Some time ago I learned about the p-adic numbers which, simply put, differ from the reals in that their decimal representations (base $p$) can have infinitely many digits to the left of the decimal point instead of to the right. I have been wondering whether the concept of a number system where both ends of a decimal representation may extend indefinitely makes sense and/or has any meaningful applications but I couldn't find any useful sources.
The usually stated flaw of such a number system (let's call it $\mathbb{M}$) is that it wouldn't have any well-defined absolute value, since the Ostrowski's theorem (1916) states that every non-trivial absolute value on the rational numbers $\mathbb{Q}$ is equivalent to either the usual real absolute value or a p-adic absolute value. Since we probably want $\mathbb{Q}$ to be the part of $\mathbb{M}$, we can from that conclude that $\mathbb{M}$ can also have no non-trivial absolute value other than that (otherwise it would also be an absolute value on $\mathbb{Q}$), and both real and p-adic absolute values don't work on numbers that have two infinite ends.
But do we care? I mean, the juicy stuff like limits and continuity and connectedness come from a metric (or, more generally, from a topology) anyway, and these may, but don't have to be induced by an absolute value. Addition also seems well-defined, since formally, by commutativity and associativity (which we hope to hold true in $\mathbb{M}$), we can add any two numbers of the form $x.\!y$ and $u.\!v$ as
$$x.\!y+u.\!v=x.0+0.y+u.0+0.v=(x.0+u.0)+(0.y+0.v)$$
(here $.$ represents the decimal point; $x,u$ and $y,v$ are sequences of digits base $p$, possibly infinite on their respectful ends)
And from that, multiplication and probably subtraction can also be constructed.
In conclusion, even if the number system $\mathbb{M}$ is impossible, inconsistent or just plain useless, the reasons of that don't seem immediately obvious to me and I would greatly appreciate any clarification on the topic!
Addition certainly works, but multiplication is trickier than you may expect. The issue is that breaking into "left" and "right" pieces helps addition but runs into real issues with multiplication: because of the distributive law, we wind up needing to evaluate the product of a "left-infinite" number and a "right-infinite" number with each other.
To see that this is a problem, consider the simple example of $$...111.0\cdot 0.111...$$ (which arises when, for instance, we try to compute $...111.111...^2$). A reasonable first attempt at making sense of this is to break the second term into $0.1+0.01+0.001+...$ and then distribute, which yields $$...111.1 + ...111.11+...111.111+...,$$ which in turn yields ... what precisely? We wind up summing infinitely many $1$s in each place.
Indeed, this can be turned into an impossibility result: there is no way to make sense of multiplication of such "bi-infinite" numbers in a way that is continuous with respect to what seems the most natural topology, namely the topology generated by taking the $\mathbb{Z}$-fold product topology of the discrete topology on the set of digits $\{0,1,...,9\}$. This seems like a pretty fundamental limitation.