Section 1.1 Introduction to Algebra and the Set of Real Numbers 49 pq We also say that a rational number is a ratio of two integers, p and q, where q is not equal to zero. Identifying Rational Numbers Example 4 Show that the numbers are rational numbers by finding an equivalent ratio of two integers. 2 3 a. b. c. 0.5 d. 12 0.6 Solution: a. The fraction is a rational number because it can be expressed as the ratio of 2 and 3. 2 3 b. The number is a rational number because it can be expressed as the 12 12 12 ratio of 12 and 1, that is, 1 . In this example, we see that an integer is also a rational number. c. The terminating decimal 0.5 is a rational number because it can be 0.5 5 expressed as the ratio of 5 and 10. That is, 10 . In this example, we see that a terminating decimal is also a rational number. d. Table R-1 on page 23 shows that is the decimal form of the fraction 23 0.6 . The repeating decimal 0.6 is a rational number because it can be expressed as the ratio of 2 and 3. In this example, we see that a repeating decimal is also a rational number. Some real numbers, such as the number , cannot be represented by the ratio of two integers. These numbers are called irrational numbers and in decimal form are nonterminating, nonrepeating decimals. The value of , for example, can be approximated as However, the decimal p 3.1415926535897932. p digits continue forever with no repeated pattern. Another example of an irrational 13 number is (read as “the positive square root of 3”). The expression is a number that when multiplied by itself is 3. There is no rational 13 number that satisfies this condition. Thus, 13 is an irrational number. The set of real numbers consists of both the rational and the irrational numbers. The relationship among these important sets of numbers is illustrated in Figure 1-2. Skill Practice Show that each number is rational by finding an equivalent ratio of two integers. 3 5. 6. 5 7 7. 0.3 8. 0.3 TIP: Any rational number can be represented by a terminating decimal or by a repeating decimal. Irrational Numbers The set of irrational numbers is a subset of the real numbers whose elements cannot be written as a ratio of two integers. Note: An irrational number cannot be written as a terminating decimal or as a repeating decimal. Classroom Examples: p. 56, Exercises 20 and 22 Answers 5. ratio of 3 and 7 6. ratio of 5 and 1 7. ratio of 3 and 10 8. ratio of 1 and 3
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