Postulate: The Ruler Postulate

The points on any line can be paired with the real numbers in such a way that:

   1. Any two chosen points can be paired with 0 and 1.
   2. The distance between any two points in a number line is the absolute. value of the difference of the real numbers corresponding to the points.

By virtue of the Ruler Postulate, a system to determine the length of a segment, which is equal to the distance between its endpoints, can be formulated. Every point on a number line can be paired to a real number; it is called the coordinate of the point. The length of a segment is found by determining the difference between the coordinates of its endpoints and taking the absolute value.

Ex. 

The coordinate of
A = -4.
B = -2
O = 0
C = 3
P = 5

The length of segment BC is determined as follows:

B has coordinate -2 , C has coordinate 3
-2 - ( 3 ) = -5
| -5 | = 5 \ the length of segment BC is 5.