236 CHAPTER 5 Number Theory and the Real Number System A fraction is said to be in its lowest terms (or reduced) when the numerator and denominator are relatively prime (that is, have no common divisors other than 1). To reduce a fraction to its lowest terms, divide both the numerator and the denominator by the greatest common divisor. Recall that a procedure for determining the greatest common divisor was discussed in Section 5.1. The fraction 6 10 is reduced to its lowest terms as follows. = ÷ ÷ = 6 10 6 2 10 2 3 5 Example 1 Reducing a Fraction to Lowest Terms Reduce 36 90 to its lowest terms. Solution In Example 4 of Section 5.1, we determined that the GCD of 36 and 90 is 18. Divide the numerator and the denominator by GCD, 18. = ÷ ÷ = 36 90 36 18 90 18 2 5 Since there are no common divisors of 2 and 5 other than 1, the fraction 2 5 is in its lowest terms. 7 Now try Exercise 11 Mixed Numbers and Improper Fractions Consider the number 2 . 3 4 It is an example of a mixed number. It is called a mixed number because it consists of an integer, 2, and a fraction, . 3 4 The mixed number 2 3 4 means + 2 . 3 4 The mixed number 4 1 4 − means ( ) − + 4 . 1 4 Rational numbers greater than 1 or less than −1 that are not integers may be represented as mixed numbers, or as improper fractions. An improper fraction is a fraction whose numerator is greater than its denominator. An example of an improper fraction is . 8 5 Fig. 5.6 shows both mixed numbers and the corresponding improper fractions indicated on a number line. In this section, we show how to convert mixed numbers to improper fractions and vice versa. 25 24 23 22 21 0 1 2 3 17 4 2 1 4 24 2 3 22 3 4 2 1 2 21 1 2 3 8 3 2 3 2 2 11 4 7 2 mixed numbers improper fractions Figure 5.6 We begin by limiting our discussion to positive mixed numbers and positive improper fractions. CONVERTING A POSITIVE MIXED NUMBER TO AN IMPROPER FRACTION 1. Multiply the denominator of the fraction in the mixed number by the integer preceding it. 2. Add the product obtained in Step 1 to the numerator of the fraction in the mixed number. This sum is the numerator of the improper fraction we are seeking. The denominator of the improper fraction we are seeking is the same as the denominator of the fraction in the mixed number. PROCEDURE
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