The decimal expansion of a number is its representation in base10 (i.e., in the decimal system). In this system, each "decimal place" consists of a digit 09 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 .