1.
4 Multiplication and Division of Real Numbers (1-25) 25
1.4
MULTIPLICATION AND DIVISION
OF REAL NUMBERS
In this section we will complete the study of the four basic operations with real
In this numbers.
section
Multiplication of Real Numbers
● Multiplication of Real
Numbers The result of multiplying two numbers is referred to as the product of the numbers.
● Division of Real Numbers The numbers multiplied are called factors. In algebra we use a raised dot between
● Division by Zero
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 3, we may write it as 5 3, 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
3 3 3 3 3 15.
So we have the multiplication fact 5 3 15. Adding together five 3’s gives
helpful hint
The product of two numbers
(3) (3) (3) (3) (3) 15.
with like signs is positive, but
So we should have 5(3) 15. We can think of 5(3) 15 as saying that
the product of three numbers
with like signs could be nega-
taking on five debts of $3 each is equivalent to a debt of $15. Losing five debts of
tive. For example, $3 each is equivalent to gaining $15, so we should have (5)(3) 15.
(2)(2)(2) 4(2) These examples illustrate the rule for multiplying signed numbers.
8.
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)(3) b) 3(6) c) 5 10
1 1
d) e) (0.02)(0.08) f) (300)(0.06)
3 2
Solution
a) First find the product of the absolute values:
2 3 2 3 6
Because 2 and 3 have the same sign, we get (2)(3) 6.
b) First find the product of the absolute values:
3 6 3 6 18
Because 3 and 6 have unlike signs, we get 3(6) 18.
26 (1-26) Chapter 1 Real Numbers and Their Properties
calculator c) 5 10 50 Unlike signs, negative result
1 1 1
d) Like signs, positive result
3 2 6
close-up 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
Try finding the products in Ex-
ample 1 with your calculator. (0.02)(0.08) 0.0016.
f) (300)(0.06) 18 ■
Division of Real Numbers
We say that 10 5 2 because 2 5 10. This example illustrates how division
is defined in terms of multiplication.
Division of Real Numbers
If a, b, and c are any real numbers with b 0, then
abc provided that c b a.
helpful hint Using the definition of division, we get
Some people remember that 10 (2) 5
“two positives make a positive,
a negative and a positive because (5)(2) 10;
make a negative, and two
10 2 5
negatives make a positive.” Of
course that is true only for because (5)(2) 10; and
multiplication, division, and
cute stories like the following: 10 (2) 5
If a good person comes to
because (5)(2) 10. From these examples we see that the rule for dividing
town, that’s good. If a bad per-
son comes to town, that’s bad.
signed numbers is similar to that for multiplying signed numbers.
If a good person leaves town,
that’s bad. If a bad person Division of Signed Numbers
leaves town, that’s good. 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 0.05 f) 0 (6)
3
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 Division of Real Numbers (1-27) 27
1 3
study tip d) 4 4 Invert and multiply.
3 1
Read the material in the text 4 3
before it is discussed in class,
even if you do not totally un- 12
derstand it. The classroom dis- 2.5
cussion will be the second e) 2.5 0.05 Write in fraction form.
0.05
time you have seen the mate-
2.5 100
rial and it will be easier to Multiply by 100 to eliminate the decimals.
question points that you do 0.05 100
not understand. 250
Simplify.
5
50 Divide.
f) 0 (6) 0 ■
Division can also be indicated by a fraction bar. For example,
24
24 6 4.
6
If signed numbers occur in a fraction, we use the rules for dividing signed numbers.
For example,
9 9 1 1 1 4
3, 3, , and 2.
3 3 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.
study tip Division by Zero
If you don’t know how to get Why do we exclude division by zero from the definition of division? If we write
started on the exercises, go
10 0 c, we need to find a number c such that c 0 10. This is impossible. If
back to the examples. Cover
we write 0 0 c, we need to find a number c such that c 0 0. In fact, c 0 0
the solution in the text with a
piece of paper and see if you
is true for any value of c. Having 0 0 equal to any number would be confusing in
can solve the example. After doing computations. Thus a b is defined only for b 0. Quotients such as
you have mastered the exam- 8 0
ples, then try the exercises 8 0, 0 0, , and
again.
0 0
are said to be undefined.
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 10. False
3. The quotient of x and 3 can be written as x 3 or x. True
3
4. 0 6 is undefined. False
5. (9) (3) 3 True 6. 6 (2) 3 True
28 (1-28) Chapter 1 Real Numbers and Their Properties
WARM-UPS
(continued)
4 True
1 1 1
7. 8. (0.2)(0.2) 0.4 False
2 2
1 True
1 1 0
9. 10. 0 False
2 2 0
1. 4 EXERCISES
1
Reading and Writing After reading this section write out the
29. 0 0 30. 0 43.568 0
answers to these questions. Use complete sentences. 3
1. What operations did we study in this section? 31. 40 (0.5) 80 32. 3 (0.1) 30
We learned to multiply and divide signed numbers.
33. 0.5 (2) 0.25 34. 0.75 (0.5) 1.5
2. What is a product?
Perform the indicated operations.
A product is the result of multiplication. The product of a
and b is ab. The product of 2 and 4 is 8. 35. (25)(4) 100 36. (5)(4) 20
3. How do you find the product of two signed numbers? 37. (3)(9) 27 38. (51) (3) 17
To find the product of signed numbers, multiply their 39. 9 3 3 40. 86 (2) 43
absolute values and then affix a negative sign if the two
original numbers have opposite signs. 41. 20 (5) 4 42. (8)(6) 48
4. What is the relationship between division and 43. (6)(5) 30 44. (18) 3 6
multiplication? 45. (57) (3) 19 46. (30) (4) 120
Division is defined in terms of multiplication as a b c
47. (0.6)(0.3) 0.18 48. (0.2) (0.5) 0.1
provided c b a.
49. (0.03)(10) 0.3 50. (0.05) (1.5) 0.075
5. How do you find the quotient of nonzero real numbers?
To find the quotient of nonzero numbers divide their 51. (0.6) (0.1) 6 52. 8 (0.5) 16
absolute values and then affix a negative sign if the two 53. (0.6) (0.4) 1.5 54. (63) (0.9) 70
original numbers have opposite signs. 9 4 6
12 55
6. Why is division by zero undefined? 55. 22 56.
5 6 10 3 5
Division by zero is undefined because it cannot be made
3 1 1
1 1
consistent with the definition of division: a b c 57. 2 8 58. 9 3 3
provided c b a. 4 4 3 2 6
Evaluate. See Example 1. 59. (0.45)(365) 60. 8.5 (0.15)
7. 3 9 27 8. 6(4) 24 164.25 56.667
61. (52) (0.034) 62. (4.8)(5.6)
9. (12)(11) 132 10. (9)(15) 135
1529.41 26.88
3 4 1 2 6 4
11.
4 9
3
12.
3 7
7
Perform the indicated operations. Use a calculator to check.
63. (4)(4) 16 64. 4 4 8
13. 0.5(0.6) 0.3 14. (0.3)(0.3) 0.09
65. 4 (4) 8 66. 4 (4) 1
15. (12)(12) 144 16. (11)(11) 121
67. 4 4 0 68. 4 4 16
17. 3 0 0 18. 0(7) 0
69. 4 (4) 0 70. 0 (4) 0
Evaluate. See Example 2.
71. 0.1 4 3.9 72. (0.1)(4) 0.4
19. 8 (8) 1 20. 6 2 3
73. (4) (0.1) 40 74. 0.1 4 4.1
1
21. (90) (30) 3 22. (20) (40)
2 75. (0.1)(4) 0.4 76. 0.1 4 3.9
44 2 33 11
23. 24. 77. 0.4 0.4 78. 0.4 0.4
66 3 36 12
2 4 5 1 4 3 0.06 2
79. 0.2 50
25.
3 5 6
26.
3 9 4 0.3
80.
0.04
125 3 1.2
27. undefined 28. 37 0 undefined 81. 7.5 82. 40
0 0.4 0.03
1-5 Exponential Expression and the Order of Operations (1-29) 29
1 1 1 3 1 17 GET TING MORE INVOLVED
83. 84.
5 6 30 5 4 20
95. Discussion. If you divide $0 among five people, how
3 2 1 1 much does each person get? If you divide $5 among zero
85. 86. 1 4
4 15 10 4 people, how much does each person get? What do these
questions illustrate?
Use a calculator to perform the indicated operation. 96. Discussion. What is the difference between the non-
Round answers to three decimal places. negative numbers and the positive numbers?
45.37 97. Writing. Why do we learn multiplication of signed num-
87. 7.562 88. (345) (28) 12.321
6 bers before division?
12.34 98. Writing. Try to rewrite the rules for multiplying and
89. (4.3)(4.5) 19.35 90. 4.113
3 dividing signed numbers without using the idea of absolute
0 value. Are your rewritten rules clearer than the original
91. 0 92. 0 (34.51) 0 rules?
6.345
23.44
93. 199.4 0 undefined 94. undefined
0
1.5 EXPONENTIAL EXPRESSIONS AND
THE ORDER OF OPERATIONS
In Sections 1.3 and 1.4 you learned how to perform operations with a pair of real
In this numbers to obtain a third real number. In this section you will learn to evaluate ex-
section pressions involving several numbers and operations.
● Arithmetic Expressions
● Exponential Expressions Arithmetic Expressions
● The Order of Operations The result of writing numbers in a meaningful combination with the ordinary oper-
ations of arithmetic is called an arithmetic expression or simply an expression.
Consider the expressions
(3 2) 5 and 3 (2 5).
The parentheses are used as grouping symbols and indicate which operation to per-
form first. Because of the parentheses, these expressions have different values:
(3 2) 5 5 5 25
3 (2 5) 3 10 13
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) (3 6)(3 6)
b) 3 4 5 9
4 (8)
c)
59
