Prove that $0.\overline{123} = 0.123123123123...$ is rational

Question

How are we to go about proving that $0.\overline{123} = 0.123123123123...$ can be expressed in the form $\frac{p}{q}$ where $p,q \in \mathbb{Z}$, i.e is rational?

Is this to be done using arithmetic or geometric progressions?

Answer 1

Pretty sure this has been asked before, but directly answering your question :

$$0.\overline{123} = 0.123123123123... = 0.123 + 0.000123 + 0.000000123 + \ldots$$ $$= 123\times10^{-3} + 123\times10^{-6} + 123\times10^{-9} + \ldots $$ $$ = 123 \cdot (10^{-3} + 10^{-6} + 10^{-9} + \ldots)$$ $$ = 123\cdot \left( \frac{10^{-3}}{1-10^{-3}} \right) $$ $$ = \frac{41}{333}$$

Answer 2

Hint: notice that if $a = 0.\overline{123}$. Then $1000a = 123 + a$