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 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 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,
 d(a, b) = | a − 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) = | x1x2 |
= xM x1,  since x1 < xM                                                   = x2x1 since x1 < x2
by plugging into above equation,     xM x1 = ½ · ( x2x1)
 and solving for xM gives: xM  = ½ · ( x1+ x2) - the midpoint formula for the real line
Beginning Algebra Contents A