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 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).

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 .

REFERENCES:

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, 1949.

de 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 ." 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 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.