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

84 INTRODUCTION TO MATHEMATICS
multiplied and otherwise combined. We have
explained above that positive and negative
numbers are operations. They have also
been called "steps." Thus +3 is the step
by which we go from 2 to 5, and -3 is the
step backwards by which we go from 5 to 2.
Consider the line OX divided in the way explained in the earlier part of the chapter, so
that its points represent numbers. Then +2
D' C' B' A' +1 +2 +3
-3 -2 -1 0 A B C D E
is the step from O to B, or from A to C, or
(if the divisions are taken backwards along
OX') from C' to A', or from D' to B', and so
on. Similarly -2 is the step from O to B',
or from B' to D', or from B to 0, or from C
to A.
We may consider the point which is reached
by a step from O, as representative of that
step. Thus A represents +1, B represents
+2, A' represents -1, B' represents -2, and
so on. It will be noted that, whereas previously with the mere "unsigned" real numbers
the points on one side of O only, namely along
OX, were representative of numbers, now
with steps every point on the whole line
stretching on both sides of O is representative
of a step. This is a pictorial representation
of the superior generality introduced by the
positive and negative numbers, namely the