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

GENERALIZATIONS OF NUMBERS 73
to V2 as closely as we like, but we never
49
exactly reach its value. For example, 4- is
just less than 2, and 9 is greater than 2, so
4
7 3
that V2 lies between - and - But the best
5 2
systematic way of approximating to 2/ in
obtaining a series of decimal fractions, each
bigger than the last, is by the ordinary
method of extracting the square root; thus
14 141 1414
the series is 1 14 141 141 and so on.
'10I' 100' 1000'
Ratios of this sort are called by the Greeks
incommensurable. They have excited from
the time of the Greeks onwards a great deal
of philosophic discussion, and the difficulties
connected with them have only recently been
cleared up.
We will put the incommensurable ratios
with the fractions, and consider the whole
set of integral numbers, fractional numbers,
and incommensurable numbers as forming
one class of numbers which we will call "real
numbers." We always think of the real
numbers as arranged in order of magnitude,
starting from zero and going upwards, and
becoming indefinitely larger and larger as we
proceed. The real numbers are conveniently