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

GENERALIZATIONS OF NUMBERS 75
of size, starting from zero and continually
increasing as we go on.
All this seems simple enough, but even at
this stage there are some interesting ideas to
be got at by dwelling on these obvious facts.
Consider the series of points which represent
the integral numbers only, namely, the points
0, A, B, C, D, etc. Here there is a first point
O, a definite next point, A, and each point,
such as A or B, has one definite immediate
predecessor and one definite immediate successor, with the exception of 0, which has no
predecessor; also the series goes on indefinitely without end. This sort of order is
called the type of order of the integers; its
essence is the possession of next-door neighbours on either side with the exception of
No. 1 in the row. Again consider the integers
and fractions together, omitting the points
which correspond to the incommensurable
ratios. The sort of serial order which we
now obtain is quite different. There is a first
term 0; but no term has any immediate predecessor or immediate successor. This is
easily seen to be the case, for between any
two fractions we can always find another
fraction intermediate in value. One very
simple way of doing this is to add the
fractions together and to halve the result.
For example, between 2 and -;, the fraction
2(3 + I), that is -1 7 lies; and between 3 and
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