# Repeating digits in fractions: Is there a maximal repetition of the same digit?

Rational numbers, or fractions, either terminate or go on in repetitive patterns. The repetitive pattern-length cannot be greater than the denominator. My question is about the same digit repeating itself (after the decimal point) in a terminating rational number that does not terminate in this digit - is there an upper limit to the times a digit can repeat itself if the number is not endlessly repeating?

• Not sure what you mean. $0.555555555555555555$ is a terminating rational fraction. You can make the digits appear as often as you want. – Martin R Sep 20 '18 at 11:41
• @MartinR you are right. I forgot to include that the digit-repeats i am asking about do not terminate the number themselves. eg. 3889/2500 = 1.5556 – bukwyrm Sep 20 '18 at 13:10
• Then choose $1.5555555555555555555555555555555555556$, with as many 5s as you like. – Martin R Sep 20 '18 at 13:17
• Every terminating decimal, and every repeating decimal, represents a rational number, and that should answer your question. – Gerry Myerson Sep 20 '18 at 13:39
• Indeed it does. Care to put it as such? – bukwyrm Sep 20 '18 at 15:11

As Martin R notes in a comment, there is no limit to the number of repeats of a digit in a terminating decimal expansion of a rational number, the expansion not terminating in said digit.

More generally, every terminating decimal and every repeating decimal represents a rational number. In particular, the terminating decimal

$$.d_1d_2\dots d_n$$

represents the rational

$$(d_1\times10^{n-1}+d_2\times10^{n-2}+\cdots+d_n)/10^n$$,

and the (eventually) repeating decimal

$$.d_1d_2\dots d_n\dot e_1\dot e_2\dots\dot e_m$$

where the dots indicate the repeating portion) represents the rational number $${d_1\times10^{n-1}+d_2\times10^{n-2}+\cdots+d_n\over10^n}+{e_1\times10^{m-1}+e_2\times10^{m-2}+\cdots+e_m\over10^n(10^m-1)}$$

• Thank you very much, that is a very nice write-up. – bukwyrm Sep 25 '18 at 8:36