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

a

As the division does not terminate; the remainder at each step
of the division is greater than 0, but less than the denominator.
Consequently, there cannot be more different remainders than one
less than the denominator, while the number is without limit. Hence,
after as many remainders as are equal to the number in the denominator less by unity, a remainder will recur the same as some former
one; after this, the circumstances producing the succeeding remainders are the same as those which produced the preceding ones,
and therefore the second period of remainders will be the same as
the first. And, similarly, the third, fourth, &c., periods will also be
identical to the first period. From that step in the division where a
remainder becomes the same as a former one, it is evident from the
nature of the division that the former figures of the quotient must be
repeated in the same order.
Decimals are called repeating, recurring, or circulating decimals,
in which one or more figures continually recur, and the figure or
period of figures that recur, or are repeated, are called repetends.
The periods or repeating figures in decimals are distinguished by
placing a point over the first and last figures of the periods. If one
figure only be repeated, one point only is placed over the repeating
figure.
If the decimal consist only of figures which recur, it is named a
pure repetend; if it consist partly of figures which do not, and which
do recur, it is named a mixed repetend:-thus, 3, -45 are pure repetends, and '16, '345667 are mixed repetends.
12. Pror. To conviert any given termi)natitn decimcal into an ordinary
fraction.
A given decimal can be expressed as a decimal fraction by writing
under the numerator the suppressed denominator. And if the numerator and denominator of this decimal fraction be divided by their
greatest common measure, the quotients will be the numerator and
denominator of the fraction equivalent to the given decimal.
13. PRop. To clhange a pztre repeating decinal to its equivalent
ordinary fraction.
If 1 with ciphers annexed as decimals be divided by 9, 99, 999,
9999, &c., respectively,
Then = 11111111....... or '1
9'- = *01010101....... or i01
3J) = - -'001001001. or ')01
— i-, = '00010001. or OOOi
&c., &c., &c.
and any other numbers, 2, 3, 4, &c., when divided by 9, 99, 999,
9999, &., respectively, will give the same number of figures continually repeated. Hence, conversely: Every pure repeating decimal