
The
Set of Real Numbers  The Real Number System 


Interval definition and notation 
Distance
and absolute value 
Distance between two numbers 
Properties of absolute value 
Midpoint formula for the real
number line 





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
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
x_{1}and
x_{2}
are given numbers and x_{1}<
x_{2},
and let x_{M}
be the midpoint. Therefore, 
x_{1}<
x_{M}
< x_{2
}and d(x_{1},
x_{M})
= ½
·
d(x_{1},
x_{2}). 
Then,
d(x_{1},
x_{M})
=  x_{1}−
x_{M
}
and d(x_{1},
x_{2})
=  x_{1}− x_{2
} 
= x_{M
}−
x_{1},
since x_{1
}
< x_{M
}= x_{2}− x_{1},
since x_{1
}
< x_{2} 
by
plugging into above equation, x_{M
}−
x_{1}
= ½
· ( x_{2}− x_{1}) 
and
solving for x_{M
}gives: 
x_{M }
= ½
· ( x_{1}+
x_{2}) 

the midpoint formula for the real line 









Beginning
Algebra Contents A 



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