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

64 INTRODUCTION TO MATHEMATICS
value of a digit depends on the position in
which it occurs. Consider, for example, the
digit 5, as occurring in the numbers 25, 51,
3512, 5213. In the first number 5 stands for
five, in the second number 5 stands for fifty,
in the third number for five hundred, and in
the fourth number for five thousand. Now,
when we write the number fifty-one in the
symbolic form 51, the digit 1 pushes the digit
5 along to the second place (reckoning from
right to left) and thus gives it the value fifty.
But when we want to symbolize fifty by itself,
we can have no digit 1 to perform this service;
we want a digit in the units place to add
nothing to the total and yet to push the 5
along to the second place. This service is
performed by 0, the symbol for zero. It is
extremely probable that the men who introduced O for this purpose had no definite conception in their minds of the number zero.
They simply wanted a mark to symbolize the
fact that nothing was contributed by the
digit's place in which it occurs. The idea of
zero probably took shape gradually from a
desire to assimilate the meaning of this mark
to that of the marks, 1, 2,... 9, which do represent cardinal numbers. This would not
represent the only case in which a subtle idea
has been introduced into mathematics by a
symbolism which in its origin was dictated by
practical convenience.