Section 1.1 Real Numbers
Types of Real Numbers
1.
Natural numbers
(
N
):
1
,
2
,
3
,
4
,
5
, . . .
2.
Integer numbers
(
Z
):
0
,
±
1
,
±
2
,
±
3
,
±
4
,
±
5
, . . .
REMARK: Any natural number is an integer number, but not any integer number is a natural
number.
3.
Rational numbers
(
Q
):
r
=
m
n
,
where
m
∈
Z
, n
∈
N
EXAMPLES:
1
2
7
3

11
53
2 =
2
1
0
.
2 =
2
10
=
1
5
0
.
222
. . .
= 0
.
2 =
2
9
0
.
999
. . .
= 0
.
9 = 1
REMARK: Any integer number is a rational number, but not any rational number is an integer.
4.
Irrational Numbers
. These are numbers that cannot be expressed as a ratio of integers.
EXAMPLES:
√
2
√
3
√
2 +
√
3
1 +
√
5
2

3
√
2
π
√
π
2
√
2
The set of all real numbers is denoted by
R
.
The real numbers can be represented by points on
a line which is called a
coordinate line
, or a
real number line
, or simply a
real line
:
1
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Every real number has a decimal representation. If the number is rational, then its correspond
ing decimal is repeating. For example,
1
2
= 0
.
500
. . .
= 0
.
5
0 = 0
.
4
9
,
2
3
= 0
.
66666
. . .
= 0
.
6
157
495
= 0
.
3171717
. . .
= 0
.
3
17
,
9
7
= 1
.
285714285714
. . .
= 1
.
285714
If the number is irrational, the decimal representation is nonrepeating:
√
2 = 1
.
414213562373095
. . .
π
= 3
.
141592653589793
. . .
Operations on Real Numbers
Real numbers can be combined using the familiar operations of addition, subtraction, multi
plication, and division. When evaluating arithmetic expressions that contain several of these
operations, we use the following conventions to determine the order in which the operations are
performed:
1. Perform operations inside parentheses first, beginning with the innermost pair. In dividing
two expressions, the numerator and denominator of the quotient are treated as if they are
within parentheses.
2. Perform all multiplications and divisions, working from left to right.
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 Spring '12
 KIRYLTSISHCHANKA
 Calculus, Algebra, Real Numbers, Natural Numbers, 2k, 4k, π, 2 1 m

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