| |
| The
Set of Real Numbers - The Real Number System |
| |
|
|
| |
|
Sets
of
Numbers: |
Natural numbers |
|
Whole numbers |
|
Integers |
|
Rational numbers |
|
Irrational numbers |
|
Real numbers |
|
The
real number line and relations |
|
One-dimensional coordinate system |
|
Unit
interval |
|
Relations,
less than and greater than |
|
Interval definition and notation |
|
Distance
and absolute value |
|
Distance between two numbers |
|
Properties of absolute value |
|
Midpoint formula for the real
number line |
|
|
|
|
|
|
| Sets
of
Numbers: |
| The set of
natural numbers |
|
N = {1, 2, 3, . . . , n,
n
+ 1, . . . }, the positive integers used for counting. |
|
| The
set of whole numbers |
|
N0
= {
0, 1, 2, 3, . .
. }, is just like the set of natural numbers except that it also
includes zero. |
|
| The set of
integers |
|
Z
= {
. . . , −3,
−2,
−1,
0, 1, 2, 3, . . . }, consists of all natural numbers, negative whole numbers and |
|
zero. This means that the set of natural numbers is a subset of
integers, i.e., N
is a subset
Z. |
|
| The set of
rational numbers |
|
Q
= {
a/b
| a,
b
Î Z,
b
is not
0 }, is the set of all proper and improper fractions. That is, a ratio or |
| quotient of two
integers a
and b, where
b
is not zero. |
| All integers are in this set since every integer a
can be expressed as the fraction a/1
= a.
Thus, the set of all natural numbers
N is proper subset of integers
Z
and the set of integers is proper subset of the set of rational
numbers, N is a subset
Z is a subset
Q.
Rational numbers can be represented as integers, fractions, terminating decimals and recurring or repeating decimals. |
| |
| The set of
irrational numbers, denoted I, is the set of numbers that
cannot be written as ratio of two |
| integers.
An irrational number expressed as a decimal never repeat or terminate. The irrational numbers are precisely those numbers whose decimal expansion never ends and never enters
a periodic pattern, such as 0.1020030004..., p,
Ö2, Ö3, or any root of any natural number
that is not a perfect root is an irrational number.
|
| |
| The set of
real numbers, denoted R,
|
R =
Q U I |
| is
the set of all rational and irrational numbers, R =
Q U I.
The real numbers or the reals are either rational or
irrational and are
intuitively defined as numbers that are in one-to-one correspondence
with the points on an infinite line, the number line. |
 |
|
|
|
The Real Number Line, and Relations |
|
|
 |
|
| The real number line is an infinite line on which points are taken to represent the real numbers by their |
|
distance from a fixed point labeled
O and called the
origin.
|
| We
use the variable x
to denote a one-dimensional coordinate system, in
this case the number line is |
|
called the x-axis.
|
| The line segment
OE
denotes
unit length, that is, | OE
|
= 1. The absolute value (or modulus) of a real |
|
number
x,
denoted | x | is its numerical value without regard to its sign.
|
| For example,
| + 5 | = 5 and | −
5 | = 5, and 0 is the only absolute value of 0.
The absolute value of a real |
|
number a
is its distance from the origin.
|
|
|
| The
unit interval is the interval
[ 0, 1 ] that is the set of all real
numbers x such that
0 ≤
x ≤
1, we say x
is
|
|
greater than or equal to zero
and x
is less than or equal to one, meaning x
is between 0 and 1 including the
|
|
endpoints.
|
|
|
| A rational number
-
a/b,
in the above picture,
corresponds to the point which is symmetrical regarding the |
|
origin to the point
A, which corresponds to the rational number
a/b.
|
| Therefore,
| OA | = |
OA' |
= a/b
· | OE
|
= a/b.
|
| For
each real number x,
there is a unique real number, denoted −x,
such that x
+ ( − x
) = 0. In
other words, |
|
by adding a number to its negative or opposite, the
result is 0.
|
| Every point of the number line corresponds to one real number. |
|
|
| Relations,
less than and greater than |
| Let
a
and b
are distinct real numbers. We
say that a
is less than b
if a
− b
is a negative number, and write
|
|
a
< b, i.e.,
a
< b
means a
− b
is negative.
|
| On
the x-axis,
a
< b
is represented as the number a
lies to the left of b. |
| We
say that a
is greater than b
if a
− b
is a positive number, and write
a
> b, i.e.,
a
> b means a
− b
is |
|
positive. On
the x-axis,
a
> b
is represented as the number a
lies to the right of b.
|
| Similarly,
a
≤
b
denotes that a
is less than or equal to b,
and a
>
b,
a
is greater than or equal to b. |
|
|
Interval
Definition and Notation |
| An
interval is the set containing all real numbers (or points)
between two given real numbers, a
and
b,
where |
|
a
<
b.
|
| A
closed interval [ a,
b ]
includes the endpoints, a
and
b,
and it corresponds to a set notation |
|
{ x
| a
<
x
<
b },
while an open interval (a,
b) does
not include the endpoints.
|
| On
the real line, the half-closed (or half-opened) interval
from a
to b
is written [a,
b)
or (a,
b],
where square |
|
brackets indicate inclusion of the endpoint, while
round parentheses denote its exclusion.
|
| Thus,
[ a,
b ]
= { x
| a
<
x
<
b }
- a closed interval |
|
[ a,
b )
= { x
| a
<
x
<
b }
- an interval closed on left, open on right |
|
( a,
b ]
= { x
| a
< x
<
b }
- an interval open on left closed on right |
|
( a,
b )
= { x
| a
<
x
<
b }
- an open interval |
|
| Unbounded
intervals or intervals of infinite length are also written
in this notation, thus [ a,
oo
) is |
|
unbounded interval x
> a,
which is regarded as closed, while (a,
oo
) is the open interval x
> a
and where oo
denotes infinity.
|
|
Hence,
(
− oo
, a
]
= { x
| x
<
a }
and [ a,
oo )
= { x
| x
>
a },
|
|
(
− oo
, a
= { x
| x
< a
} and ( a,
oo )
= { x
| x >
a }.
|
|
The real line R
= (
−
oo
, oo
).
|
|
|
| Distance
and Absolute Value |
| The
distance between a number x
and 0 equals x
if x
>
0, and equals −
x
if x
<
0,
therefore the absolute |
|
value of a number x, denoted
| x |, is x
if x
>
0, and −
x
if x
<
0.
|
| |
| From
the definition of the absolute value it follows that the
distance between two numbers a
and b, |
|
|
| |
| Properties of absolute value |
Examples: |
| 1 |
 |
1 |
 |
| 2 |
 |
2 |
 |
| 3 |
 |
3 |
 |
| 4 |
 |
4 |
 |
| 5 |
 |
5 |
 |
| 6 |
 |
6 |
 |
| 7 |
 |
7 |
 |
| 8 |
 |
8 |
 |
|
|
| The
Midpoint Between Two Numbers |
| Suppose
x1and
x2
are given numbers and x1<
x2,
and let xM
be the midpoint. Therefore, |
|
x1<
xM
< x2
and d(x1,
xM)
= ½
·
d(x1,
x2). |
| Then,
d(x1,
xM)
= | x1−
xM
|
and
d(x1,
x2)
= | x1− x2
| |
|
= xM
−
x1,
since x1
< xM
= x2− x1,
since x1
< x2 |
| by
plugging into above equation, xM
−
x1
= ½
· ( x2− x1) |
| and
solving for xM
gives: |
xM
= ½
· ( x1+
x2) |
-
the midpoint formula for the real line |
|
|
|
|
|
|
|
|
|
|
|
|
| Beginning
Algebra Contents |
|
 |
|
| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
