3: Integers: Addition and Subtraction
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1.3 • Integers: Addition and Subtraction
{0, 1, 2, 3, 4, . . .}
15
Nonnegative integers
{. . . , Ϫ3, Ϫ2, Ϫ1}
Negative integers
{. . . , Ϫ3, Ϫ2, Ϫ1, 0}
Nonpositive integers
The symbol Ϫ1 can be read as “negative one,” “opposite of one,” or “additive inverse of one.”
The opposite-of and additive-inverse-of terminology is very helpful when working with variables.
The symbol Ϫx, read as “opposite of x” or “additive inverse of x,” emphasizes an important issue:
Since x can be any integer, Ϫx (the opposite of x) can be zero, positive, or negative. If x is a positive integer, then Ϫx is negative. If x is a negative integer, then Ϫx is positive. If x is zero, then Ϫx
is zero. These statements are written as follows and illustrated on the number lines in Figure 1.3.
If x ϭ 3,
then Ϫx ϭ Ϫ(3) ϭ Ϫ3.
x
−4 −3 −2 −1
If x ϭ Ϫ3,
then Ϫx ϭ Ϫ(Ϫ3) ϭ 3.
0
1
2
3
4
0
1
2
3
4
1
2
3
4
x
−4 −3 −2 −1
If x ϭ 0,
then Ϫx ϭ Ϫ(0) ϭ 0.
x
−4 −3 −2 −1
0
Figure 1.3
From this discussion we also need to recognize the following general property.
Property 1.1
If a is any integer, then
Ϫ(Ϫa) ϭ a
(The opposite of the opposite of any integer is the integer itself.)
Addition of Integers
The number line is also a convenient visual aid for interpreting the addition of integers. In
Figure 1.4 we see number line interpretations for the following examples.
Problem
Number line interpretation
3
3ϩ2
3 ϩ (Ϫ2)
Ϫ3 ϩ (Ϫ2)
Ϫ3 ϩ 2 ϭ Ϫ1
−3
−5 − 4 −3 −2 −1 0 1 2 3 4 5
Figure 1.4
3 ϩ (Ϫ2) ϭ 1
−3
−5 − 4 −3 −2 −1 0 1 2 3 4 5
−2
3ϩ2ϭ5
−2
−5 − 4 −3 −2 −1 0 1 2 3 4 5
2
Ϫ3 ϩ 2
2
−5 − 4 −3 −2 −1 0 1 2 3 4 5
3
Sum
Ϫ3 ϩ (Ϫ2) ϭ Ϫ5
16
Chapter 1 • Some Basic Concepts of Arithmetic and Algebra
Once you acquire a feeling of movement on the number line, a mental image of this movement is sufficient. Consider the following addition problems, and mentally picture the number line interpretation. Be sure that you agree with all of our answers.
5 ϩ (Ϫ2) ϭ 3
Ϫ6 ϩ 4 ϭ Ϫ2
Ϫ8 ϩ 11 ϭ 3
Ϫ7 ϩ (Ϫ4) ϭ Ϫ11
Ϫ5 ϩ 9 ϭ 4
9 ϩ (Ϫ2) ϭ 7
14 ϩ (Ϫ17) ϭ Ϫ3
0 ϩ (Ϫ4) ϭ Ϫ4
6 ϩ (Ϫ6) ϭ 0
The last example illustrates a general property that should be noted: Any integer plus its
opposite equals zero.
Remark: Profits and losses pertaining to investments also provide a good physical model for
interpreting addition of integers. A loss of $25 on one investment along with a profit of $60 on a
second investment produces an overall profit of $35. This can be expressed as Ϫ25 ϩ 60 ϭ 35.
Perhaps it would be helpful for you to check the previous examples using a profit and loss interpretation.
Even though all problems involving the addition of integers could be done by using the
number line interpretation, it is sometimes convenient to give a more precise description
of the addition process. For this purpose we need to briefly consider the concept of
absolute value. The absolute value of a number is the distance between the number and
0 on the number line. For example, the absolute value of 6 is 6. The absolute value of Ϫ6
is also 6. The absolute value of 0 is 0. Symbolically, absolute value is denoted with vertical bars. Thus we write
͉6͉ ϭ 6
͉Ϫ6͉ ϭ 6
͉0͉ ϭ 0
Notice that the absolute value of a positive number is the number itself, but the absolute value
of a negative number is its opposite. Thus the absolute value of any number except 0 is positive, and the absolute value of 0 is 0.
We can describe precisely the addition of integers by using the concept of absolute
value as follows.
Two Positive Integers
The sum of two positive integers is the sum of their absolute values. (The sum of two
positive integers is a positive integer.)
43 ϩ 54 ϭ ͉43͉ ϩ ͉54͉ ϭ 43 ϩ 54 ϭ 97
Two Negative Integers
The sum of two negative integers is the opposite of the sum of their absolute values.
(The sum of two negative integers is a negative integer.)
(Ϫ67) ϩ (Ϫ93) ϭ Ϫ(͉Ϫ67͉ ϩ ͉Ϫ93͉)
ϭ Ϫ(67 ϩ 93)
ϭ Ϫ160
One Positive and One Negative Integer
We can find the sum of a positive and a negative integer by subtracting the smaller
absolute value from the larger absolute value and giving the result the sign of the original number that has the larger absolute value. If the integers have the same absolute
value, then their sum is 0.
1.3 • Integers: Addition and Subtraction
17
82 ϩ (Ϫ40) ϭ ͉82͉ Ϫ ͉Ϫ40͉ ϭ 82 Ϫ 40 ϭ 42
74 ϩ (Ϫ90) ϭ Ϫ(͉Ϫ90͉ Ϫ ͉74͉)
ϭ Ϫ(90 Ϫ 74) ϭ Ϫ16
(Ϫ17) ϩ 17 ϭ ͉Ϫ17͉ Ϫ ͉17͉
ϭ 17 Ϫ 17 ϭ 0
Zero and Another Integer
The sum of 0 and any integer is the integer itself.
0 ϩ (Ϫ46) ϭ Ϫ46
72 ϩ 0 ϭ 72
The following examples further demonstrate how to add integers. Be sure that you agree
with each of the results.
Ϫ18 ϩ (Ϫ56) ϭ Ϫ(0 Ϫ18 0 ϩ 0 Ϫ56 0 ) ϭ Ϫ(18 ϩ 56) ϭ Ϫ74
Ϫ71 ϩ (Ϫ32) ϭ Ϫ( 0 Ϫ71 0 ϩ 0 Ϫ32 0 ) ϭ Ϫ(71 ϩ 32) ϭ Ϫ103
64 ϩ (Ϫ49) ϭ 0 64 0 Ϫ 0 Ϫ49 0 ϭ 64 Ϫ 49 ϭ 15
Ϫ56 ϩ 93 ϭ 0 93 0 Ϫ 0 Ϫ56 0 ϭ 93 Ϫ 56 ϭ 37
Ϫ114 ϩ 48 ϭ Ϫ( 0 Ϫ114 0 Ϫ 0 48 0 ) ϭ Ϫ(114 Ϫ 48) ϭ Ϫ66
45 ϩ (Ϫ73) ϭ Ϫ(0 Ϫ73 0 Ϫ 0 45 0 ) ϭ Ϫ(73 Ϫ 45) ϭ Ϫ28
46 ϩ (Ϫ46) ϭ 0 Ϫ48 ϩ 0 ϭ Ϫ48
(Ϫ73) ϩ 73 ϭ 0 0 ϩ (Ϫ81) ϭ Ϫ81
It is true that this absolute value approach does precisely describe the process of adding integers,
but don’t forget about the number line interpretation. Included in the next problem set are other
physical models for interpreting the addition of integers. You may find these models helpful.
Subtraction of Integers
The following examples illustrate a relationship between addition and subtraction of whole
numbers.
7 Ϫ 2 ϭ 5 because 2 ϩ 5 ϭ 7
9 Ϫ 6 ϭ 3 because 6 ϩ 3 ϭ 9
5 Ϫ 1 ϭ 4 because 1 ϩ 4 ϭ 5
This same relationship between addition and subtraction holds for all integers.
5 Ϫ 6 ϭ Ϫ1 because 6 ϩ (Ϫ1) ϭ 5
Ϫ4 Ϫ 9 ϭ Ϫ13 because 9 ϩ (Ϫ13) ϭ Ϫ4
Ϫ3 Ϫ (Ϫ7) ϭ 4
8 Ϫ (Ϫ3) ϭ 11
because Ϫ7 ϩ 4 ϭ Ϫ3
because Ϫ3 ϩ 11 ϭ 8
Now consider a further observation:
5 Ϫ 6 ϭ Ϫ1
and
Ϫ4 Ϫ 9 ϭ Ϫ13 and
Ϫ3 Ϫ (Ϫ7) ϭ 4
8 Ϫ (Ϫ3) ϭ 11
and
and
5 ϩ (Ϫ6) ϭ Ϫ1
Ϫ4 ϩ (Ϫ9) ϭ Ϫ13
Ϫ3 ϩ 7 ϭ 4
8 ϩ 3 ϭ 11
18
Chapter 1 • Some Basic Concepts of Arithmetic and Algebra
The previous examples help us realize that we can state the subtraction of integers in terms
of the addition of integers. A general description for the subtraction of integers follows.
Subtraction of Integers
If a and b are integers, then a Ϫ b ϭ a ϩ (Ϫb).
It may be helpful for you to read a Ϫ b ϭ a ϩ (Ϫb) as “a minus b is equal to a plus the
opposite of b.” Every subtraction problem can be changed to an equivalent addition problem
as illustrated by the following examples.
6 Ϫ 13 ϭ 6 ϩ (Ϫ13) ϭ Ϫ7
9 Ϫ (Ϫ12) ϭ 9 ϩ 12 ϭ 21
Ϫ8 Ϫ 13 ϭ Ϫ8 ϩ (Ϫ13) ϭ Ϫ21
Ϫ7 Ϫ (Ϫ8) ϭ Ϫ7 ϩ 8 ϭ 1
It should be apparent that the addition of integers is a key operation. The ability to effectively
add integers is a necessary skill for further algebraic work.
Evaluating Algebraic Expressions
Let’s conclude this section by evaluating some algebraic expressions using negative and positive integers.
Classroom Example
Evaluate each algebraic expression
for the given values of the variables.
(a) m Ϫ n for m ϭ Ϫ10 and n ϭ 23
(b) Ϫ x ϩ y for x ϭ Ϫ11 and y ϭ Ϫ2
(c) Ϫc Ϫ d for c ϭ Ϫ57 and d ϭ Ϫ4
EXAMPLE 1
Evaluate each algebraic expression for the given values of the variables.
(a) x Ϫ y for x ϭ Ϫ12 and y ϭ 20
(b) Ϫa ϩ b for a ϭ Ϫ8 and b ϭ Ϫ6
(c) Ϫx Ϫ y for x ϭ 14 and y ϭ Ϫ7
Solution
(a) x Ϫ y ϭ Ϫ12 Ϫ 20 when x ϭ Ϫ12 and y ϭ 20
ϭ Ϫ12 ϩ (Ϫ20)
ϭ Ϫ32
Change to addition
(b) Ϫa ϩ b ϭ Ϫ(Ϫ8) ϩ (Ϫ6) when a ϭ Ϫ8 and b ϭ Ϫ6
ϭ 8 ϩ (Ϫ6)
ϭ2
Note the use of parentheses when substituting the values
(c) Ϫx Ϫ y ϭ Ϫ(14) Ϫ (Ϫ7) when x ϭ 14 and y ϭ Ϫ7
ϭ Ϫ14 ϩ 7
ϭ Ϫ7
Change to addition
Concept Quiz 1.3
For Problems 1– 4, match the letters of the description with the set of numbers.
1.
2.
3.
4.
{…, Ϫ3, Ϫ2, Ϫ1}
{1, 2, 3,…}
{0, 1, 2, 3,…}
{…, Ϫ3, Ϫ2, Ϫ1, 0}
A.
B.
C.
D.
Positive integers
Negative integers
Nonnegative integers
Nonpositive integers
1.3 • Integers: Addition and Subtraction
19
For Problems 5–10, answer true or false.
5. The number zero is considered to be a positive integer.
6. The number zero is considered to be a negative integer.
7. The absolute value of a number is the distance between the number and one on the
number line.
8. The ͉Ϫ4͉ is Ϫ4.
9. The opposite of Ϫ5 is 5.
10. a minus b is equivalent to a plus the opposite of b.
Problem Set 1.3
For Problems 1–10, use the number line interpretation to
find each sum. (Objective 2)
1. 5 ϩ (Ϫ3)
2. 7 ϩ (Ϫ4)
3. Ϫ6 ϩ 2
4. Ϫ9 ϩ 4
5. Ϫ3 ϩ (Ϫ4)
6. Ϫ5 ϩ (Ϫ6)
7. 8 ϩ (Ϫ2)
8. 12 ϩ (Ϫ7)
9. 5 ϩ (Ϫ11)
10. 4 ϩ (Ϫ13)
For Problems 51– 66, add or subtract as indicated. (Objective 2)
51. 6 Ϫ 8 Ϫ 9
52. 5 Ϫ 9 Ϫ 4
53. Ϫ4 Ϫ (Ϫ6) ϩ 5 Ϫ 8
54. Ϫ3 Ϫ 8 ϩ 9 Ϫ (Ϫ6)
55. 5 ϩ 7 Ϫ 8 Ϫ 12
56. Ϫ7 ϩ 9 Ϫ 4 Ϫ 12
57. Ϫ6 Ϫ 4 Ϫ (Ϫ2) ϩ (Ϫ5)
58. Ϫ8 Ϫ 11 Ϫ (Ϫ6) ϩ (Ϫ4)
59. Ϫ6 Ϫ 5 Ϫ 9 Ϫ 8 Ϫ 7
For Problems 11–30, find each sum. (Objective 2)
60. Ϫ4 Ϫ 3 Ϫ 7 Ϫ 8 Ϫ 6
11. 17 ϩ (Ϫ9)
12. 16 ϩ (Ϫ5)
61. 7 Ϫ 12 ϩ 14 Ϫ 15 Ϫ 9
13. 8 ϩ (Ϫ19)
14. 9 ϩ (Ϫ14)
62. 8 Ϫ 13 ϩ 17 Ϫ 15 Ϫ 19
15. Ϫ7 ϩ (Ϫ8)
16. Ϫ6 ϩ (Ϫ9)
63. Ϫ11 Ϫ (Ϫ14) ϩ (Ϫ17) Ϫ 18
17. Ϫ15 ϩ 8
18. Ϫ22 ϩ 14
64. Ϫ15 ϩ 20 Ϫ 14 Ϫ 18 ϩ 9
19. Ϫ13 ϩ (Ϫ18)
20. Ϫ15 ϩ (Ϫ19)
65. 16 Ϫ 21 ϩ (Ϫ15) Ϫ (Ϫ22)
21. Ϫ27 ϩ 8
22. Ϫ29 ϩ 12
23. 32 ϩ (Ϫ23)
24. 27 ϩ (Ϫ14)
66. 17 Ϫ 23 Ϫ 14 Ϫ (Ϫ18)
25. Ϫ25 ϩ (Ϫ36)
26. Ϫ34 ϩ (Ϫ49)
27. 54 ϩ (Ϫ72)
28. 48 ϩ (Ϫ76)
29. Ϫ34 ϩ (Ϫ58)
30. Ϫ27 ϩ (Ϫ36)
For Problems 31– 50, subtract as indicated. (Objective 2)
31. 3 Ϫ 8
32. 5 Ϫ 11
33. Ϫ4 Ϫ 9
34. Ϫ7 Ϫ 8
35. 5 Ϫ (Ϫ7)
36. 9 Ϫ (Ϫ4)
37. Ϫ6 Ϫ (Ϫ12)
38. Ϫ7 Ϫ (Ϫ15)
39. Ϫ11 Ϫ (Ϫ10)
40. Ϫ14 Ϫ (Ϫ19)
41. Ϫ18 Ϫ 27
The horizontal format is used extensively in algebra, but
occasionally the vertical format shows up. Some exposure to
the vertical format is therefore needed. Find the following
sums for Problems 67–78. (Objective 2)
67.
5
Ϫ9
68.
8
Ϫ13
69. Ϫ13
Ϫ18
70. Ϫ14
Ϫ28
71. Ϫ18
9
72. Ϫ17
9
42. Ϫ16 Ϫ 25
73. Ϫ21
39
74. Ϫ15
32
43. 34 Ϫ 63
44. 25 Ϫ 58
75.
76.
45. 45 Ϫ 18
46. 52 Ϫ 38
27
Ϫ19
31
Ϫ18
47. Ϫ21 Ϫ 44
48. Ϫ26 Ϫ 54
50. Ϫ76 Ϫ (Ϫ39)
77. Ϫ53
24
78.
49. Ϫ53 Ϫ (Ϫ24)
47
Ϫ28
20
Chapter 1 • Some Basic Concepts of Arithmetic and Algebra
For Problems 79 – 90, do the subtraction problems in vertical
format. (Objective 2)
79. 5
12
80.
81.
6
Ϫ9
82. 13
Ϫ7
83. Ϫ7
Ϫ8
84. Ϫ6
Ϫ5
85.
86.
17
Ϫ19
8
19
18
Ϫ14
87. Ϫ23
16
88. Ϫ27
15
89. Ϫ12
12
90. Ϫ13
Ϫ13
For Problems 91–100, evaluate each algebraic expression
for the given values of the variables. (Objective 3)
91. x Ϫ y for x ϭ Ϫ6 and y ϭ Ϫ13
92. Ϫx Ϫ y for x ϭ Ϫ7 and y ϭ Ϫ9
93. Ϫx ϩ y Ϫ z for x ϭ 3, y ϭ Ϫ4, and z ϭ Ϫ6
94. x Ϫ y ϩ z for x ϭ 5, y ϭ 6, and z ϭ Ϫ9
95. Ϫx Ϫ y Ϫ z for x ϭ Ϫ2, y ϭ 3, and z ϭ Ϫ11
96. Ϫx Ϫ y ϩ z for x ϭ Ϫ8, y ϭ Ϫ7, and z ϭ Ϫ14
97. Ϫx ϩ y ϩ z for x ϭ Ϫ11, y ϭ 7, and z ϭ Ϫ9
98. Ϫx Ϫ y Ϫ z for x ϭ 12, y ϭ Ϫ6, and z ϭ Ϫ14
99. x Ϫ y Ϫ z for x ϭ Ϫ15, y ϭ 12, and z ϭ Ϫ10
100. x ϩ y Ϫ z for x ϭ Ϫ18, y ϭ 13, and z ϭ 8
A game such as football can be used to interpret addition
of integers. A gain of 3 yards on one play followed by a
loss of 5 yards on the next play places the ball 2 yards
behind the initial line of scrimmage; this could be
expressed as 3 ϩ (Ϫ5) ϭ Ϫ2. Use this football interpretation to find the following sums for Problems 101–110.
(Objective 4)
101. 4 ϩ (Ϫ7)
102. 3 ϩ (Ϫ5)
103. Ϫ4 ϩ (Ϫ6)
104. Ϫ2 ϩ (Ϫ5)
105. Ϫ5 ϩ 2
106. Ϫ10 ϩ 6
107. Ϫ4 ϩ 15
108. Ϫ3 ϩ 22
109. Ϫ12 ϩ 17
110. Ϫ9 ϩ 21
For Problems 111–120, refer to the Remark on page 16 and
use the profit and loss interpretation for the addition of
integers. (Objective 4)
111. 60 ϩ (Ϫ125)
112. 50 ϩ (Ϫ85)
113. Ϫ55 ϩ (Ϫ45)
114. Ϫ120 ϩ (Ϫ220)
115. Ϫ70 ϩ 45
116. Ϫ125 ϩ 45
117. Ϫ120 ϩ 250
118. Ϫ75 ϩ 165
119. 145 ϩ (Ϫ65)
120. 275 ϩ (Ϫ195)
121. The temperature at 5 A.M. was Ϫ17ЊF.
By noon the temperature had increased by 14ЊF. Use the addition of
integers to describe this situation and
to determine the temperature at noon
(see Figure 1.5).
120°
100°
80°
60°
40°
20°
0°
−20°
− 40°
122. The temperature at 6 P.M. was Ϫ6ЊF,
and by 11 P.M. the temperature had
dropped 5ЊF. Use the subtraction of
integers to describe this situation and
to determine the temperature at 11 P.M.
Figure 1.5
(see Figure 1.5).
123. Megan shot rounds of 3 over par, 2 under par, 3 under
par, and 5 under par for a four-day golf tournament.
Use the addition of integers to describe this situation
and to determine how much over or under par she was
for the tournament.
124. The annual report of a company contained the following figures: a loss of $615,000 for 2007, a loss of
$275,000 for 2008, a loss of $70,000 for 2009, and a
profit of $115,000 for 2010. Use the addition of integers to describe this situation and to determine the company’s total loss or profit for the four-year period.
125. Suppose that during a five-day period, a share of Dell’s
stock recorded the following gains and losses:
Monday
lost $2
Tuesday
gained $1
Thursday
gained $1
Friday
lost $2
Wednesday
gained $3
Use the addition of integers to describe this situation
and to determine the amount of gain or loss for the fiveday period.
126. The Dead Sea is approximately thirteen hundred
eighty-five feet below sea level. Suppose that you
are standing eight hundred five feet above the Dead
Sea. Use the addition of integers to describe this situation and to determine your elevation.
127. Use your calculator to check your answers for
Problems 51– 66.
1.4 • Integers: Multiplication and Division
21
Thoughts Into Words
128. The statement Ϫ6 Ϫ (Ϫ2) ϭ Ϫ6 ϩ 2 ϭ Ϫ4 can be
read as “negative six minus negative two equals negative six plus two, which equals negative four.” Express
in words each of the following.
(a)
(b)
(c)
(d)
129. The algebraic expression Ϫx Ϫ y can be read as “the
opposite of x minus y.” Express in words each of the
following.
(a) Ϫx ϩ y
(b) x Ϫ y
(c) Ϫx Ϫ y ϩ z
8 ϩ (Ϫ10) ϭ Ϫ2
Ϫ7 Ϫ 4 ϭ Ϫ7 ϩ (Ϫ4) ϭ Ϫ11
9 Ϫ (Ϫ12) ϭ 9 ϩ 12 ϭ 21
Ϫ5 ϩ (Ϫ6) ϭ Ϫ11
Answers to the Concept Quiz
1. B
2. A
3. C
4. D
5. False
1.4
6. False
7. False
8. False
9. True
10. True
Integers: Multiplication and Division
OBJECTIVES
1
Multiply and divide integers
2
Evaluate algebraic expressions involving the multiplication and division of integers
3
Apply the concepts of multiplying and dividing integers to model problems
Multiplication of whole numbers may be interpreted as repeated addition. For example, 3 и 4
means the sum of three 4s; thus, 3 и 4 ϭ 4 ϩ 4 ϩ 4 ϭ 12. Consider the following examples
that use the repeated addition idea to find the product of a positive integer and a negative integer:
3(Ϫ2) ϭ Ϫ2 ϩ (Ϫ2) ϩ (Ϫ2) ϭ Ϫ6
2(Ϫ4) ϭ Ϫ4 ϩ (Ϫ4) ϭ Ϫ8
4(Ϫ1) ϭ Ϫ1 ϩ (Ϫ1) ϩ (Ϫ1) ϩ (Ϫ1) ϭ Ϫ4
Note the use of parentheses to indicate multiplication. Sometimes both numbers are enclosed
in parentheses so that we have (3)(Ϫ2) .
When multiplying whole numbers, the order in which we multiply two factors does not
change the product: 2(3) ϭ 6 and 3(2) ϭ 6. Using this idea we can now handle a negative
number times a positive integer as follows:
(Ϫ2)(3) ϭ (3)(Ϫ2) ϭ (Ϫ2) ϩ (Ϫ2) ϩ (Ϫ2) ϭ Ϫ6
(Ϫ3)(2) ϭ (2)(Ϫ3) ϭ (Ϫ3) ϩ (Ϫ3) ϭ Ϫ6
(Ϫ4)(3) ϭ (3)(Ϫ4) ϭ (Ϫ4) ϩ (Ϫ4) ϩ (Ϫ4) ϭ Ϫ12
Finally, let’s consider the product of two negative integers. The following pattern helps
us with the reasoning for this situation:
4(Ϫ3) ϭ Ϫ12
3(Ϫ3) ϭ Ϫ9
2(Ϫ3) ϭ Ϫ6
1(Ϫ3) ϭ Ϫ3
0(Ϫ3) ϭ 0
(Ϫ1)(Ϫ3) ϭ ?
The product of 0 and any integer is 0
22
Chapter 1 • Some Basic Concepts of Arithmetic and Algebra
Certainly, to continue this pattern, the product of Ϫ1 and Ϫ3 has to be 3. In general, this type of
reasoning helps us to realize that the product of any two negative integers is a positive integer.
Using the concept of absolute value, these three facts precisely describe the multiplication of integers:
1. The product of two positive integers or two negative integers is the product of their
absolute values.
2. The product of a positive and a negative integer (either order) is the opposite of the
product of their absolute values.
3. The product of zero and any integer is zero.
The following are examples of the multiplication of integers:
(Ϫ5)(Ϫ2) ϭ 0 Ϫ5 0 и 0 Ϫ2 0 ϭ 5 и 2 ϭ 10
(7)(Ϫ6) ϭ Ϫ( 0 7 0 и 0 Ϫ6 0 ) ϭ Ϫ(7 и 6) ϭ Ϫ42
(Ϫ8)(9) ϭ Ϫ(0 Ϫ8 0 и 0 9 0 ) ϭ Ϫ(8 и 9) ϭ Ϫ72
(Ϫ14)(0) ϭ 0
(0)(Ϫ28) ϭ 0
These examples show a step-by-step process for multiplying integers. In reality, however,
the key issue is to remember whether the product is positive or negative. In other words, we
need to remember that the product of two positive integers or two negative integers is a
positive integer; and the product of a positive integer and a negative integer (in either
order) is a negative integer. Then we can avoid the step-by-step analysis and simply write
the results as follows:
(7)(Ϫ9) ϭ Ϫ63
(8)(7) ϭ 56
(Ϫ5)(Ϫ6) ϭ 30
(Ϫ4)(12) ϭ Ϫ48
Division of Integers
By looking back at our knowledge of whole numbers, we can get some guidance for our work
8
with integers. We know, for example, that ϭ 4, because 2 и 4 ϭ 8. In other words, we can
2
find the quotient of two whole numbers by looking at a related multiplication problem. In
the following examples we use this same link between multiplication and division to determine the quotients.
8
ϭ Ϫ4 because (Ϫ2)(Ϫ4) ϭ 8
Ϫ2
Ϫ10
ϭ Ϫ2 because (5)(Ϫ2) ϭ Ϫ10
5
Ϫ12
ϭ 3 because (Ϫ4)(3) ϭ Ϫ12
Ϫ4
0
ϭ 0 because (Ϫ6)(0) ϭ 0
Ϫ6
Ϫ9
0
0
0
is undefined because no number times 0 produces Ϫ9
is indeterminate because any number times 0 equals 0