Fractional bases are weird, and I think there’s even competing standards. What I was thinking is that you can write any number in base n like this:

\sum_{k= -∞}^{∞} a_k * n^k

where a_k are what we would call the digits of a number. To make this work (exists and is unique) for a given positive integer base, you need exactly n different symbols.

For a base 1/n, turns out you also need n different symbols, using this definition. It’s fairly easy to show that using 1/n just mirrors the number around the decimal point (e.g. 13.7 becomes 7.31)

I am not very well versed in bases tho (unbased, even), so all of this could be wrong.