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CONCEPTS 5.3 THE RATIONAL NUMBERS Rational numbers. Convert a repeating decimal to a fraction. 5.4 IRRATIONALS Pythagorean theorem square root +, -, ×, ÷ rationalizing the denominator 5.3 THE RATIONAL NUMBERS are numbers that can be expressed as the quotient of two integers. ex. Express the rational number in decimal form. a) 3 4 b) 3 11 How can you tell if a fraction will be a terminating or a repeating decimal? A fraction can be written as a terminating decimal iff the only prime factors of the reduced denominator are 2s or 5s. ex. a) 2 3 b) 7 20 c) 4 35 d) 7 35 ex. Convert to a fraction. a) 0.00045 b) 6.82 c) 26.0037 d) 0.6 e) 21.5342 Denseness Property: A set is dense iff between any two members of the set there exists a third member of the set. ex. Which sets are dense? a) integers b) rational numbers ex. Find a rational number between a) 2 5 and 3 5 b) 0.07 and 0.076 5.4 THE IRRATIONAL NUMBERS AND THE REAL NUMBER SYSTEM Pythagorean theorem: Given a right triangle with sides a and b, and hypotenuse c, ex. Given a right triangle with sides a and b, find the hypotenuse. a) a = 3 b=4 b) a = 1 b=1 Irrational numbers: a real number whose decimal representation is nonterminating and nonrepeating. Square root (principle square root), √ n, is the positive number that when squared equals n. Product rule for radicals: √ a·b= ex. Simplify. √ 3 40 = Addition and Subtraction ex. Simplify. √ √ a) 4 15 + 7 15 = √ √ √ b) 3 20 − 7 45 + 8 = Division √ a √ = b √ 15 ex. √ = 5 Rationalizing the denominator (eliminate the radical in the denominator. √ 3 a) √ 5 = √ 2 b) √ 6 = ex. A doorway is 12 feet x 9 feet. Find the diameter of the largest circular mirror that will fit through the door.