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

76 INTRODUCTION TO MATHEMATICS
2 the fraction (- + -4-), that is, lies;
and so on indefinitely. Because of this property the series is said to be "compact."
There is no end-point to the series, which increases indefinitely without limit as we go
along the line OX. It would seem at first
sight as though the type of series got in this
way from the fractions, always including the
integers, would be the same as that got from
all the real numbers, integers, fractions, and
incommensurables taken together, that is,
from all the points on the line OX. All that
we have hitherto said about the series of
fractions applies equally well to the series of
all real numbers. But there are important
differences which we now proceed to develop.
The absence of the incommensurables from
the series of fractions leaves an absence of
end points to certain classes. Thus, consider
the incommensurable V2. In the series of
real numbers this stands between all the
numbers whose squares are less than 2, and
all the numbers whose squares are greater
than 2. But keeping to the series of fractions
alone and not thinking of the incommensurables, so that we cannot bring in </, there
is no fraction which has the property of
dividing off the series into two parts in this
way, i. e. so that all the members on one side
have their squares less than 2, and on the
other side greater than 2. Hence in the