42 THE FASCINATION OF NUMBERS

This method is justified by the principle that the root of
the number x is the same as the root of (x+9). No such
difficulties arise in addition or multiplication.

The digital process can still be applied to a division, even
where the division leaves a remainder. If we divide the
number X by the number 7’5o as to obtain the whole number
< plus a remainder R, then the root of X minus the root of R

will equal the root of ¥ multiplied by the root of >l
Do dohatiiss
D¢ r _er : r X Z:r
The division of 362 by 179 gives the number 2 and a re-
mainder of 4. Therefore:

3621"'"4-7': 179, X2,
or 2,~4.= 8,%2,
or 2—4+9= 16,=%
In the third line of working above it was necessary to add g
to the left-hand side to avoid the appearance of a negative
root.

- The reason why the digit g should apparently have such a
remarkable significance as is shown in the last few pages will
be found in the study of the various stages of its multiplica-
tion. Consecutive multiples of g are:

9, 18, 27, 36, 45, 54, 63, 72, 81, 9o, etc.
That is, in the tens column the digits 1 to g appear in ascend-
ing order, whereas in the units column they appear in the
reverse order. Thus, for each additional increment of 9, the
digit in the tens column increases by 1 and the digit in the
unit column decreases by 1. This is because g equals 10 —1,
so that, in effect, the addition of g is the same as adding 10
and deducting 1.

This now gives a further explanation of why the digital
root of x is the same as the root of (x+9g). When we add 10
to a number, we only add 1 to its digital root, so that if we
add g to a number, we only add 1 —1, or nought to its root.

It is now instructive to study a table of numbers (A) and