The ordinary process of long division is rather difficult, owing to
the necessity of guessing at the successive figures which form the
divisor. In case the repeating decimal expressing the _exact_ quotient
is required, the following method will be found convenient:
_Rule for division_.
_First._ Treat the divisor as follows:
If its last figure is a 0, strike this off, and treat what is left
as the divisor.
If its last figure is a 5, multiply the whole by 2, and treat the
product as the divisor.
If its last figure is an even number, multiply the whole by 5, and
treat the product as a divisor.
Repeat this treatment until these precepts cease to be applicable.
Call the result the _prepared divisor_.
_Second._ From the prepared divisor cut off the last figure: and, if
this be a 9, change it to a 1, or if it be a 1, change it to a 9;
otherwise keep it unchanged. Call this figure the _extraneous
multiplier_.
Multiply the extraneous multiplier into the divisor thus truncated,
and increase the product by 1, unless the extraneous multiplier be 7,
when increase the product by 5. Call the result the _current
multiplier_.
_Third._ Multiply together the extraneous multiplier and all the
multipliers used in the process of obtaining the prepared divisor. Use
the product to multiply the dividend, calling the result the _prepared
dividend_.
_Fourth._ From the prepared dividend cut off the last figure, multiply
this by the current multiplier, and add the product to the truncated
dividend. Call the sum the _modified dividend_, and treat this in the
same way. Continue this process until a modified dividend is reached
which equals the original prepared dividend or some previous modified
dividend; so that, were the process continued, the same figures would
recur.
_Fifth._ Consider the series of last figures which have been
successively cut off from the prepared dividend and from the modified
dividends as constituting a number, the figure first cut off being in
the units' place, the next in the tens' place, and so on. Call this
the _first infinite number_, because its left-hand portion consists of
a series of figures repeating itself indefinitely toward the left.
Imagine another infinite number, identical with the first in the
repeating part of the latter, but differing from this in that the same
series is repeated uninterruptedly and indefinitely toward the right
into the decimal places.
Subtract the first infinite number from the second, and shift the
decimal point as many places to the left as there were zeros dropped
in the process of obtaining the prepared divisor.
The result is the quotient sought.
_Examples._