The decimal expansion of a number is its representation in base-10 (i.e., in the decimal system). In this system, each "decimal
place" consists of a digit 0-9 arranged such that each digit is multiplied by
a power of 10, decreasing from left to right, and with a decimal place indicating
the 
s
place. For example, the number with decimal expansion 1234.56 is defined as
 |  |  |
(1)
|
 |  |  |
(2)
|
Expressions written in this form 
(where negative 
are allowed as exemplified above but usually not considered
in elementary education contexts) are said to be in expanded
notation.
Other examples include the decimal expansion of 
given by 625, of 
given by 3.14159..., and of 
given by 0.1111.... The decimal expansion of a number can
be found in the Wolfram Language using
the command RealDigits[n],
or equivalently, RealDigits[n,
10].
The decimal expansion of a number may terminate (in which case the number is called a regular number or finite decimal, e.g., 
), eventually become periodic (in
which case the number is called a repeating decimal,
e.g., 
),
or continue infinitely without repeating (in which case the number is called irrational).
The following table summarizes the decimal expansions of the first few unit fractions. As usual, the repeating portion of a decimal expansion is conventionally denoted with a vinculum.
| fraction | decimal expansion | fraction | decimal expansion |
| 1 | 1 |  |  |
 | 0.5 |  |  |
 |  |  |  |
 | 0.25 |  |  |
 | 0.2 |  |  |
 |  |  | 0.0625 |
 |  |  |  |
 | 0.125 |  |  |
 |  |  |  |
 | 0.1 |  | 0.05 |
If 
has a finite decimal expansion (i.e., 
is a regular number), then
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
Factoring possible common multiples gives
 |
(6)
|
where 
(mod 2, 5). Therefore, the numbers with finite decimal expansions are fractions of
this form. The first few regular numbers are 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40,
50, ... (OEIS A003592).
Any nonregular fraction 
is periodic, and has a decimal
period 
independent of 
,
which is at most 
digits long. If 
is relatively prime to
10, then the period 
of 
is a divisor of 
and has at most 
digits, where 
is the totient function.
It turns out that 
is the multiplicative order of 10 (mod 
) (Glaisher 1878, Lehmer 1941). 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.
When a rational number 
with 
is expanded, the period begins after 
terms and has length 
, where 
and 
are the smallest numbers satisfying
 |
(7)
|
When 
(mod 2, 5), 
,
and this becomes a purely periodic decimal with
 |
(8)
|
As an example, consider 
.
 |
(9)
|
so 
,
.
The decimal representation is 
. When the denominator
of a fraction 
has the form 
with 
,
then the period begins after 
terms and the length of the period is the exponent
to which 10 belongs (mod 
), i.e., the number 
such that 
. If 
is prime and 
is even, then breaking
the repeating digits into two equal halves and adding gives
all 9s. For example, 
,
and 
.
For 
with a prime denominator
other than 2 or 5, all cycles 
have the same length (Conway and Guy 1996).
If 
is a prime and 10 is a primitive
root of 
,
then the period 
of the repeating decimal 
is given by
 |
(10)
|
where 
is the totient function. Furthermore, the decimal
expansions for 
,
with 
,
2, ..., 
have periods of length 
and differ only by a cyclic permutation. Such numbers 
are called full
reptend primes.
To find denominators with short periods, note that
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
 |  |  |
(14)
|
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
 |  |  |
(18)
|
 |  |  |
(19)
|
 |  |  |
(20)
|
 |  |  |
(21)
|
 |  |  |
(22)
|
The decimal period of a fraction with denominator equal to a prime factor
above is therefore the power of 10 in which the factor
first appears. For example, 37 appears in the factorization of 
and 
, so its period is 3. Multiplication of any factor
by a 
still gives the same period as the factor alone. A denominator obtained by a multiplication of two factors
has a period equal to the first power of 10 in which both
factors appear. The following table gives the primes
having small periods (OEIS A007138, A046107,
and A046108; Ogilvy and Anderson 1988).
| period | primes |
| 1 | 3 |
| 2 | 11 |
| 3 | 37 |
| 4 | 101 |
| 5 | 41, 271 |
| 6 | 7, 13 |
| 7 | 239, 4649 |
| 8 | 73, 137 |
| 9 | 333667 |
| 10 | 9091 |
| 11 | 21649, 513239 |
| 12 | 9901 |
| 13 | 53, 79, 265371653 |
| 14 | 909091 |
| 15 | 31, 2906161 |
| 16 | 17, 5882353 |
| 17 | 2071723, 5363222357 |
| 18 | 19, 52579 |
| 19 | 1111111111111111111 |
| 20 | 3541, 27961 |
A table of the periods 
of small primes other than
the special 
,
for which the decimal expansion is not periodic, follows (OEIS A002371).
 |  |  |  |  |  |
| 3 | 1 | 31 | 15 | 67 | 33 |
| 7 | 6 | 37 | 3 | 71 | 35 |
| 11 | 2 | 41 | 5 | 73 | 8 |
| 13 | 6 | 43 | 21 | 79 | 13 |
| 17 | 16 | 47 | 46 | 83 | 41 |
| 19 | 18 | 53 | 13 | 89 | 44 |
| 23 | 22 | 59 | 58 | 97 | 96 |
| 29 | 28 | 61 | 60 | 101 | 4 |
Shanks (1873ab) computed the periods for all primes up to 
and published those up to 
.
." Proc. Roy. Soc. London 22, 200, 1873a.Shanks,
W. "On the Number of Figures in the Period of the Reciprocal of Every Prime
Number Between 
and 
."
Proc. Roy. Soc. London 22, 384, 1873b.Shiller, J. K.
"A Theorem in the Decimal Representation of Rationals." Amer. Math.
Monthly 66, 797-798, 1959.Sloane, N. J. A. Sequences