THE MAKING OF NUMBERS 15

rational numbers—a rational number being one which can
be expressed by a finite number of digits—and in each cate-
gory there is an infinitude of numbers. There is no limit to
them at all. So far as positive integral numbers are concerned
there is a first number—one—but there is no such thing as a
last number, since we can add one to any number and so

obtain a larger number. If we take into consideration the

fractional numbers as well, then we are faced by the pro-
position that there is not even a first number for, as we have
already shown, there is an infinitude of numbers between
o and 1. And in the same way as the positive numbers
stretch interminably in one direction, so do the negative
numbers extend in the opposite direction.

In addition to the rational numbers there are also irra-
tional numbers or ‘inexpressibles’ as they have been called
for the simple reason that they cannot be expressed in
normal numerical language. A rational fractional number
can be expressed in decimal form either by a finite number
of digits or by recurring cycles of digits which always repeat
in the same order, but expressions for irrational numbers
have no ending and the decimals never recur.

The square root of 2, for instance, can only be calculated
to the nearest significant figure, according to the degree of
accuracy required; it can never be calculated exactly. In the
same way, there is no exact value for z, the symbol used to
show, amongst other things, the relationship between the
diameter of a circle and its circumference. For practical pur-
poses, this is not a serious complication since the approximate
values satisfy most requirements. From the theoretical stand-
point, the existence of such numbers opens up wide fields of
study which are well beyond the scope of this work, but their
nature and basic properties are discussed in a later chapter.

DECIMALS

The behaviour of decimal expressions when multiplied is a
source of never-ending mystification to readers who have not

 

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