Editing Decimal (section)

=== Rational numbers ===
{{main|Repeating decimal}}

[[Long division]] allows computing the infinite decimal expansion of a [[rational number]]. If the rational number is a [[#decimal fraction|decimal fraction]], the division stops eventually, producing a decimal numeral, which may be prolongated into an infinite expansion by adding infinitely many zeros. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainders are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient. That is, one has a ''repeating decimal''. For example,
:{{sfrac|81}} = 0.{{thin space}}012345679{{thin space}}012... (with the group 012345679 indefinitely repeating).

The converse is also true: if, at some point in the decimal representation of a number, the same string of digits starts repeating indefinitely, the number is rational.
{|
|-
|For example, if ''x'' is || {{figure space|6}}0.4156156156...
|-
|then 10,000''x'' is || {{figure space|3}}4156.156156156...
|-
|and 10''x'' is|| {{figure space|6}}4.156156156...
|-
|so 10,000''x'' − 10''x'', i.e. 9,990''x'', is||{{figure space|3}}4152.000000000...
|-
|and ''x'' is|| {{figure space|3}}{{sfrac|4152|9990}}
|}
or, dividing both numerator and denominator by 6, {{sfrac|692|1665}}.