
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 
Onedimensional coordinate system 
Unit
interval 
Relations,
less than and greater than 





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 
N_{0}
= {
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 onetoone 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 onedimensional coordinate system, in
this case the number line is called the xaxis. 
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 A'
which is symmetrical regarding the origin to the point
A, which
denotes 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 xaxis,
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 xaxis,
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. 









Beginning
Algebra Contents A 



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