

Irrational numbers 
Rationalizing
a denominator 
The
real number line, and relations 
Inequalities of real
numbers 
Relations,
less than and greater than 
Interval
definition and notation 
Closed and open intervals
(unbounded
intervals) 
Distance
and absolute value 
Properties of absolute
value 





Irrational numbers

Irrational numbers are numbers that can be written as
decimals but not as fractions.
Irrational numbers have decimal expansion that neither terminate
nor become periodic.

Any number on a number line that isn't
a rational number is irrational. For example, Ö2,
Ö3,
and Ö5
are irrational numbers because they can't be written as a ratio of two
integers. Only the square roots of square numbers are rational.

For any integers m
and n, ^{
}^{n}Öm
is irrational unless m
is the
nth
power of an integer. Or, any
root that is not a perfect root is an irrational number.


Rationalizing
a denominator 
Rationalizing a denominator is a method for changing an irrational denominator into a rational one. 

To
rationalize a denominator or numerator of the form 

multiply both numerator 

and denominator by a
conjugate, where 

are conjugates of each other. 


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 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. 

Inequalities of real
numbers 
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. 

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 halfclosed (or halfopened)
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. 
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 










Intermediate
algebra contents 



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