CHAPTER 5 Number Theory and the Real Number

5. 4 The Irrational Numbers ALWAYS LEARNING 2 Copyright © 2019, 2015, 2011 Pearson

1. 2. 3. 4. Objectives Define the irrational numbers. Simplify square roots. Perform operations with square roots. Rationalize the denominator. ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 3

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: π ≈ 3. 1415926535897932384626433832795… ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 4

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. ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 5

Square Roots A perfect square is a number that is the square of a whole number. For example, here a few perfect squares: 0 = 02 1 = 12 4 = 22 9 = 32 The square root of a perfect square is a whole number: ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 6

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. ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 7

Example: Simplifying Square Roots Simplify, if possible: a. c. b. Because 17 has no perfect square factors (other than 1), it cannot be simplified. ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 8

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. ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 9

Example: Multiplying Square Roots Multiply: a. b. c. ALWAYS LEARNING Copyright © 2019, 2015,

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. ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 11

Example: Dividing Square Roots Find the quotient: a. b. ALWAYS LEARNING Copyright © 2019,

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: ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 13

Example: Adding and Subtracting Square Roots Add or subtract as indicated: a. b. Solution:

Rationalizing the Denominator We rationalize the denominator to rewrite the expression so that the denominator no longer contains any radicals. ALWAYS LEARNING Copyright © 2019, 2015, 2011 Pearson Education, Inc. Slide 15