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?
Repeating digits in fractions: Is there a maximal repetition of the same digit
elementary-number-theory
Best Answer
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 $${d_1\times10^{n-1}+d_2\times10^{n-2}+\cdots+d_n\over10^n}+{e_1\times10^{m-1}+e_2\times10^{m-2}+\cdots+e_m\over10^n(10^m-1)}$$