Fix a base $b \neq 0$. To a sequence of non-negative integers $(c_i), i \in \mathbb{Z}$ such that $c_i = 0$ for $i \gg 0$, associate the sum
$$\sum_{i=-\infty}^{\infty} c_i b^i$$
We can order such sequences lexicographically: if $c_i = 0$ for $i \geq n$ and $d_n \neq 0$ then $(c_i) < (d_i)$, and otherwise we can compare $c_{n-1}$ to $d_{n-1}$, etc.
Among all such sequences with sum $x$, we can declare the base-$b$ representation of $x$ to be the one which is lexicographically last.
This seems to have pretty nice properties:
If $b\geq 2$ is an integer then this gives the (or at least a) usual base $b$ representation.
We take $1.000\ldots$ rather than $0.999\ldots$ as the base ten representation of 1, and similarly for other numbers and bases.
We don't have to artificially declare that the digits are smaller than the base, meaning that everything smoothly generalizes to negative and/or non-integer bases.
Two questions:
Does this have a name or has it been studied before?
It seems like with a slight modification we should be able to get the base-1 representation of a positive integer to reduce to tallies, but I don't see a way to do this that feels natural. (E.g. you could sort by the largest digit appearing and then lexicographically or something but that seems like forcing the issue.) Is there a clean way to tweak the definition to get this to work?
(Tags aren't quite right but couldn't find anything better... Feel free to suggest improvements.)