Section 1.2 Introduction to Algebra and the Set of Real Numbers 25 c. The terminating decimal 0.5 is a rational number because it can be expressed as the ratio of 5 and 10, that is, . In this example, we see that a terminating decimal is also a rational number. d. The numeral represents the nonterminating, repeating decimal 0.6666666 . . . .The number is a rational number because it can be expressed as the ratio of 2 and 3, that is, . In this example, we see that a repeating decimal is also a rational number. Skill Practice Show that each number is rational by finding an equivalent ratio of two integers. 5. 6. 5 7. 0.3 8. 0.3 3 7 0.6 23 0.6 0.6 0.5 5 10 TIP: A rational number can be represented by a terminating decimal or by a repeating decimal. 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 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. digits continue forever with no repeated pattern. Another example of an 13 irrational 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 number 13 that satisfies this condition. Thus, 13 is an irrational number. p, p, 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. 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 1-6: Rational numbers Integers . . . 3, 2, 1 Whole numbers 0 Natural numbers 1, 2, 3, . . . Real Numbers 0.25 √2 0.121122111222 . . . 0.3 Irrational numbers √17 2 7 Figure 1-6
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