A graph-based approach to repeating decimals

Abstract

In this paper we deal with a classical problem in elementary number theory, namely repeating decimals. We show how the digits of the period of the decimal representation of any fraction $\frac{k}{m}$, where $k$ and $m$ are positive integers arbitrarily chosen, can be obtained relying upon the graphs associated with the iteration of a certain map over the finite set $\{0, 1, \dots, 10n-2 \}$ for a suitable integer $n$, which depends on $m$. In the last section of the paper we generalize the results to any arbitrary choice of the base $B \geq 2$ for the representation of the fraction $\frac{k}{m}$.