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 5.4 The Irrational Numbers

3. © 2010 Pearson Prentice Hall. All rights reserved. Objectives 1.Define the irrational numbers. 2.Simplify square roots. 3.Perform operations with square roots. 4.Rationalize the denominator.

4. © 2010 Pearson Prentice Hall. All rights reserved. The Irrational Numbers The set of irrational numbers is the set of numbers whose decimal representations are neither terminating nor repeating. For example, a well-known irrational number is π because there is no last digit in its decimal representation, and it is not a repeating decimal: 4 π ≈ 3.1415926535897932384626433832795…

5. © 2010 Pearson Prentice Hall. All rights reserved. Square Roots The principal square root of a nonnegative number n, written, is the positive number that when multiplied by itself gives n. For example, because 6 · 6 = 36. Notice that is a rational number because 6 is a terminating decimal. Not all square roots are irrational. 5

6. © 2010 Pearson Prentice Hall. All rights reserved. Square Roots A perfect square is a number that is the square of a whole number. For example, here are a few perfect squares: 0 = 0 2 1 = 1 2 4 = 2 2 9 = 3 2 The square root of a perfect square is a whole number: 6

7. © 2010 Pearson Prentice Hall. All rights reserved. The Product Rule For Square Roots If a and b represent nonnegative numbers, then The square root of a product is the product of the square roots. 7

8. © 2010 Pearson Prentice Hall. All rights reserved. Example 1: Simplifying Square Roots Simplify, if possible: a.b. c. 8 Because 17 has no perfect square factors (other than 1), it cannot be simplified.

9. © 2010 Pearson Prentice Hall. All rights reserved. Multiplying Square Roots If a and b are nonnegative, then we can use the product rule to multiply square roots. The product of the square roots is the square root of the product. 9

10. © 2010 Pearson Prentice Hall. All rights reserved. Example 2: Multiplying Square Roots Multiply: a. b. c. 10

11. © 2010 Pearson Prentice Hall. All rights reserved. Dividing Square Roots The Quotient Rule If a and b represent nonnegative real numbers and b ≠ 0, then The quotient of two square roots is the square root of the quotient. 11

12. © 2010 Pearson Prentice Hall. All rights reserved. Example 3: Dividing Square Roots Find the quotient: a. b. 12

13. © 2010 Pearson Prentice Hall. All rights reserved. Adding and Subtracting Square Roots The number that multiplies a square root is called the square root’s coefficient. Square roots with the same radicand can be added or subtracted by adding or subtracting their coefficients: 13

14. © 2010 Pearson Prentice Hall. All rights reserved. Example 4: Adding and Subtracting Square Roots Add or subtract as indicated: a.b. Solution: a. b. 14

15. © 2010 Pearson Prentice Hall. All rights reserved. Rationalizing the Denominator We rationalize the denominator to rewrite the expression so that the denominator no longer contains any radicals. 15

16. © 2010 Pearson Prentice Hall. All rights reserved. Example 6: Rationalizing Denominators Rationalize the denominator: a. b. 16