1. © 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 5 Number Theory and the Real Number System

2. © 2010 Pearson Prentice Hall. All rights reserved. 2 5.5 Real Numbers and Their Properties

3. © 2010 Pearson Prentice Hall. All rights reserved. 3 Objectives 1.Recognize the subsets of the real numbers. 2.Recognize the properties of real numbers.

4. © 2010 Pearson Prentice Hall. All rights reserved. 4 The Set Real Numbers The union of the rational numbers and the irrational numbers is the set of real numbers. The sets that make up the real numbers are called subsets of the real numbers.

5. © 2010 Pearson Prentice Hall. All rights reserved. 5 Example 1: Classifying Real Numbers Consider the following set of numbers: List the numbers in the set that are a.natural numbers b. whole numbers c. Integers d. rational numbers e. irrational numbers f. real numbers

6. © 2010 Pearson Prentice Hall. All rights reserved. 6 Solution: Example 1: Classifying Real Numbers (continued) a. natural numbersBecause = 9 b. whole numbers0, because whole numbers include 0 and the natural numbers c. integers0,, -7 because integers include whole numbers and the negative natural numbers d. rational numbers0,, -7, -¾, 0.6, & 7.3 because these numbers can be expressed as a quotient or as a terminating or repeating decimal e. irrational numbers, π because neither terminate nor have blocks of repeating digits f. real numbers0,, -7, -¾, 0.6,, 7.3, & π because real numbers have all the above numbers as subsets

7. © 2010 Pearson Prentice Hall. All rights reserved. 7 Properties of the Real Numbers

8. © 2010 Pearson Prentice Hall. All rights reserved. 8 Properties of the Real Numbers

9. © 2010 Pearson Prentice Hall. All rights reserved. 9 Properties of the Real Numbers

10. © 2010 Pearson Prentice Hall. All rights reserved. 10 Name the property illustrated: a. b.(4 + 7) + 6 = 4 + (7 + 6) c. d. Example 2: Identifying Properties of Real Numbers Commutative property of multiplication Associative property of addition Distributive property of multiplication over addition Inverse property of addition