Decimal Expansion

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 10^0=1s place. For example, the number with decimal expansion 1234.56 is defined as


Expressions written in this form sum_(k)b_k10^k (where negative k 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 25^2 given by 625, of pi given by 3.14159..., and of 1/9 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., 1/2=0.5), eventually become periodic (in which case the number is called a repeating decimal, e.g., 1/3=0.3^_), 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.

fractiondecimal expansionfractiondecimal expansion

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


Factoring possible common multiples gives


where p≢0 (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 m/n is periodic, and has a decimal period lambda(n) independent of m, which is at most n-1 digits long. If n is relatively prime to 10, then the period lambda(n) of m/n is a divisor of phi(n) and has at most phi(n) digits, where phi is the totient function. It turns out that lambda(n) 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 m/n with (m,n)=1 is expanded, the period begins after s terms and has length t, where s and t are the smallest numbers satisfying

 10^s=10^(s+t) (mod n).

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

 10^t=1 (mod n).

As an example, consider n=84.

 10^0=1 10^1=10 10^2=16 10^3=-8; 10^4=4 10^5=40 10^6=-20 10^7=-32; 10^8=16,

so s=2, t=6. The decimal representation is 1/84=0.01190476^_. When the denominator of a fraction m/n has the form n=n_02^alpha5^beta with (n_0,10)=1, then the period begins after max(alpha,beta) terms and the length of the period is the exponent to which 10 belongs (mod n_0), i.e., the number x such that 10^x=1 (mod n_0). If q is prime and lambda(q) is even, then breaking the repeating digits into two equal halves and adding gives all 9s. For example, 1/7=0.142857^_, and 142+857=999. For 1/q with a prime denominator other than 2 or 5, all cycles n/q have the same length (Conway and Guy 1996).

If n is a prime and 10 is a primitive root of n, then the period lambda(n) of the repeating decimal 1/n is given by


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

To find denominators with short periods, note that


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 10^3-1 and 10^9-1, so its period is 3. Multiplication of any factor by a 2^alpha5^beta 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).

541, 271
67, 13
7239, 4649
873, 137
1121649, 513239
1353, 79, 265371653
1531, 2906161
1617, 5882353
172071723, 5363222357
1819, 52579
203541, 27961

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


Shanks (1873ab) computed the periods for all primes up to 120000 and published those up to 29989.

See also

10, Base, Binary, Decimal, Decimal Period, Decimal Point, Expanded Notation, Fraction, Midy's Theorem, Multiplicative Order, Regular Number, Repeating Decimal, Unique Prime Explore this topic in the MathWorld classroom

Portions of this entry contributed by Christopher Stover

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Conway, J. H. and Guy, R. K. "Fractions Cycle into Decimals." In The Book of Numbers. New York: Springer-Verlag, pp. 157-163 and 166-171, 1996.Das, R. C. "On Bose Numbers." Amer. Math. Monthly 56, 87-89, Polignac, A. "Note sur la divisibilité des nombres." Nouv. Ann. Math. 14, 118-120, 1855.Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, pp. 159-179, 2005.Glaisher, J. W. L. "Periods of Reciprocals of Integers Prime to 10." Proc. Cambridge Philos. Soc. 3, 185-206, 1878.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 25, 2003.Lehmer, D. H. "Guide to Tables in the Theory of Numbers." Bulletin No. 105. Washington, DC: National Research Council, pp. 7-12, 1941.Lehmer, D. H. "A Note on Primitive Roots." Scripta Math. 26, 117-119, 1963.Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, p. 60, 1988.Rademacher, H. and Toeplitz, O. The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 147-163, 1957.Rao, K. S. "A Note on the Recurring Period of the Reciprocal of an Odd Number." Amer. Math. Monthly 62, 484-487, 1955.Shanks, W. "On the Number of Figures in the Period of the Reciprocal of Every Prime Number Below 20000." 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 20000 and 30000." 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 A002329/M4045, A002371/M4050, A003592, A007138/M2888, A046107, and A046108 in "The On-Line Encyclopedia of Integer Sequences."Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 60, 1986.

Referenced on Wolfram|Alpha

Decimal Expansion

Cite this as:

Stover, Christopher and Weisstein, Eric W. "Decimal Expansion." From MathWorld--A Wolfram Web Resource.

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