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

GENERALIZATIONS OF NUMBERS 77
series of fractions there is a quasi-gap where
V/2 ought to come. This presence of quasigaps in the series of fractions may seem a
small matter; but any mathematician, who
happens to read this, knows that the possible
absence of limits or maxima to a class of
numbers, which yet does not spread over
the whole series of numbers, is no small evil.
It is to avoid this difficulty that recourse is
had to the incommensurables, so as to obtain a complete series with no gaps.
There is another even more fundamental
difference between the two series. We can
rearrange the fractions in a series like that of
the integers, that is, with a first term, and
such that each term has an immediate successor and (except the first term) an immediate predecessor. We can show how this
can be done. Let every term in the series of
fractions and integers be written in the fractional form by writing 1 for 1, 2 for 2, and so
on for all the integers, excluding 0. Also for
the moment we will reckon fractions which
are equal in value but not reduced to their
lowest terms as distinct; so that, for example,
until further notice, 3, -, 182, etc., are ail
reckoned as distinct. Now group the fractions into classes by adding together the
numerator and denominator of each term.
For the sake of brevity call this sum of the
numerator and denominator of a fraction its