (12 + 4)b – 9 – 7b + 1 = 16b – 9 – 7b + 1 = (16b – 7b) – 9 + 1 = (16 – 7)b – 9 + 1 = 9b – 8 Copyright © 2011 Pearson Education, Inc. Slide 1.1- 15

Copyright © 2011 Pearson Education, Inc. EXAMPLE 4 Simplify each expression. a. 12b – 9b + 5b – 7b = (12 – 9 + 5 – 7)b Distributive property. = b Combine like terms. b. 6 – (2x + 7) – 3 = 6 – 2x – 7 – 3 Distributive property. = –2x + 6 – 7 – 3 Commutative property. = –2x – 4 Combine like terms. Copyright © 2011 Pearson Education, Inc. Slide 1.1- 16

Copyright © 2011 Pearson Education, Inc. continued Simplify each expression. c. 4m(2n) = (4)(2)mn = 8mn Copyright © 2011 Pearson Education, Inc. Slide 1.1- 17

Copyright © 2011 Pearson Education, Inc. Objective 5 Use the multiplication property of 0. Copyright © 2011 Pearson Education, Inc. Slide 1.1- 18

Copyright © 2011 Pearson Education, Inc. Multiplication Property of 0 For any real number a, a  0 = 0 and 0  a = 0. Copyright © 2011 Pearson Education, Inc. Slide 1.1- 19