Slide Copyright © 2009 Pearson Education, Inc. 5.4 The Irrational Numbers and the Real Number System

Slide Copyright © 2009 Pearson Education, Inc. Pythagorean Theorem 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

Slide Copyright © 2009 Pearson Education, Inc. Irrational Numbers An irrational number is a real number whose decimal representation is a nonterminating, nonrepeating decimal number. Examples of irrational numbers:

Slide Copyright © 2009 Pearson Education, Inc. are all irrational numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand. Radicals

Slide Copyright © 2009 Pearson Education, Inc. Principal Square Root The principal (or positive) square root of a number n, written is the positive number that when multiplied by itself, gives n. For example,

Slide Copyright © 2009 Pearson Education, Inc. Perfect Square 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.

Slide Copyright © 2009 Pearson Education, Inc. Product Rule for Radicals Simplify: a) b)

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Slide Copyright © 2009 Pearson Education, Inc. Addition and Subtraction of Irrational Numbers 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.

Slide Copyright © 2009 Pearson Education, Inc. Example: Adding or Subtracting Irrational Numbers Simplify:

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Slide Copyright © 2009 Pearson Education, Inc. Multiplication of Irrational Numbers Simplify:

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Slide Copyright © 2009 Pearson Education, Inc. Quotient Rule for Radicals

Slide Copyright © 2009 Pearson Education, Inc. Example: Division Divide: Solution: Divide: Solution:

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Slide Copyright © 2009 Pearson Education, Inc. Rationalizing the Denominator 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.

Slide Copyright © 2009 Pearson Education, Inc. Example: Rationalize Rationalize the denominator of Solution:

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Slide Copyright © 2009 Pearson Education, Inc. Homework P. 249 # 9 – 66 (x3)