Repeating DecimalA 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,
Repeating decimal notation was implemented in versions of Mathematica 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., 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, ... (Sloane's A085837). The repeating portion of a rational number can be found in Mathematica 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 If |
) or repeating decimals (e.g., 
). However, irrational numbers, such as 
neither terminate nor become periodic.
.
for 
, 2, ... are 0, 0, 1, 0, 0, 1, 6, 0, 1, 0, 2, 1, 6, 6, 1, 0, 16, 1, 18, ... (Sloane's A051626), where 0 indicates that the number is regular.
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).