5 4 The Irrational Numbers and the Real
5. 4 The Irrational Numbers and the Real Number System Copyright © 2009 Pearson Education, Inc. Slide 5 - 1
Pythagorean Theorem n n Pythagoras, a Greek mathematician, is credited with proving that in any right triangle, the square of the length of one side (a 2) added to the square of the length of the other side (b 2) equals the square of the length of the hypotenuse (c 2). a 2 + b 2 = c 2 Copyright © 2009 Pearson Education, Inc. Slide 5 - 2
Irrational Numbers n n An irrational number is a real number whose decimal representation is a nonterminating, nonrepeating decimal number. Examples of irrational numbers: Copyright © 2009 Pearson Education, Inc. Slide 5 - 3
Radicals n are all irrational numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand. Copyright © 2009 Pearson Education, Inc. Slide 5 - 4
Principal Square Root n n The principal (or positive) square root of a number n, written is the positive number that when multiplied by itself, gives n. For example, Copyright © 2009 Pearson Education, Inc. Slide 5 - 5
Perfect Square n n Any number that is the square of a natural number is said to be a perfect square. The numbers 1, 4, 9, 16, 25, 36, and 49 are the first few perfect squares. Copyright © 2009 Pearson Education, Inc. Slide 5 - 6
Product Rule for Radicals n Simplify: a) b) Copyright © 2009 Pearson Education, Inc. Slide 5 - 7
Copyright © 2009 Pearson Education, Inc. Slide 5 - 8
Addition and Subtraction of Irrational Numbers n n To add or subtract two or more square roots with the same radicand, add or subtract their coefficients. The answer is the sum or difference of the coefficients multiplied by the common radical. Copyright © 2009 Pearson Education, Inc. Slide 5 - 9
Example: Adding or Subtracting Irrational Numbers n Simplify: Copyright © 2009 Pearson Education, Inc. n Simplify: Slide 5 - 10
Copyright © 2009 Pearson Education, Inc. Slide 5 - 11
Multiplication of Irrational Numbers n Simplify: Copyright © 2009 Pearson Education, Inc. Slide 5 - 12
Copyright © 2009 Pearson Education, Inc. Slide 5 - 13
Quotient Rule for Radicals Copyright © 2009 Pearson Education, Inc. Slide 5 - 14
Example: Division n Divide: n Solution: Copyright © 2009 Pearson Education, Inc. Slide 5 - 15
Copyright © 2009 Pearson Education, Inc. Slide 5 - 16
Rationalizing the Denominator n n A denominator is rationalized when it contains no radical expressions. To rationalize the denominator, multiply BOTH the numerator and the denominator by a number that will result in the radicand in the denominator becoming a perfect square. Then simplify the result. Copyright © 2009 Pearson Education, Inc. Slide 5 - 17
Example: Rationalize n Rationalize the denominator of n Solution: Copyright © 2009 Pearson Education, Inc. Slide 5 - 18
Copyright © 2009 Pearson Education, Inc. Slide 5 - 19
Homework n P. 249 # 9 – 66 (x 3) Copyright © 2009 Pearson Education, Inc. Slide 5 - 20
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