SECT
ION
0.4
0.4
0.4
OBJECTIVES
1. Multiply signed
numbers
2. Use the commutative property of
multiplication
3. Use the associative property of
multiplication
4. Divide signed
numbers
5. Use the distributive property
Example 1
Multiplying and Dividing
Signed Numbers
“Man’s mind, once stretched by a new idea, never regains its original dimensions.”
–Oliver Wendell Holmes
Multiplication can be seen as repeated addition. We can interpret
3 4 4 4 4 12
We can use this interpretation together with the work of Section 0.3 to find the product of two signed numbers.
Finding the Product of Two Signed Numbers
Multiply.
Note that we use parentheses
( ) to indicate multiplication
when negative numbers are
involved.
(a) (3)(4) (4) (4) (4) 12
(b) (4)(5) (5) (5) (5) (5) 20
✓ CHECK YOURSELF 1
■
Find the product by writing as repeated addition.
4(3)
Looking at the products we found by repeated addition in Example 1 should suggest our first rule for multiplying signed numbers.
To Multiply Signed Numbers
28
1. The product of two numbers with different signs is negative.
Section 0.4
Multiplying and Dividing Signed Numbers
■
29
The rule is easy to use. To multiply two numbers with different signs, just multiply
their absolute values and attach a minus sign to the product.
Example 2
Finding the Product of Two Signed Numbers
Find each product.
(5)(6) 30
(10)(12) 120
(7)(9) 63
The product must have two
decimal places.
(1.5)(0.3) 0.45
The product is negative. You
can simplify as before in
finding the product.
1
1
5 4
5
4
8 15
8
15
2
3
1
6
✓ CHECK YOURSELF 2
■
Find each product.
(a) (15)(5)
(b) (0.8)(0.2)
2 6
(c) 3 7
The product of two negative numbers is harder to visualize. The following pattern
may help you see how we can determine the sign of the product.
(3)(2) 6
(2)(2) 4
(1)(2) 2
(0)(2) 0
(1)(2) 2
Do you see that the product
is increasing by 2 each time
the first number decreases by 1?
(2)(2) 4
We already know that the
product of two positive
numbers is positive.
This suggests that the product of two negative numbers is positive, and this is in fact
the case. To extend our multiplication rule, we have the following.
30
Chapter 0
The Arithmetic of Signed Numbers
■
To Multiply Signed Numbers
2. The product of two numbers with the same sign is positive.
Finding the Product of Two Signed Numbers
Example 3
Find each product.
8 7 56
Since the numbers have the
same sign, the product is positive.
(9)(6) 54
(0.5)(2) 1
✓ CHECK YOURSELF 3
■
Find each product.
(a) (5)(7)
Caution
Example 4
!
(b) (8)(6)
(c) (9)(6)
(d) (1.5)(4)
Be Careful! (8)(6) tells you to multiply. The parentheses are next to one another. The expression 8 6 tells you to subtract. The numbers are separated by the
operation sign.
To multiply more than two signed numbers, apply the multiplication rule repeatedly.
Finding the Product of a Set of Signed Numbers
Multiply.
(5)(7)(3)(2)









(35)(3)(2)
(105)(2)
210
(5)(7) 35
(35)(3) 105
✓ CHECK YOURSELF 4
■
Find the product.
(4)(3)(2)(5)
Section 0.4
■
Multiplying and Dividing Signed Numbers
31
We saw in Section 0.3 that the commutative and associative properties for addition could be extended to signed numbers. The same is true for multiplication. What
about the order in which we multiply? Look at the following examples.
Example 5
Using the Commutative Property of Multiplication
Find the products.
(5)(7) (7)(5) 35
(6)(7) (7)(6) 42
The order in which we multiply does not affect the product. This gives us the following rule.
The Commutative Property of Multiplication
The centered dot represents
multiplication. This could have
been written as
The order in which we multiply does not change the product. Multiplication is
commutative. In symbols, for any a and b,
abba
abba
✓ CHECK YOURSELF 5
■
Show that (8)(5) (5)(8).
What about the way we group numbers in multiplication? Look at Example 6.
Example 6
Using the Associative Property of Multiplication
Multiply.
The symbols [ ] are called
brackets and are used to
group numbers in the same
way as parentheses.
[(3)(7)](2)
or
(3)[(7)(2)]
(21)(2)
(3)(14)
42
42
We group the first two numbers on the left and the second two numbers on the right.
Note that the product is the same in either case.
32
Chapter 0
■
The Arithmetic of Signed Numbers
The Associative Property of Multiplication
The way we group the numbers does not change the product. Multiplication is
associative. In symbols, for any a, b, and c,
(a b) c a (b c)
✓ CHECK YOURSELF 6
■
Show that [(2)(6)](3) (2)[(6)(3)].
Two numbers, 0 and 1, have special properties in multiplication.
The Multiplicative Identity
The product of 1 and any number is that number. We call 1 the multiplicative identity. In symbols, for any a,
a1a
Example 7
Multiplying Signed Numbers by 1
Find the products.
(8)(1) 8
(1)(15) 15
✓ CHECK YOURSELF 7
■
Find the product.
(10)(1)
What about multiplication by 0?
Multiplying by Zero
The product of 0 and any number is 0. In symbols, for any a,
a00
Section 0.4
Example 8
■
Multiplying and Dividing Signed Numbers
33
Multiplying Signed Numbers by Zero
Find the products.
(9)(0) 0
(0)(23) 0
✓ CHECK YOURSELF 8
■
Find the product.
(0)(12)
Another important property in mathematics is the distributive property. The distributive property involves addition and multiplication together. We can illustrate the
property with an application.
Remember: The area of a
rectangle is the product of its
length and width:
30
10
Area 1
ALW
15
We can find the total area by multiplying
the length by the overall width, which is
found by adding the two widths.
We can find the total area as a sum
of the two areas.
(Area 1)
Length Width
(Area 2)
Length Width







30 10
30 15
30 25
300
450
750
750
30
(10 15)
or







{





Length Overall Width
Area 2
So
30 (10 15) 30 10 30 15
34
Chapter 0
■
The Arithmetic of Signed Numbers
This leads us to the following property.
Note the pattern.
a(b c) a b a c
We “distributed” the
multiplication “over” the
addition.
Example 9
The Distributive Property
If a, b, and c are any numbers,
a(b c) a b a c
and
(b c)a b a c a
Using the Distributive Property
Use the distributive property to simplify (remove the parentheses in) the following.
(a) 5(3 4)
Note: It is also true that
5(3 4) 5 7 35
Note: It is also true that
1
1
(9 12) (21) 7
3
3
5(3 4) 5 3 5 4 35
1
1
1
(b) (9 12) 9 12
3
3
3
347
✓ CHECK YOURSELF 9
■
Use the distributive property to simplify (remove the parentheses).
(a) 4(6 7)
1
(b) (10 15)
5
The distributive property applies to all signed numbers. First let us look at an example of multiplication distributed over addition.
Example 10
Distributing Multiplication over Addition
Use the distributive property to remove the parentheses and simplify the following.
(a) 4(2 5) 4(2) 4(5) 8 20 12
(b) 5(3 2) (5)(3) (5)(2) 15 (10) 5
✓ CHECK YOURSELF 10
■
Use the distributive property to remove the parentheses and simplify the following.
(a) 7(3 5)
(b) 2(6 3)
Section 0.4
■
Multiplying and Dividing Signed Numbers
35
The distributive property can also be used to distribute multiplication over
subtraction.
Example 11
Distributing Multiplication over Subtraction
Use the distributive property to remove the parentheses and simplify the following.
(a) 4(3 6) 4(3) 4(6) 12 24 36
(b) 7(3 2) 7(3) (7)(2) 21 (14) 21 14 35
✓ CHECK YOURSELF 11
■
Use the distributive property to remove the parentheses and simplify the following.
(a) 7(3 4)
(b) 2(4 3)
A detailed explanation of why the product of two negative numbers must be positive concludes our discussion of multiplying signed numbers.
The Product of Two Negative Numbers
The following argument shows why the product of two negative numbers must be
positive.
From our earlier work, we know that a
number added to its opposite is 0.
5 (5) 0
Multiply both sides of the statement by 3.
(3)[5 (5)] (3)(0)
A number multiplied by 0 is 0, so on the
right we have 0.
(3)[5 (5)] 0
We can now use the distributive property
on the left.
(3)(5) (3)(5) 0
Since we know that (3)(5) 15, the
statement becomes
15 (3)(5) 0
We now have a statement of the form 15 must we add to 15 to get 0, where
0. This asks, “What number
is the value of (3)(5)?” The answer
is, of course, 15. This means that
(3)(5) 15
The product must be positive.
It doesn’t matter what numbers we use in the argument. The product of two negative numbers will always be positive.
36
Chapter 0
■
The Arithmetic of Signed Numbers
Multiplication and division are related operations. So every division problem can
be stated as an equivalent multiplication problem.
842
12
4
3
Since 8 4 2
Since 12 3 4
Since the operations are related, the rules of signs for multiplication are also true
for division.
To Divide Signed Numbers
1. If two numbers have the same sign, the quotient is positive.
2. If two numbers have different signs, the quotient is negative.
Example 12
Dividing Two Signed Numbers
Divide 20 by 5.
The numbers 20 and 5
have different signs, and so
the quotient is negative.
20 (5) 4
Since 20 (5)(4)
✓ CHECK YOURSELF 12
■
Write the multiplication statement that is equivalent to
36 (4) 9
Example 13
Dividing Two Signed Numbers
Divide 20 by 5.
The two numbers have the
same sign, and so the
quotient is positive.
20
4
5
✓ CHECK YOURSELF 13
■
Find each quotient.
48
(a) 6
(b) (50) (5)
Since 20 (5)(4)
Section 0.4
■
Multiplying and Dividing Signed Numbers
37
As you would expect, division with fractions or decimals uses the same rules for
signs. Example 14 illustrates this concept.
Example 14
Dividing Two Signed Numbers
Divide.
1
First note that the quotient is
positive. Then invert the
divisor and multiply.
4
3
9
3 20
4
5
20
5 9
3
1
3
✓ CHECK YOURSELF 14
■
Find each quotient.
5
3
(a) 8
4
(b) 4.2 (0.6)
Be very careful when 0 is involved in a division problem. Remember that 0 divided by any nonzero number is 0. However, division by 0 is not allowed and will be
described as undefined.
Example 15
Dividing Signed Numbers When Zero Is Involved
Divide.
A statement like 9 0 has
no meaning. There is no
answer to the problem. Just
write “undefined.”
(a) 0 7 0
0
(b) 0
4
(c) 9 0 is undefined.
5
(d) is undefined.
0
✓ CHECK YOURSELF 15
■
Find the quotient if possible.
0
(a) 7
12
(b) 0
The result of Example 15 can be confirmed on your calculator. That will be included in the next example.
38
Chapter 0
■
Example 16
The Arithmetic of Signed Numbers
Dividing with a Calculator
Use your calculator to find each quotient.
12.567
(a) 0
The key stroke sequence on a graphing calculator
()
12.567
0
Enter
results in a “Divide by 0” error message. The calculator recognizes that it cannot divide by zero.
On a scientific calculator, 12.567 /
0
results in an error
message.
(b) 10.992 4.58
The key stroke sequence
()
10.992
or 10.992
/
()
4.58
4.58
/
Enter
yields 2.4
✓ CHECK YOURSELF 16
■
Find each quotient
31.44
(a) 6.55
(b) 23.6 0
✓ CHECK YOURSELF ANSWERS
■
4
2. (a) 75; (b) 0.16; (c) .
7
4. 120.
5. 40 40.
6. 36 36.
1.
(3) (3) (3) (3) 12.
3.
(a) 35; (b) 48; (c) 54; (d) 6.
11. (a) 49; (b) 14.
5
12. 36 (4)(9).
13. (a) 8; (b) 10.
14. (a) ; (b) 7.
6
15. (a) 0; (b) undefined.
16. (a) 4.8; (b) undefined.
7.
10.
8. 0.
9. (a) 52; (b) 5.
10. (a) 14; (b) 6.
E xercises
1. 56
2. 72
3. 12
4. 75
5. 72
6. 24
7. 42
8. 24
9. 0
10. 100
■
0.4
Multiply.
1. 7 8
2. (6)(12)
3. (4)(3)
4. 15 5
5. (8)(9)
6. (8)(3)
7. (7)(6)
8. (12)(2)
9. (10)(0)
10. (10)(10)
11. (8)(8)
12. (0)(50)
13. (20)(4)
14. (25)(8)
15. (9)(12)
16. (9)(9)
17. (20)(1)
18. (1)(30)
19. (40)(5)
20. (25)(5)
21. (10)(15)
22. (5)(6)
11. 64
12. 0
13. 80
14. 200
15. 108
16. 81
17. 20
18. 30
19. 200
20. 125
21. 150
22. 30
1
23. 4
1
24. 6
25. 22
26. 0
1
27. 6
2
28. 3
29. 120
30. 60
5
4
27. 8
15
8
7
28. 21
4
31. 80
32. 70
29. (5)(3)(8)
30. (4)(3)(5)
33. 150
34. 300
31. (2)(8)(5)
32. (7)(5)(2)
35. 144
36. 240
33. (2)(5)(3)(5)
34. (2)(5)(5)(6)
37. 15
38. 48
35. (4)(3)(6)(2)
36. (8)(3)(2)(5)
39. 48
40. 55
41. 70
42. 36
37. 5(6 9)
38. 12(5 9)
39. 8(9 15)
40. 11(8 3)
43. 36
44. 35
45. 5
41. 5(8 6)
42. 2(7 11)
43. 4(6 3)
44. 7(2 3)
46. 5
47. 6
48. 10
49. 10
50. 6
7
5
23. 10
14
3 4
24. 8 9
25. (11)(2)
15
26. (0)
4
Use the distributive property to remove parentheses and simplify the following.
Divide.
45. 15 (3)
35
46. 7
48
47. 8
48. 20 (2)
50
49. 5
50. 36 6
39
40
Chapter 0
■
The Arithmetic of Signed Numbers
24
51. 3
42
52. 6
60
53. 15
54. 70 (10)
55. 18 (1)
250
56. 25
62. 25
0
57. 9
12
58. 0
63. 25
59. 144 (12)
0
60. 10
65. 5
61. 7 0
25
62. 1
3
66. 2
150
63. 6
80
64. 16
65. 45 (9)
2
4
66. 3
9
51. 8
52. 7
53. 4
54. 7
55. 18
56. 10
57. 0
58. Undefined
59. 12
60. 0
61. Undefined
64. 5
1
67. 6
68. 2
5
69. 4
70. 15
71. 5
72. 2
7
14
67. 9
3
68. (8) (4)
7
14
69. 10
25
75
70. 5
75
71. 15
5
5
72. 8
16
73. 2.349
Divide using a graphing calculator. Round answers to the nearest thousandth.
74. 0.213
73. 5.634 2.398
74. 1.897 8.912
75. 13.859 4.148
75. 3.341
76. 39.476 17.629
77. 32.245 48.298
78. 43.198 56.249
76. 2.239
79. Dieting. A woman lost 42 pounds (lb). If she lost 3 lb each week, how long has
she been dieting?
77. 0.668
79. 14 weeks
80. Mowing lawns. Patrick worked all day mowing lawns and was paid $9 per hour.
If he had $125 at the end of a 9-hour day, how much did he have before he started
working?
80. $44
81. Unit pricing. A 4.5-lb can of food costs $8.91. What is the cost per pound?
81. $1.98
82. Investment. Suppose that you and your two brothers bought equal shares of an
investment for a total of $20,000 and sold it later for $16,232. How much did each
person lose?
78. 0.768
82. $1256
83. 2.4°F
83. Temperature. Suppose that the temperature outside is dropping at a constant rate.
At noon, the temperature is 70°F and it drops to 58°F at 5:00 P.M. How much did
the temperature change each hour?
Section 0.4
84. 126
85. 2
86. 4
87. 2
88. 5
■
Multiplying and Dividing Signed Numbers
41
84. Test tube count. A chemist has 84 ounces (oz) of a solution. He pours the solu2
tion into test tubes. Each test tube holds oz. How many test tubes can he fill?
3
To evaluate an expression involving a fraction (indicating division), we evaluate the
numerator and then the denominator. We then divide the numerator by the denominator as the last step. Using this approach, find the value of each of the following expressions.
89. 4
90. 3
5 15
85. 23
4 (8)
86. 25
6 18
87. 2 4
4 21
88. 38
(5)(12)
89. (3)(5)
(8)(3)
90. (2)(4)
91. Create an example to show that the division of signed numbers is not commutative.
92. Create an example to show that the division of signed numbers is not associative.
93. Here is another conjecture to consider:
ab a b for all numbers a and b.
(See the discussion in Exercises 0.3, problem 87, concerning testing a conjecture.)
Test this conjecture for various values of a and b. Use positive numbers, negative numbers, and 0. Summarize your results in a rule.
94. Use a calculator (or mental calculations) to compute the following:
5
5
5
5
5
, , , , 0.1 0.01 0.001 0.0001 0.00001
In this series of problems, while the numerator is always 5, the denominator is getting
smaller (and is getting closer to 0). As this happens, what is happening to the value of
the fraction?
5
Write an argument that explains why could not have any finite value.
0