Repeating Decimal
A repeating decimal, also called a recurring decimal, is a number whose decimal representation eventually becomes periodic (i.e., the same sequence of digits repeats indefinitely). The repeating portion of a decimal expansion is conventionally denoted with a vinculum so, for example,
 |
The minimum number of digits that repeats in such a number is known as the decimal period.
Repeating decimal notation was implemented in versions of the Wolfram Language prior to 6 as PeriodicForm[RealDigits[r]] after loading the add-on package NumberTheory`ContinuedFractions`.
All rational numbers have either finite decimal expansions (i.e., are regular numbers; e.g.,
) or repeating decimals (e.g., 
). However,
irrational numbers, such as 
neither
terminate nor become periodic.
Numbers such as 0.5 are sometimes regarded as repeating decimals since 
.
The denominators of the first few unit fractions having repeating decimals are 3, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, ... (OEIS A085837).
The repeating portion of a rational number can be found in the Wolfram Language using the command RealDigits[r][[1,-1]].
The number of digits in the repeating portion of the decimal expansion of a rational
number can also be found directly from the multiplicative
order of its denominator. The periods of the
decimal expansions of 
for 
, 2, ... are
0, 0, 1, 0, 0, 1, 6, 0, 1, 0, 2, 1, 6, 6, 1, 0, 16, 1, 18, ... (OEIS A051626),
where 0 indicates that the number is regular.
If 
is a repeating decimal and 
is a terminating
decimal, them 
has a nonperiodic part whose length
is that of 
and a repeating part whose length
is that of 
(Wells 1986, p. 60).
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