1. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 1

2. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 2 The Real Number System Chapter 1

3. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 3 1.7 Properties of Real Numbers

4. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 4 Objectives 1.Use the commutative properties. 2.Use the associative properties. 3.Use the identity properties. 4.Use the inverse properties. 5.Use the distributive property. 1.7 Properties of Real Numbers

5. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 5 1.7 Properties of Real Numbers The commutative properties say that if two numbers are added or multiplied in any order, they give the same result. Addition Multiplication Using the Commutative Properties

6. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 6 Example 1 Use a commutative property to complete each statement. (a) –5 + 7 = 7 + ___ 1.7 Properties of Real Numbers Using the Commutative Properties (–5) (b) –3(8) = 8 (___)–3

7. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 7 1.7 Properties of Real Numbers The associative properties say that when we add or multiply three numbers, we can group them in any manner and get the same answer. Addition Multiplication Using the Associative Properties

8. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 8 Example 2 Use an associative property to complete each statement. (a) –6 + (–3 + 5) = (–6 + ___) + 5 1.7 Properties of Real Numbers Using the Associative Properties (–3) (b) –3(5 · –2) =(–3 ·___) · –25

9. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 9 1.7 Properties of Real Numbers Using Commutative and Associative Properties Example 3 Find each sum or product. (a) 18 + 23 + 9 + 12 + 27 + 11 =(18 + 12) + (23 + 27) + (9 + 11) = 30 + 50 + 20 = 100 (b) 75(9 · 4) =(75 · 4)9 = 300 · 9 = 2700

10. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 10 1.7 Properties of Real Numbers The identity properties say that the sum of 0 and any number equals that number, and the product of 1 and any number equals that number. Addition Multiplication Using the Identity Properties

11. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 11 1.7 Properties of Real Numbers Using the Inverse Properties Example 4 These statements are examples of identity properties.

12. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 12 1.7 Properties of Real Numbers Using the Identity Properties Example 5 Simplify each expression. The product of a number and 1 is that number.

13. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 13 The opposite of a, – a, is the additive inverse of a. The inverse property of addition says that the sum of a number and its additive inverse is 0 (the additive identity). Addition 1.7 Properties of Real Numbers Using the Inverse Properties

14. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 14 The reciprocal of a, 1/ a, is the multiplicative inverse of the nonzero number a. The inverse property of multiplication says that the product of a number and its multiplicative inverse is 1 (the multiplicative identity). Multiplication 1.7 Properties of Real Numbers Using the Inverse Properties

15. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 15 1.7 Properties of Real Numbers Using the Inverse Properties Example 6 The following statements are examples of inverse properties.

16. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 16 1.7 Properties of Real Numbers 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. a(b + c) = ab + ac The distributive property is also valid for subtraction. a(b – c) = ab – ac Using the Distributive Property

17. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 17 Example 7 Use the distributive property to rewrite each expression. (a) –6(–2 + 5) = –6(–2) + –6(5) 1.7 Properties of Real Numbers Using the Distributive Property = 12 + (–30) = –18 (b) 2(3x – 7) = 2(3x) + 2(–7) = 6x + (–14) = 6x –14

18. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.7 – Slide 18 Example 7 (concluded) Use the distributive property to rewrite each expression. (c) 3 · 9 + 3 · 6 = Here, we can use the distributive property in reverse. 1.7 Properties of Real Numbers Using the Distributive Property = 3(15) = 45 (d) 5(2r – 3s + 4t) = 5(2r) + 5(–3s) + 5(4t) = 10r – 15s + 20t 3(9 + 6)