Real Numbers and Their Properties
1-10
In general,
a
means “the opposite of
a
.” If
a
is positive,
a
is negative. If
a
is
negative,
a
is positive. Opposites have the following property.
Opposite of an Opposite
For any real number
a
,
(
a
)
a
.
Absolute Value
Remember that we have defined
a
to be the distance between 0 and
a
on the
number line. Using opposites, we can give a symbolic definition of absolute value.
a
if
a
is positive or zero
if
a
is negative
a
a
E X A M P L E
7
Warm-Ups
▼
Fill in the blank.
1.
The set of
is {. . . ,
3,
2,
1, 0, 1, 2, 3, . . .}.
2.
The set of
numbers is {1, 2, 3, . . .}.
3.
Every
number can be expressed as a ratio of
integers.
4.
and
decimal numbers are
rational numbers.
5.
A decimal number that does not repeat and does not
terminate is
.
6.
The rationals together with the irrationals form the set of
numbers.
7.
The ratio of the
and diameter of any
circle is
.
8.
The
of a number is its distance form 0
on a number line.
True or false?
9.
The natural numbers and the counting numbers are
the same.
10.
Zero is a counting number.
11.
Zero is an irrational number.
12.
The opposite of negative 3 is positive 3.
13.
The absolute value of 4 is
4.
14.
The real number
is in the interval (3, 4).
15.
The interval (4, 9) contains 8.
16.
The interval (2, 6) contains 6.
17.
The interval [3, 5] contains 3.
18.
The interval (9,
) contains 88 trillion.
Using the symbolic definition of absolute value
Evaluate.
a)
8
b)
0
c)
8
Solution
a)
From the definition,
a
a
if
a
is positive. Since 8 is positive, we replace
a
with 8
to get 8
8.
b)
From the definition,
a
a
if
a
is zero. Replacing
a
with 0, we get
0
0.
c)
From the definition,
a
a
if
a
is negative. Since
8 is negative, we replace
a
with
8 to get
8
(
8)
8.
Now do Exercises 55–60

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Exercises
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•
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1.1
U
3
V
The Number Line
Determine which number is the larger in each given pair of
numbers. See Example 1.
1.
0, 6
2.
7, 4
3.
3, 6
4.
7,
10
5.
0,
6
6.
8, 0
7.
3,
2
8.
5,
8
9.
12,
15
10.
13,
7
11.
2.9,
2.1
12.
2.1, 2.9
List the numbers described and graph them on a number line.
See Example 2.
13.
The counting numbers smaller than 6
14.
The natural numbers larger than 4
15.
The whole numbers smaller than 5
16.
The integers between
3 and 3
17.
The whole numbers between
5 and 5
18.
The integers smaller than
1
19.
The counting numbers larger than
4
20.
The natural numbers between
5 and 7
21.
The integers larger than
1
2
22.
The whole numbers smaller than
7
4
U
4
V
The Real Numbers
Determine whether each statement is true or false. Explain your
answer. See Example 3.
23.
Every integer is a rational number.
24.
Every counting number is a whole number.
25.
Zero is a counting number.