Are numbers with repeating patterns in their decimal expansion (e.g. $0.123123123\ldots$) rational?

Question

There's a question that I've been thinking about for quite some time now. We all know that numbers with infinite decimal expansion such as $0.\overline{3}$ or $0.\overline{1}$ are not necessarily irrational. $$0.\overline{3}=0.333333\dots=\frac{1}{3}$$ $$0.\overline{1}=0.111111\ldots=\frac{1}{9}$$ Therefore we can say that for instance $0.\overline{4}$ is rational because we have: $$0.\overline{4}=4\cdot 0.\overline{1}=\frac{4}{9}$$ My question is: can we generalize these results to any number that has a repeating pattern in its decimal expansion? Can we claim for instance that $0.\overline{123}=0.123123123\ldots$ is rational? This question boils down to asking whether or not expressions like $0.\overline{01}$ and $0.\overline{001}$ etcetera are rational. If so, in the case of my example we could say: $$0.\overline{123}=123\cdot 0.\overline{001}$$ A naive thought of mine was that $0.\overline{01}$ is simply $\frac{1}{10}\cdot 0.\overline{1}$ but this is of course not true. Then I thought that maybe we can represent $0.\overline{01}$ as $\sum\limits_{i=1}^{\infty}\frac{1}{10^{2i}}$ which converges by the ratio test. However this does not tell us if it is rational or not.

This is not for homework or anything, just something I've been thinking about and I was wondering if any of you have some knowledge about this issue. Thanks.

Answer 1

As you mentioned, any infinite repeating decimal with $k$ digits is form: $$q = r + 10^{-k}r+10^{-2k}r+..$$ This series always converges to: $$q = \frac{r 10^k}{10^k-1}$$ Which is of course rational. For example: $$0.123123123.. = \frac{0.123\cdot 10^3}{10^3-1}=\frac{123}{999}$$ Or a longer one: $$0.571428571428... = \frac{0.571428\cdot 10^{6}}{10^{6}-1}=\frac{4}{7}$$

Answer 2

Let's take some number with repeating decimal expansion. I'll choose $x = 0.\overline{142857}$ for this. You might recognize this a $\frac{1}{7}$, but the way we're going to find out goes for any repeating decimal expansion.

First, take the equation above and multiply by $10^6$, since there are $6$ repeating decimals (this $6$ is the only thing that depends on the specific number you're interested in; everything else is the same for every number): $$ 1,\!000,\!000x = 142,\!857.\overline{142857} $$ Then we subtract $x = 0.\overline{142857}$ from both sides, like so: $$ 999,\!999x = 142,\!857 $$ and we get $$ x = \frac{ 142,\!857}{999,\!999} $$ which is the ratio of two integers, and thus a rational number.

---

This way you can express any number with repeating decimals as a fraction where the denominator is a string of nines. Of course, most likely you can shorten the fraction to get a nicer number, but that is not neccessary if you just want to prove rationality.