Balanced based representation
The expression $10.\overline{1}$ should be interpreted as follows:
$$
1 \cdot 3^1 + 0 \cdot 3^0 + (-1) \cdot 3^{-1} = 3 - \frac{1}{3} = \frac{8}{3}.
$$
The interpretation of balanced base $b=2d+1$ is the same, with the digits ranging from $-d$ to $d$.