The decimal expansion of a number is its representation in base 10 (i.e., the decimal system). For example, the decimal expansion of 25² is 625, of pi is 3.14159..., and of is 0.1111....

The decimal expansion of a number can be found in Mathematica 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.

If has a finite decimal expansion (i.e., r is a regular number), then

Factoring possible common multiples gives

where (mod 2, 5). Therefore, the numbers with finite decimal expansions are fractions of this form. The number of decimals is given by (Wells 1986, p. 60).

Any nonregular fraction is periodic, and has a period independent of m, which is at most digits long. If n 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 n) (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 s terms and has length t, where s and t are the smallest numbers satisfying

When (mod 2, 5), s = 0, and this becomes a purely periodic decimal with

As an example, consider n = 84.

so s = 2, t = 6. 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 x such that . If q is prime and is even, then breaking the repeating digits into two equal halves and adding gives all 9s. For example, , and 142 + 857 = 999. For with a prime denominator other than 2 or 5, all cycles have the same length (Conway and Guy 1996).

If n is a prime and 10 is a primitive root of n, then the period of the repeating decimal is given by

where is the totient function. Furthermore, the decimal expansions for , with p = 1, 2, ..., have periods of length and differ only by a cyclic permutation. Such numbers n are called full reptend primes.

To find denominators with short periods, note that

The 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 (Sloane's A007138, A046107, and A046108; Ogilvy and Anderson 1988).

A table of the periods e of small primes other than the special p = 5, for which the decimal expansion is not periodic, follows (Sloane's A002371).

Shanks (1873ab) computed the periods for all primes up to and published those up to .

 

Base, Binary, Decimal, Decimal Point, Fraction, Haupt-Exponent, Midy's Theorem, Repeating Decimal, Unique Prime