Editing Decimal (section)

==Infinite decimal expansion==
{{main|Decimal representation}}

For a [[real number]] {{Mvar|x}} and an integer {{Math|''n'' ≥ 0}}, let {{Math|[''x'']<sub>''n''</sub>}} denote the (finite) decimal expansion of the greatest number that is not greater than ''{{Mvar|x}}'' that has exactly {{Mvar|n}} digits after the decimal mark. Let {{Math|''d''<sub>''i''</sub>}} denote the last digit of {{Math|[''x'']<sub>''i''</sub>}}. It is straightforward to see that {{Math|[''x'']<sub>''n''</sub>}} may be obtained by appending {{Math|''d''<sub>''n''</sub>}} to the right of {{Math|[''x'']<sub>''n''−1</sub>}}. This way one has
:{{Math|1=[''x'']<sub>''n''</sub> = [''x'']<sub>0</sub>.''d''<sub>1</sub>''d''<sub>2</sub>...''d''<sub>''n''−1</sub>''d''<sub>''n''</sub>}},

and the difference of {{Math|[''x'']<sub>''n''−1</sub>}} and {{Math|[''x'']<sub>''n''</sub>}} amounts to
:<math>\left\vert \left [ x \right ]_n-\left [ x \right ]_{n-1} \right\vert=d_n\cdot10^{-n}<10^{-n+1}</math>,

which is either 0, if {{Math|1=''d''<sub>''n''</sub> = 0}}, or gets arbitrarily small as ''{{Mvar|n}}'' tends to infinity. According to the definition of a [[limit (mathematics)|limit]], ''{{Mvar|x}}'' is the limit of {{Math|[''x'']<sub>''n''</sub>}} when ''{{Mvar|n}}'' tends to [[infinity]]. This is written as<math display="inline">\; x = \lim_{n\rightarrow\infty} [x]_n \;</math>or
: {{Math|1=''x'' = [''x'']<sub>0</sub>.''d''<sub>1</sub>''d''<sub>2</sub>...''d''<sub>''n''</sub>...}},
which is called an '''infinite decimal expansion''' of ''{{Mvar|x}}''.

Conversely, for any integer {{Math|[''x'']<sub>0</sub>}} and any sequence of digits<math display="inline">\;(d_n)_{n=1}^{\infty}</math> the (infinite) expression {{Math|[''x'']<sub>0</sub>.''d''<sub>1</sub>''d''<sub>2</sub>...''d''<sub>''n''</sub>...}} is an ''infinite decimal expansion'' of a real number ''{{Mvar|x}}''. This expansion is unique if neither all {{Math|''d''<sub>''n''</sub>}} are equal to 9 nor all {{Math|''d''<sub>''n''</sub>}} are equal to 0 for ''{{Mvar|n}}'' large enough (for all ''{{Mvar|n}}'' greater than some natural number {{Mvar|N}}).

If all {{Math|''d''<sub>''n''</sub>}} for {{Math|''n'' > ''N''}} equal to 9 and {{Math|1=[''x'']<sub>''n''</sub> = [''x'']<sub>0</sub>.''d''<sub>1</sub>''d''<sub>2</sub>...''d''<sub>''n''</sub>}}, the limit of the sequence<math display="inline">\;([x]_n)_{n=1}^{\infty}</math> is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.: {{Math|''d''<sub>''N''</sub>}}, by {{Math|''d''<sub>''N''</sub> + 1}}, and replacing all subsequent 9s by 0s (see [[0.999...]]).

Any such decimal fraction, i.e.: {{Math|1=''d''<sub>''n''</sub> = 0}} for {{Math|''n'' > ''N''}}, may be converted to its equivalent infinite decimal expansion by replacing {{Math|''d''<sub>''N''</sub>}} by  {{Math|''d''<sub>''N''</sub> − 1}} and replacing all subsequent 0s by 9s (see [[0.999...]]).

In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion. Each decimal fraction has exactly two infinite decimal expansions, one containing only 0s after some place, which is obtained by the above definition of {{Math|[''x'']<sub>''n''</sub>}}, and the other containing only 9s after some place, which is obtained by defining {{Math|[''x'']<sub>''n''</sub>}} as the greatest number that is ''less'' than {{Mvar|x}}, having exactly ''{{Mvar|n}}'' digits after the decimal mark.

=== 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}}.