5.3 The Rational Numbers 235 We introduced the number line and discussed the integers in Section 5.2. The numbers that fall between the integers on the number line are either rational or irrational numbers. In this section, we discuss the rational numbers, and in Section 5.4, we discuss the irrational numbers. Any number that can be expressed as a quotient of two integers (denominator not 0) is a rational number. Among the ingredients called for in a recipe for chocolate chip cookies are 21 4 cups flour, 1 2 teaspoon salt, 1 1 3 teaspoons baking soda, 3 4 cup sugar, and 3 4 cup brown sugar. How much of each of these ingredients would you need to use if you wish to double the recipe? In this section, we will review the use of fractions such as those in this chocolate chip cookie recipe. We will also learn about the operations of addition, subtraction, multiplication, and division using fractions. The Rational Numbers SECTION 5.3 LEARNING GOALS Upon completion of this section, you will be able to: 7 Reduce fractions to lowest terms. 7 Convert mixed numbers to improper fractions and vice versa. 7 Express rational numbers as terminating or repeating decimal numbers. 7 Convert decimal numbers to fractions. 7 Multiply and divide fractions. 7 Add and subtract fractions. Why This Is Important Fractions are everywhere in our everyday lives. Kitchen measurements, tool sizes, and many other real-life applications all involve fractions. Definition: Rational Numbers The set of rational numbers, denoted by Q, is the set of all numbers of the form p q , where p and q are integers and ≠ q 0. Recall from Section 5.2 that division by zero is undefined. For this reason, the denominator, q, of a rational number, p q , cannot be zero. The following numbers are examples of rational numbers: 1 3 , 3 4 , 7 8 , 1 2 3 , 2, 0, 15 7 − The integers 2 and 0 are rational numbers because each can be expressed as the quotient of two integers: = 2 2 1 and = 0 . 0 1 In fact, every integer n is a rational number because it can be written in the form of . n 1 Numbers such as 1 3 and − 7 8 are also called fractions. The number above the fraction line is called the numerator, and the number below the fraction line is called the denominator. Reducing Fractions Sometimes the numerator and denominator in a fraction have a common divisor (or common factor). For example, both the numerator and denominator of the fraction 6 10 have the common divisor 2. When a numerator and denominator have a common divisor, we can reduce the fraction to its lowest terms. Jaimie Duplass/Shutterstock
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