258 CHAPTER 5 Number Theory and the Real Number System Evaluate + 10 5 and then evaluate + 5 10. Evaluate × 10 5 and then evaluate × 5 10. Evaluate − 10 5 and then evaluate − 5 10. Finally, evaluate ÷ 10 5 and then evaluate ÷ 5 10. You will notice that + = + 10 5 5 10 and that × = × 10 5 5 10. On the other hand, − ≠ − 10 5 5 10 and ÷ ≠ ÷ 10 5 5 10. These simple examples demonstrate a certain property that holds for addition and multiplication, but not for subtraction and division. In this section, we will study the set of real numbers and their properties. Real Numbers and Their Properties SECTION 5.5 LEARNING GOAL Upon completion of this section, you will be able to: 7 Understand the properties of the real number system. Why This Is Important Understanding the properties of real numbers is vital to learning algebra and many other areas of mathematics. Many applications that we will see in this section rely on our understanding of real numbers. For example, building a bookcase, creating a budget, determining price per unit, and determining the percent discount of an item are just some of the ways we use the properties of real numbers. 82. The product of any two rational numbers is always a rational number. True 83. The product of an irrational and a rational number is always an irrational number. * 84. The product of any two irrational numbers is always an irrational number. False. The product of two irrational numbers may be a rational number or an irrational number. In Exercises 85–88, give an example to show that the stated case can occur. 85. The sum of two irrational numbers may be a rational number. ( ) 0 π π + − = 86. The sum of two irrational numbers may be an irrational number. 3 2 5 2 8 2 + = 87. The product of two irrational numbers may be an irrational number. 2 3 6 ⋅ = 88. The product of two irrational numbers may be a rational number. 2 8 16 4 ⋅ = = 89. Without doing any calculations, determine whether 2 1.414. = 2 1.414 ≠ since 2 is irrational and 1.414 is rational. 90. Without doing any calculations, determine whether 17 4.123. = 17 4.123 ≠ since 17 is irrational and 4.123 is rational. 91. The number π is an irrational number. Often the values 3.14 or 22 7 are used for .π Does π equal either 3.14 or ? 22 7 No. π is irrational; it cannot equal 22/7 or 3.14, both of which are rational. 92. Give an example to show that a b a b. + ≠ + 9 16 3 4, 5 7 + ≠ + ≠ 93. Give an example to show that a b a b. ⋅ = ⋅ 49 4 9, 36 23,6 6 ⋅ = ⋅ = ⋅ = 94. Give an example to show that for b 0, ≠ a b a b . = 36 9 36 9 , 4 6 3 , 2 2 = = = Challenge Problems/Group Activities 95. a) Is 0.04 rational or irrational? Explain. Rational. 0.04 0.2, = which is a rational number. b) Is 0.7 rational or irrational? Explain. * 96. One way to determine a rational number between two distinct rational numbers is to add the two distinct rational numbers and divide by 2. Do you think that this method will always work for finding an irrational number between two distinct irrational numbers? Explain. No. The sum of two irrational numbers may not be an irrational number. Recreational Mathematics 97. More Four s 4’ In the Recreational Math box on page 228 and in Exercise 82 on page 234, we introduced some of the basic rules of the game Four 4’s. We now expand our operations to include square roots. For example, one way to represent the whole number 8 is 4 4 4 4 + + + = 2 2 2 2 8. + + + = Using at least one square root of 4, 4, play Four 4’s to represent the following whole numbers: a) 11 (44 4) 4 11 ÷ ÷ = b) 13 (44 4) 4 13 ÷ + = c) 14 4 4 4 4 14 + + + = d) 18 + + = 4(4 4) 4 18; Other answers are possible. Research Activities 98. History of Irrational Numbers Write a report on the history of the development of the irrational numbers. 99. History of Pi Write a report on the history of pi. In your report, indicate when the symbol π was first used and list the first 100 digits of .π *See Instructor Answer Appendix Yanlev/123RF
RkJQdWJsaXNoZXIy NjM5ODQ=