TITLE: Is it possible to write a number in a base of less than 1?
QUESTION [18 upvotes]: Following on from this question:
https://math.stackexchange.com/a/217112/45127
If we take base 10 as an example, the granularity is 1.  I.e. we increment the digits in an increment of 1 until we reach 9 and then start a new column.
If the base is less than 1, then would any number we write require an infinite number of columns or is there a way of writing a number in a base between 0 and 1?

REPLY [3 votes]: Yipes!  I just tried base -10.
$$12345.678=\begin{aligned}
   2&\times\;^{_-}10^{4}\\
+\;8&\times\;^{_-}10^{3}\\
+\;4&\times\;^{_-}10^{2}\\
+\;6&\times\;^{_-}10^{1}\\
+\;6&\times\;^{_-}10^{0}\\
+\;4&\times\;^{_-}10^{^{_-}1}\\
+\;8&\times\;^{_-}10^{^{_-}2}\\
+\;2&\times\;^{_-}10^{^{_-}3}
\end{aligned}$$
$$12345.678_{10} = 28466.482_{^{_-}10}$$
Although, that's using positive digits with a neative base.  You may need to use negative digits and negate the whole number.