THE MAKING OF NUMBERS II

sand, two hundreds, three tens and four ones, and is really a
shorthand form of the expression:

(1 X 1000) +(2 X 100) +(3 X 10) +(4 X 1)
=(1x10%) +(2Xx102) +(3X10)+(4X1)

If we replace the digits 1, 2, 3 and 4 by the letters 4, b, ¢
and d respectively, then the number ‘abed’ is really a form of:

1000a +100b +10¢+d

The fact that all numbers have a similar structure is of para-
mount importance in all that follows. It is not, however,
- essential that counting should be done in groups of ten as
shown above, but the use of such groups was found to be
convenient and has therefore become an established prac-
tice. Our numbers are said to be in the scale of ten, but these
same numbers can be converted to quite different appear-
ances in other scales. ,

In each scale of notation the basic idea is the same. For
example, if the scale we use is x, then we count in groups of
x, x2, x3 and so on (instead of 10, 102, 10%). Thus in the scale
of 7, the number 1234 represents:

(1 x7%) +(2x7%)+(3%7)+(4x1)

and is therefore equivalent to the number 466 in the scale of
10. It is important, however, that it should be understood
that the number 1234 (in the scale of 7) and the number 466
(in the scale of 10) both represent the same thing. They are
merely translations of each other in different languages.
Each different scale has one advantage and one disadvan-
tage as against other scales. The lower the scale, so the fewer
different symbols are required. In the scale of 7, for example,
the symbols 8 and g would never be used. The number 8
equals (1 x7) -1 and would therefore be 11 in the scale of 7.
But the advantage of having less different symbols to bother
about is counter-balanced by the fact that, in the lower
scales, numbers require a greater quantity of digits for their
expression. The number 466 (in the scale of 10) is equivalent

 

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