Elementary arithmetic, with brief notices of its history... by Robert Potts.

9

consisting of 1, 2, 3, 4, &c., repeating figures respectively is equal to a
fraction having the repeating figures for the numerator and as many
nines for the denominator.
A singular exception arises when 9 is the repeating figure of the
decimal. In this case I- is not a fraction, and the repeating decimal
cannot have been produced by the division of any number with ciphers
annexed, by any other number.
14. PROP. To redace a mnixed repeati/ng decial to its equiivalent ordinary fr'action.
Every mixed repeating decimal as -123 can be separated into two
parts, of which one is, and the other is not, repeating.
Here -123 = -12 + 003.12 =1 and.00 1 of.3 -100 ' 100 900
and l12 +. 003 = 12 + 108 3+
100.900 900 900
_ 111 _ 123-12
900 900
Therefore.123 _ 123-12
900
Hence, any mixed repeating decimal is equivalent to the fraction
whose numerator is the difference between the numbers composed of the
mixed repeating decimal and of the non-repeating figures, and whose
denominator consists of the number composed of as many nines as the
figures in the repeating part prefixed to as many ciphers as figures
in the non-repeating part.
15. PROP. To add and suitbract repeating dec imals.
If the repeating periods of the given decimals do not begin with
figures of the same local value, each period must be made to begin with
a figure of the same local value and to end with one of the same lower
local value, so that the decimals may have the same number of repeating figures, and of the same local values. The sum or difference may
then be found, and the number of repeating figures in the sum or
difference will be the same as those noted in the decimals.'
1 Example.-To find the sum and difference of 12-453 and 7-276.
Here 12'453 contains two repeating figures beginning from the place of hun,
dredths, and 7-276 contains three, beginning from the place of tenths. The repeatingfigures in 7'276 must be made to begin from the place of hundredths, making the first
figure non-repeating, thus 7'2764, so that 764 are now made the repeating figures. It
will be obvious that as the repeating figures in each number begin with a figure of the
same local value, and as there are 3 in one and 2 in the other, there must be 6 repeating
figures in each, that they may end with figures of the same but lower local value.
Thus, 12-453 must be written 12'45353535 12-45353535
7276,,,, 727627627 7-27627627
19'7298116 sum. 5'1772590 difference.
It may be observed that one or two figures of each repetend must be continued
in order to secure the accuracy of the last repeating figure in the result.