Regular Number

A regular number, also called a finite decimal (Havil 2003, p. 25), is a positive number that has a finite decimal expansion. A number such as  which is not regular is said to be nonregular.

If  is a regular number, then

			(1)
			(2)
			(3)

Factoring possible common multiples gives

(4)

where  (mod 2, 5).

The denominators of the first few regular unit fractions are 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, ... (OEIS A003592), which have decimal expansions 1, , , , , , , , , , , , ....

The number of decimal digits in a regular number is given by  (Wells 1986, p. 60). The numbers of digits in the regular unit fractions are 1, 2, 1, 3, 1, 4, 2, 2, 5, 3, 2, 6, 4, 2, 3, ... (OEIS A117920).

REFERENCES:

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.

Sloane, N. J. A. Sequences A003592 and A117920 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.