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12 THE FASCINATION OF NUMBERS

to 122021 (in the scale of 3) so that, in the latter scale, six
digits are required to represent a number which, in the
former scale, required only three digits.

The scale of 10 has been found to work as a good com-
promise between these twin advantages and disadvantages,
but it should be mentioned that some mathematicians advo-
cate the adoption of a scale of 12. The adoption of this
scale would, of course, call for the invention of an entirely
new symbol for the numbers 10 and 11, since each of these
would have to be expressed as one digit instead of as two.

It is a simple matter to convert a number from the scale of
10 to any other scale. The procedure is to divide successively
by the new scale number, and the various remainders, taken
in reverse order, will give the number in the new scale.

To reduce 466 to the scale of 7, we proceed as follows:

7| 466

7| 66 remainder 4

7]‘9 3

I 5 2

and this gives the number (in the scale of 7) of 1234.

For a long time in the development of numbering, there
was no symbol for nought, the accepted principle being that,
as it was impossible to count nothing, there was no point in
having a symbol. Nought was not considered a number at all.
This view is, of course, no longer tenable if we accept nega-
tive numbers as being numbers in the same sense as we
accept positive numbers, since nought has its own definite
position between —1 and 1.

Quite apart from the consideration of nought as a separate
number by itself, the introduction of a symbol for nought was
of the utmost importance in the construction of the larger
numbers, as will be seen when one tries to express the number