An introduction to mathematics, by A. N. Whitehead.

74 INTRODUCTION TO MATHEMATICS
represented by points on a line. Let OX be
1 3 5 7
0 2 1 2 5 3 7 4
OM A N B P C Q D X
any line bounded at O and stretching away
indefinitely in the direction OX. Take any
convenient point, A, on it, so that OA represents the unit length; and divide off lengths
AB, BC, CD, and so on, each equal to OA.
Then the point O represents the number 0, A
the number 1, B the number 2, and so on.
In fact the number represented by any point
is the measure of its distance from 0, in
terms of the unit length OA. The points
between O and A represent the proper fractions and the incommensurable numbers less
than 1; the middle point of OA represents -,
3
that of AB represents 2, that of BC represents
and so on. In this way every point on
OX represents some one real number, and
every real number is represented by some
one point on OX.
The series (or row) of points along OX,
starting from O and moving regularly in the
direction from O to X, represents the real
numbers as arranged in an ascending order