jmc

First, we convert $0.\overline{81}$ to a fraction by the following trick. Let $x=0.\overline{81}$. Then $100x=81.\overline{81}$, so we can subtract:

\begin{array}{r r c r@{}l} &100x &=& 81&.818181\ldots \\ - &x &=& 0&.818181\ldots \\ \hline &99x &=& 81 & \end{array}

Therefore, $x=\frac{81}{99}=\frac{9}{11}$.

At this point, we could write $0.81$ as $\frac{81}{100}$ and subtract this from $\frac{9}{11}$. However, the following observation will save us some work: $$\begin{array}{rl}0.\overline{81}-0.81& =0.818181\dots -0.81\\ & =0.008181\dots \\ & =\frac{x}{100}.\end{array}$$ Therefore, $$0.\overline{81}-0.81=\boxed{\frac{9}{1100}}.$$