RECURRING DECIMALS 103

Not all derivatives have this property. The derivative of J;,
for example, cannot be multiplied by 2 or 3 merely by
moving its digits (though it can be thus multiplied by 4 and
some other numbers), nor can it be split into two groups so
that relative pairs of digits sum to 9. It retains, however, the
common property of having g as its digital root and, in
addition, has another property. If it is split into two groups,
the following results are obtained.

Derivative of =-023255813953488372093
Split into two groups = 2325581395
(ignoring the zero) =5 3488372093
Divide first group by 2 = 1162790697
(ignoring final remainder)
»  second group by 3= 1162790697

(ignoring final remainder)

This property is not uncommon, although it may take
slightly different forms for different derivatives. The deriva-
tive for -} is a particularly good example, and is therefore
shown in full in the table on page 104.

It will be seen that, no matter at what digit we start,
this digit, taken with the next six or seven digits (as re-
quired) in rotation will give a multiple or near-multiple of
140845.

The usual method of ascertaining the digits appearing in
a cycle of decimals is by ordinary division. These digits can,
however, be found by a reverse method, whereby it is pos-
sible to find the last digit in a cycle without knowing the first
digit. To understand this method, it must be remembered that
when obtaining a complete cycle of digits, we are really
dividing a number consisting entirely of a repetition of the
digit. 9. When dividing by 7, for instance, it is obvious that
we shall obtain a cycle which when multiplied by 7, will give
a number consisting entirely of nines and this number will
also be exactly divisible by 7. It follows therefore that the

®
Bl
g
&
e
=
=]
X
2= 1
2

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