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1.4 Multiplication and Division of Real Numbers
In this section, we will complete the study of the four basic operations with real numbers.
U 1 V 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, ab 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 3, we may write it as 53, 5(3), (5)3, or (5)(3). 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 3’s gives
3333315.
So we have the multiplication fact 5315. Adding together five3’s gives (3)(3)(3)(3)(3) 15.
So we should have 5(3) 15. Receiving five debts of $3 each is the same as a
$15 debt. If you have five debts of $3 each and they are forgiven, then you have gained $15. So we should have (5)(3)15.
These examples illustrate the rule for multiplying signed numbers.
In This Section U1VMultiplication of Real
Numbers
U2VDivision of Real Numbers
U3VDivision by Zero
UHelpful Hint V
The product of two numbers with like signs is positive, but the product of three numbers with like signs can be positive or negative. For example,
2228 and
(2)(2)(2) 8.
105. Discussion
Aimee and Joni are traveling south in separate cars on Interstate 5 near Stockton. While they are speaking to each other on cellular telephones, Aimee gives her location as mile marker x and Joni gives her location as mile marker y. Which of the following expressions gives the distance between them? Explain your answer.
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.
1-35 1.4 Multiplication and Division of Real Numbers 35
U 2 V Division of Real Numbers
We say that 10 2 5 because 5 2 10. This example illustrates how division is defined in terms of multiplication.
Using the definition of division, we can make the following table:
Division of Real Numbers
If a, b, and c are any real numbers with b0, then
a b c provided that c ba.
E X A M P L E 1 Multiplying signed numbers Evaluate each product.
a) (2)(3) b) 3(6) c) 5 10
d) 1312 e) (0.02)(0.08) f) (300)(0.06)
Solution
a) First find the product of the absolute values:
232 3 6
Because 2 and 3 have the same sign, we get (2)(3)6.
b) First find the product of the absolute values:
363 6 18
Because 3 and 6 have unlike signs, we get 3(6) 18.
c) 5 10 50 Unlike signs, negative result d) 1312 16 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,
(0.02)(0.08) 0.0016.
f ) (300)(0.06)18 Like signs, positive result
Now do Exercises 1–12 UCalculator Close-Up V
Try finding the products in Example 1 with your calculator.
10 25 because 5 210 10 (2)5 because 5(2) 10 10 (2) 5 because 5(2)10
10 2 5 because 5 2 10
Negative quotient
Positive quotient
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Zero divided by any nonzero real number is zero.
Division of Signed Numbers
To find the quotient of two nonzero real numbers, divide their absolute values.
• The quotient is positive if the two numbers have like signs.
• The quotient is negative if the two numbers have unlike signs.
E X A M P L E 2 Dividing signed numbers Evaluate.
a) (8) (4) b) (8) 8 c) 8 (4)
d) 4 1
3 e) 2.5 0.05 f ) 0 (6)
Solution
a) (8) (4) 8
4 2 Same sign, positive result b) (8) 8
8
8 1 Unlike signs, negative result c) 8 (4)
8
4 2 Unlike signs, negative result
d) 4 1
3 4 3
1 Invert and multiply.
4 3 12 e) 2.50.05
0.
2 0 . 5
5 Write in fraction form.
0.
2 0
. 5 5
1 1 0 0 0
0 Multiply by 100 to eliminate the decimals.
2 5
50 Simplify.
50 Divide.
f ) 0 (6)
0
6 0 Zero divided by a nonzero number is zero.
Now do Exercises 13–26 UHelpful Hint V
Do not use negative numbers in long division. To find 3787, divide 378 by 7:
54 7378 35
28 28 0
Since a negative divided by a positive is negative,
3787 54.
Notice that in this table, the quotient for two numbers with the same sign is posi- tive and the quotient for two numbers with opposite signs is negative. These exam- ples illustrate the rule for dividing signed numbers. The rule for dividing signed numbers is similar to that for multiplying signed numbers because of the definition of division.
1-37 1.4 Multiplication and Division of Real Numbers 37
Division can also be indicated by a fraction bar. For example,
246 2
6 4 4.
If signed numbers occur in a fraction, we use the rules for dividing signed numbers.
For example,
3
9 3, 9
3 3, 2
1
1 2 1
2, and
4 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.
U 3 V Division by Zero
Why do we exclude division by zero from the definition of division? If we write 100 c, we need to find a number c such that c010. This is impossible. If we write 00c, we need to find a number c such that c00. In fact, c00 is true for any value of c. Having 00 equal to any number would be confusing in doing computations. Thus, ab is defined only for b0. Quotients such as
8 0, 0 0, 8
0, and 0 0 are said to be undefined.
E X A M P L E 3 Division involving zero
Evaluate. If the operation is undefined, say so.
a) 0 1 b) 3
4 0 c)
0
12 d)
0 9
Solution
a) The result of 0 divided by a nonzero number is zero. So 0 10.
b) Since division by zero is not allowed,3
4 0 is an undefined operation.
c) Since division by zero is not allowed, 0
12is undefined.
d) The result of 0 divided by a nonzero number is zero. So
0 9 0.
Now do Exercises 27–34
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Warm-Ups ▼
Fill in the blank.
1. The result of multiplication is a .
2. To find the product of two signed numbers multiply their and use a negative sign if the original numbers have opposite signs.
3. To find the of two signed numbers divide their absolute values and use a negative sign if the original numbers have opposite signs.
4. Division is defined in terms of as a bc provided c ba and b 0.
True or false?
5. The product of 7 and y is 7y.
6. The product of 2 and 5 is 10.
7. The quotient of x and 3 is x 3 or 3 x. 8. 0 6 is undefined.
9. 9 (3)3 10. 6 (2) 3 11. (0.2)(0.2) 0.4 12. 0 00
U1V Multiplication of Real Numbers Evaluate. See Example 1.
1. 3 9 2. 6(4)
3. (12)(11) 4. (9)(15) 5. 3
4 4
9 6. 2367
7. 0.5(0.6) 8. (0.3)(0.3)
9. (12)(12) 10. (11)(11)
11. 3 0 12. 0(7)
U2V Division of Real Numbers Evaluate. See Example 2.
13. 8 (8) 14. 6 2
15. (90) (30) 16. (20) (40)
17.
4 6 4
6 18.
3 3 3 6 19. 23 45 20. 13 49
21. 0 13 22. 0 43.568
23. 40 (0.5) 24. 3 (0.1) 25. 0.5 (2) 26. 0.75 (0.5)
U3V Division by Zero
Evaluate. If the operation is undefined, say so. See Example 3.
27. 0 125 28. 0 (99)
29. 1 0
25 30. 3 0
.5 31. 1
2 0 32. 0.236 0
33. 0
2 34.
0 5
Exercises
UStudy Tips V
• If you don’t know how to get started on the exercises, go back to the examples. Read the solution in the text, and then cover it with a piece of paper and see if you can solve the example.
• If you need help, don’t hesitate to get it. If you don’t straighten out problems in a timely manner, you can get hopelessly lost.
1.4
1-39 1.4 Multiplication and Division of Real Numbers 39
Miscellaneous
Perform the indicated operations.
35. (25)(4) 36. (5)(4)
37. (3)(9) 38. (51) (3)
39. 9 3 40. 86 (2)
41. 20 (5) 42. (8)(6)
43. (6)(5) 44. (18) 3
45. (57) (3) 46. (30)(4) 47. (0.6)(0.3) 48. (0.2)(0.5) 49. (0.03)(10) 50. (0.05)(1.5) 51. (0.6) (0.1) 52. 8 (0.5) 53. (0.6) (0.4) 54. (63) (0.9) 55. 1
5
2565 56. 190 43
57. 23 4 81
4 58. 91
2 316
Use a calculator to perform the indicated operations. Round approximate answers to two decimal places.
59. (0.45)(365) 60. 8.5 (0.15) 61. (52) (0.034) 62. (4.8)(5.6)
Fill in the parentheses so that each equation is correct.
63. 5 ( ) 60 64. 9 ( ) 54
65. 12 ( ) 96 66. 11 ( ) 44
67. 24 ( ) 4 68. 51 ( ) 17
69. 36 ( ) 36 70. 48 ( ) 6
71. 40 ( ) 8 72. 13 ( ) 1
Perform the indicated operations. Use a calculator to check.
73. (4)(4) 74. 44
75. 4 (4) 76. 4 (4)
77. 4 4 78. 4 4
79. 4 (4) 80. 0 (4)
81. 0.1 4 82. (0.1)(4)
83. (4) (0.1) 84. 0.14
85. (0.1)(4) 86. 0.14
87. 0.4 88. 0.4
89.
0 0 . . 3
06 90.
0 2
.04
91.
3
0.4 92.
0
1 . . 0
2 3 93. 1
5 1
6 94.3
5 1 4 95. 34125 96.1 14
Use a calculator to perform the indicated operations.
Round approximate answers to three decimal places.
97. 45 6
.37 98. (345) (28)
99. (4.3)(4.5) 100. 1
2 3 .34 101. 6.3
0
45 102. 0 (34.51)
103. 199.4 0 104. 23
0 .44
Applications
105. Big loss. Ford Motor Company’s profit for 2008 was $14.6 billion. Find the rate in dollars per minute (to the nearest dollar) at which Ford was “making” money in 2008.
106. Negative divided by a positive. In 2009, the national debt was $11.27 trillion dollars and the U.S. population was 306.2 million people. Find the amount of the debt per person to the nearest dollar.
Getting More Involved 107. Discussion
If you divide $0 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?
108. Discussion
What is the difference between the nonnegative numbers and the positive numbers?
109. Writing
Why do we learn multiplication of signed numbers before division?
110. 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?
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In This Section
U1VArithmetic Expressions
U2VExponential Expressions
U3VThe Order of Operations
U4VApplications