Download Chapter 2 Integers

Chapter 2
Integers
2.1 Order, Opposites, and Absolute Value
We now will expand our number system to include negative numbers. If you think about owing
people (or banks) money, losing yards in football, losing money in the stock market; these are all
examples of negative numbers. A good place to start is with the real number line:
Note that the whole numbers {0,1,2,3,…} have been extended in reverse order to {–1,–2,–3,…}.
This complete group of numbers {…–3,–2,–1,0,1,2,3,…} is called the integers. There are
additional numbers on this number line, which we will study later in Chapter 3. The number line
consists of numbers less than zero (called negative numbers), 0, and numbers greater than zero
(called positive numbers).
We first consider the concept of order on the number line. Recall that 5 < 7 refers to 5 being less
than 7. Similarly, 8 ≥ 4 means that 8 is greater than or equal to 4. Using the number line, we can
interpret < as meaning “to the left of” and > as “to the right of”. Looking at negative numbers,
that means –6 < –4, since –6 is to the left of –4. If you owe someone $6, you have less money
than if you owe them $4, so this interpretation is consistent with the meaning of negative
numbers. Here are some more examples:
symbol
meaning
example
<
less than
!8 < !5
"
less than or equal to
!6 " !6
>
greater than
2>–3
#
greater than or equal to !1 # !3
=
equal to
!5 = !5
Note that –6 ≤ –6, since the symbol is read as “less than or equal to”.
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Example 1
Solution
Determine the appropriate inequality symbols which will make each statement
true (more than one symbol may apply).
a.
–3 ____ –1
b.
–100 ____ –101
c.
–12 ____ –12
d.
–1,000 ____ –999
e.
–1 ____ 0
a.
Since –3 is to the left of –1 on a number line, both the < and ≤ symbols will
make the statement true.
b.
Since –100 is to the right of –101 on a number line, both the > and ≥
symbols will make the statement true.
c.
Since –12 = –12, the symbols =, ≤, and ≥ will make the statement true.
Remember that ≤ and ≥ both include the “or equal to” portion of the
inequality, which is why they produce a true statement.
d.
Since –1,000 is to the left of –999 on a number line, both the < and ≤
symbols will make the statement true.
e.
Since –1 is to the left of 0 on a number line, both the < and ≤ symbols will
make the statement true.
Note that the number line has symmetry about the number 0. That is, the number line to the left
of 0 is a mirror image to that of the right of 0. This idea is due to the fact that numbers on
opposite sides of 0, but the same distance from 0, are called opposites. Thus 2 and –2 are
opposites, since they lie on opposite sides of 0 but are the same distance from 0. We say that –2
is the opposite of 2, and that 2 is the opposite of –2. Using the symbol – for opposite:
“the opposite of 2 is –2”
“the opposite of –2 is 2”
–(2) = –2
–(–2) = 2
This can get a bit confusing. When we write –4, we mean the number “negative 4”. When we
write –(4), we mean the opposite of 4 (which, of course, is –4). When we write –(–5), we mean
the opposite of –5, which is 5. The next example should help to clarify this.
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Example 2
Write the simplified form of each opposite.
a.
b.
c.
d.
Solution
–(15)
–(–99)
–(–140)
–(0)
a.
We are asked to find the opposite of 15, which is –15.
Thus –(15) = –15.
b.
We are asked to find the opposite of –99, which is 99.
Thus –(–99) = 99.
c.
We are asked to find the opposite of –140, which is 140.
Thus –(–140) = 140.
d.
We are asked to find the opposite of 0, which is still 0.
Thus –(0) = 0.
Another way to look at the symmetry of the number line is with the concept of absolute value.
The absolute value of a number refers to its unsigned distance from 0. That is, since –4 is 4
unsigned units away from 0, its absolute value is 4. Symbolically, we write:
“the absolute value of –4 is 4”
!4 =4
“the absolute value of 3 is 3”
3 =3
Example 3
Solution
Write the simplified form of each expression.
a.
!8
b.
5
c.
! 4
d.
! ! 12
a.
We are asked to find the absolute value of –8, which is 8.
So ! 8 = 8 .
b.
We are asked to find the absolute value of 5, which is 5.
So 5 = 5 .
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c.
This question is asking us to find the opposite of the absolute value of 4.
Since 4 = 4 , its opposite must be ! 4 = !4 .
d.
This question is asking us to find the opposite of the absolute value of –12.
Since ! 12 = 12 , its opposite must be ! ! 12 = !12 .
Note that there is a difference between opposites and absolute values. The opposite of a number
has a different sign (unless the number is 0), whereas the absolute value of a number is always 0.
The final example in this section involves computation with absolute value and opposite
symbols. After finding the indicated absolute value or opposite, these are just computations with
whole numbers.
Example 4
Solution
Compute the value of the following expressions.
a.
! 13 ! !5
a.
b.
!7 • 4
c.
! 24 ÷ !6
d.
5 !6 +4 3
First note that ! 13 = 13 and ! 5 = 5 . Thus the expression computes as:
! 13 ! !5 = 13 ! 5
=8
b.
First note that ! 7 = 7 and 4 = 4 . Thus the expression computes as:
!7 • 4 = 7•4
= 28
c.
First note that ! 24 = 24 and ! 6 = 6 . Thus the expression computes as:
! 24 ÷ !6 = 24 ÷ 6
=4
d.
First note that ! 6 = 6 and 3 = 3 . Thus the expression computes as:
5 !6 +4 3 = 5•6+4•3
= 30 + 12
= 42
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Terminology
integers
positive numbers
absolute value
negative numbers
opposite
Exercise Set 2.1
Determine the appropriate inequality symbols which will make each statement true (more than
one symbol may apply).
1.
3.
5.
7.
9.
11.
–5 ____ –4
–15 ____ –15
–1,000 ____ –999
–259,989 ____ –260,011
–5 ____ 0
0 ____ –1
2.
4.
6.
8.
10.
12.
–8 ____ –7
–9 ____ –9
–5,999,999 ____ –6,000,000
–198,999 ____ –189,999
0 ____ –3
–8 ____ 0
Write the simplified form for each opposite.
13. –(9)
15. –(–6)
17. –(0)
14. –(15)
16. –(–4)
18. –(–0)
Write the simplified form for each absolute value.
19.
6
20.
9
21.
!7
22.
!9
23. ! 12
24. ! 23
25. ! ! 18
26. ! ! 3
27. ! ( ! ! 7
)
28. ! ( ! ! 16
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)
Write the simplified form for each expression. If a computation is involved, perform the
computation and simplify your answer.
29. ! ! 9
31. ! ( !9 )
30. ! ! 13
32. ! ( !13)
33.
! 24
34.
! 42
37.
!6 + !9
38.
! 8 + ! 13
39.
! 15 ! 6
40.
! 26 ! 4
41.
! 5 • ! 12
42.
! 20 • ! 8
35. ! ( 7 )
36. ! ( 42 )
43. ! ( !15 ) • ! 4
45.
44. ! ( !9 ) • 7
! 40 ÷ ! 2
46.
60 ÷ ! 4
47. 3 • ! 5 + 8 • ! 4
48. 6 • ! 7 + 5 • ! 9
49. 8 • ! 9 ! 4 • ! 6
50. 12 • ! 5 ! 3 • 7
For the following statements, find all integers x which will make the statement true.
51. ! ( x ) = 6
53. ! ( x ) = !7
52. ! ( x ) = 8
54. ! ( x ) = !9
55.
x =5
56.
x = 10
57.
x = !3
58.
x = !9
59. ! x = !13
60. ! x = !19
61. ! x = 13
62. ! x = 19
Determine whether each statement is true or false. If it is false, give a specific example which
shows it is false. If it is true, explain why.
63. For every integer x, x > 0 .
64. For every integer x, x ! 0 .
65. There are no integers x such that x < 0 .
66. There are no integers x such that x ! 0 .
67. For every integer x, ! ( x ) " 0 .
68. For every integer x, ! ( x ) < 0 .
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69. If x and y are opposites, then x = y .
70. If x = y , then x and y are opposites.
71. If x < y , then x < y .
72. If x > y , then x > y .
73. If x < y , then x < y .
74. If x > y , then x > y .
75. For every integer x, ! x = x .
76. For every integer x, ! x = ! x .
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