1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with like signs could be negative. For example, (2)(2)(2) 4(2) 8. 1.4 MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers. Multiplication of Real Numbers The result of multiplying two numbers is referred to as the product of the numbers. The numbers multiplied are called factors. In algebra we use a raised dot between the factors to indicate multiplication, or we place symbols next to one another to indicate multiplication. Thus a b or ab are both referred to as the product of a and b. When multiplying numbers, we may enclose them in parentheses to make the meaning clear. To write 5 times, we may write it as 5, 5(), (5), or (5)(). In multiplying a number and a variable, no sign is used between them. Thus 5x is used to represent the product of 5 and x. Multiplication is just a short way to do repeated additions. Adding together five s gives 15. So we have the multiplication fact 5 15. Adding together five s gives () () () () () 15. So we should have 5() 15. We can think of 5() 15 as saying that taking on five debts of $ each is equivalent to a debt of $15. Losing five debts of $ each is equivalent to gaining $15, so we should have (5)() 15. These examples illustrate the rule for multiplying signed numbers. Product of Signed Numbers To find the product of two nonzero real numbers, multiply their absolute values. The product is positive if the numbers have like signs. The product is negative if the numbers have unlike signs. E X A M P L E 1 Multiplying signed numbers Evaluate each product. a) (2)() b) (6) c) 5 1 1 1 2 d) e) (.2)(.8) f) ()(.6) Solution a) First find the product of the absolute values: 2 2 6 Because 2 and have the same sign, we get (2)() 6. b) First find the product of the absolute values: 6 6 18 Because and 6 have unlike signs, we get (6) 18.
26 (1-26) Chapter 1 Real Numbers and Their Properties calculator close-up Try finding the products in Example 1 with your calculator. c) 5 1 5 Unlike signs, negative result d) 1 1 1 6 2 Like signs, positive result e) When multiplying decimals, we total the number of decimal places in the factors to get the number of decimal places in the product. Thus (.2)(.8).16. f) ()(.6) 18 We say that 1 5 2 because 2 5 1. This example illustrates how division is defined in terms of multiplication. If a, b, and c are any real numbers with b, then a b c provided that c b a. helpful hint Some people remember that two positives make a positive, a negative and a positive make a negative, and two negatives make a positive. Of course that is true only for multiplication, division, and cute stories like the following: If a good person comes to town, that s good. If a bad person comes to town, that s bad. If a good person leaves town, that s bad. If a bad person leaves town, that s good. Using the definition of division, we get 1 (2) 5 because (5)(2) 1; 1 2 5 because (5)(2) 1; and 1 (2) 5 because (5)(2) 1. From these examples we see that the rule for dividing signed numbers is similar to that for multiplying signed numbers. Division of Signed Numbers To find the quotient of nonzero real numbers, divide their absolute values. The quotient is positive if the numbers have like signs. The quotient is negative if the numbers have unlike signs. Zero divided by any nonzero real number is zero. E X A M P L E 2 Dividing signed numbers Evaluate. a) (8) (4) b) (8) 8 c) 8 (4) 1 d) 4 e) 2.5.5 f) (6) Solution a) (8) (4) 2 Same sign, positive result b) (8) 8 1 Unlike signs, negative result c) 8 (4) 2
1.4 Multiplication and (1-27) 27 study tip Read the material in the text before it is discussed in class, even if you do not totally understand it. The classroom discussion will be the second time you have seen the material and it will be easier to question points that you do not understand. 1 d) 4 4 1 Invert and multiply. 4 12 2.5 e) 2.5.5 Write in fraction form..5 2.5 1.5 1 25 5 Multiply by 1 to eliminate the decimals. Simplify. 5 Divide. f) (6) Division can also be indicated by a fraction bar. For example, 24 24 6 4. 6 study tip If you don t know how to get started on the exercises, go back to the examples. Cover the solution in the text with a piece of paper and see if you can solve the example. After you have mastered the examples, then try the exercises again. If signed numbers occur in a fraction, we use the rules for dividing signed numbers. For example, 9 9 1 1,, 1 4, and 2. 2 2 2 2 Note that if one negative sign appears in a fraction, the fraction has the same value whether the negative sign is in the numerator, in the denominator, or in front of the fraction. If the numerator and denominator of a fraction are both negative, then the fraction has a positive value. Division by Zero Why do we exclude division by zero from the definition of division? If we write 1 c, we need to find a number c such that c 1. This is impossible. If we write c, we need to find a number c such that c. In fact, c is true for any value of c. Having equal to any number would be confusing in doing computations. Thus a b is defined only for b. Quotients such as are said to be undefined. 8,, 8, and WARM-UPS True or false? Explain your answer. 1. The product of 7 and y is written as 7y. True 2. The product of 2 and 5 is 1. False x. The quotient of x and can be written as x or. True 4. 6 is undefined. False 5. (9) () True 6. 6 (2) True
28 (1-28) Chapter 1 Real Numbers and Their Properties WARM-UPS (continued) 7. 1 2 1 2 1 True 8. (.2)(.2).4 False 4 9. 1 2 1 2 1 True 1. False 1. 4 EXERCISES Reading and Writing After reading this section write out the answers to these questions. Use complete sentences. 1. What operations did we study in this section? We learned to multiply and divide signed numbers. 2. What is a product? A product is the result of multiplication. The product of a and b is ab. The product of 2 and 4 is 8.. How do you find the product of two signed numbers? To find the product of signed numbers, multiply their absolute values and then affix a negative sign if the two original numbers have opposite signs. 4. What is the relationship between division and multiplication? Division is defined in terms of multiplication as a b c provided c b a. 5. How do you find the quotient of nonzero real numbers? To find the quotient of nonzero numbers divide their absolute values and then affix a negative sign if the two original numbers have opposite signs. 6. Why is division by zero undefined? Division by zero is undefined because it cannot be made consistent with the definition of division: a b c provided c b a. Evaluate. See Example 1. 7. 9 27 8. 6(4) 24 9. (12)(11) 12 1. (9)(15) 15 4 4 9 1 11. 12. 1..5(.6). 14. (.)(.).9 15. (12)(12) 144 16. (11)(11) 121 17. 18. (7) Evaluate. See Example 2. 19. 8 (8) 1 2. 6 2 21. (9) () 22. (2) (4) 1 2 44 2. 2 66 24. 1 1 6 12 2 4 5 5 6 1 4 9 4 25. 26. 27. 1 25 undefined 28. 7 undefined 2 6 7 4 7 29.. 4.568 1. 4 (.5) 8 2. (.1)..5 (2).25 4..75 (.5) 1.5 Perform the indicated operations. 5. (25)(4) 1 6. (5)(4) 2 7. ()(9) 27 8. (51) () 17 9. 9 4. 86 (2) 4 41. 2 (5) 4 42. (8)(6) 48 4. (6)(5) 44. (18) 6 45. (57) () 19 46. () (4) 12 47. (.6)(.).18 48. (.2) (.5).1 49. (.)(1). 5. (.5) (1.5).75 51. (.6) (.1) 6 52. 8 (.5) 16 5. (.6) (.4) 1.5 54. (6) (.9) 7 55. 1 2 5 5 5 9 4 6 6 22 56. 1 5 4 1 1 4 1 57. 2 8 58. 9 1 2 1 6 59. (.45)(65) 6. 8.5 (.15) 164.25 56.667 61. (52) (.4) 62. (4.8)(5.6) 1529.41 26.88 Perform the indicated operations. Use a calculator to check. 6. (4)(4) 16 64. 4 4 8 65. 4 (4) 8 66. 4 (4) 1 67. 4 4 68. 4 4 16 69. 4 (4) 7. (4) 71..1 4.9 72. (.1)(4).4 7. (4) (.1) 4 74..1 4 4.1 75. (.1)(4).4 76..1 4.9 77..4.4 78..4.4 79.. 6 2.2 8...4 5 1. 2 81. 7.5 82..4. 4
1-5 Exponential Expression and the Order of Operations (1-29) 29 8. 1 5 1 6 1 84. 5 1 4 1 7 2 85. 4 2 1 15 1 86. 1 1 4 4 Use a calculator to perform the indicated operation. Round answers to three decimal places. 45.7 87. 7.562 88. (45) (28) 12.21 6 89. (4.)(4.5) 19.5 9. 1 2.4 4.11 91. 92. (4.51) 6. 45 2.44 9. 199.4 undefined 94. undefined GETTING MORE INVOLVED 95. Discussion. If you divide $ among five people, how much does each person get? If you divide $5 among zero people, how much does each person get? What do these questions illustrate? 96. Discussion. What is the difference between the nonnegative numbers and the positive numbers? 97. Writing. Why do we learn multiplication of signed numbers before division? 98. Writing. Try to rewrite the rules for multiplying and dividing signed numbers without using the idea of absolute value. Are your rewritten rules clearer than the original rules? In this section 1.5 EXPONENTIAL EXPRESSIONS AND THE ORDER OF OPERATIONS In Sections 1. and 1.4 you learned how to perform operations with a pair of real numbers to obtain a third real number. In this section you will learn to evaluate expressions involving several numbers and operations. Arithmetic Expressions Exponential Expressions The Order of Operations Arithmetic Expressions The result of writing numbers in a meaningful combination with the ordinary operations of arithmetic is called an arithmetic expression or simply an expression. Consider the expressions ( 2) 5 and (2 5). The parentheses are used as grouping symbols and indicate which operation to perform first. Because of the parentheses, these expressions have different values: ( 2) 5 5 5 25 (2 5) 1 1 Absolute value symbols and fraction bars are also used as grouping symbols. The numerator and denominator of a fraction are treated as if each is in parentheses. E X A M P L E 1 Using grouping symbols Evaluate each expression. a) ( 6)( 6) b) 4 5 9 c) 4 (8) 5 9