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106 THE FASCINATION OF NUMBERS

Next, two columns are constructed and numbers are
entered as explained at each step:

Digit Carry-forward
Column Column

(a) The last digit of the cycle is known 7
(6) Multiply this digit (77) by the Relative (5)

=35. Write the digit 5 in the digit

column and the digit g in the carry-

forward column 5 3
(¢) Multiply the last digit in the digit column

by the Relative and add in the carry-

forward. 5 X 5+ 9 =28. Write the digit 8

in the digit column and 2 in the carry-

forward column 8 2

(d) Proceed as in (c), thus:
5X8+2=42 2 4
() 5x2+4=14 4 I
(f) 5X4+1=21 I 2
(8) 5X1+2= 7 7 o

This process is continuous, and the digits of the cycle can be
found in reverse order in the digit column.

One of the basic points outstanding is the question of why
decimals recur at all, but this is easily shown. We have
already seen that when unity is divided by a number larger
than itself, we have to ‘bring down’ a succession of noughts
to give a decimal result. We are dividing 10 or 100 or any
other power of 10 (according to the number of noughts
employed) and then shifting the decimal point. If the divisor
is a factor of any power of 10, then the decimal will have a
definite period and the digits will not recur. If, however, the
divisor is not such a factor then it will not divide into 10 (or
a power thereof) an exact integral number of times and
there will always be a remainder. There can, therefore, be no
end to the decimal expression.

But the number of different remainders possible is limited
by relation to the divisor. When we divide by 7, the only
possible remainders are 1, 2, 3, 4, 5 and 6. This clearly
applies to any divisor, so that if we divide by the number ¥,
then there are only N —1 possible remainders. When all these
different remainders have appeared once in the course of the