Expressing recurring decimals as rational numbers

A recurring decimal fraction can always be written as a rational number, e.g. $0.\overline6=\dfrac23$; this can also be verified using the infinite G.P. sum formula.

Example (i): $0.66666\ldots=0.6+0.06+0.006+\cdots$; these terms are a G.P. with $a=0.6, |r|=|0.1|<1$, so the sum to infinity is $\dfrac{a}{1-r}=\dfrac{0.6}{1-0.1}=\dfrac{0.6}{0.9}=\dfrac69=\dfrac23$.

Example (ii): $0.\overline{46}=0.46+0.0046+0.000046+\cdots$; a G.P. with $a=0.46, |r|=|0.01|<1$, so the sum to infinity is $\dfrac{0.46}{1-0.01}=\dfrac{0.46}{0.99}=\dfrac{46}{99}$. …