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

GENERALIZATIONS OF NUMBERS 79
n-2 1 2 3 n-1 1
1 ' n-1' n-2' n-3' * ' n'
and so on.
Now we can get rid of all repetitions of
fractions of the same value by simply striking
them out whenever they appear after their
first occurrence. In the few initial terms
written down above, |- which is enclosed above
in square brackets is the only fraction not in
its lowest terms. It has occurred before as. Thus this must be struck out. But the
series is still left with the same properties,
namely, (a) there is a first term, (b) each term
has next-door neighbours, (c) the series goes
on without end.
It can be proved that it is not possible to
arrange the whole series of real numbers in
this way. This curious fact was discovered
by Georg Cantor, a German mathematician
still living; it is of the utmost importance in
the philosophy of mathematical ideas. We
are here in fact touching on the fringe of the
great problems of the meaning of continuity
and of infinity.
Another extension of number comes from
the introduction of the idea of what has been
variously named an operation or a step,
names which are respectively appropriate
from slightly different points of view. We
will start with a particular case. Consider