Are there "numbers" with infinite number of digits (to the left) and are they useful?

Question

Are there "numbers" with infinite amount of digits (to the left) and are they useful?(not talking about p-adic numbers) By useful I mean used in math (or something) and not a dead end idea. I guess I want a "reasonable" number system that has the reals as a subset, addition, multiplication, and ideally is ordered. For example something like $123;...3415$ would be a number in the system where the ";" separates an infinite number of digits. If there is no such a "reasonable" number system is there a reason? References would be nice.

I was asked something along this line in an email exchange. I also came up with a rough idea on how I would go about constructing the "numbers" but it is sort of besides the point. My rough idea (which is only presented to give a better idea of what I am asking about and not to be taken as an actual construction of what I am looking for) is this: Consider $\mathbb{N} \times (\mathbb{Z} \setminus \{0\})$ with a dictionary order and $\mathbf{10}=\{0,1,2,...,9\}$. Each number can be considered as a function $f: \mathbb{N} \times (\mathbb{Z} \setminus \{0\}) \to \mathbf{10}$ where $f(0,n)$ for negative $n$ corresponds to the $n$th digit to the right of the decimal place and for positive $n$ is the $n$th digit to the left. When you change the "0" part in the function corresponds "looking" infinitely far or an infinite "shift". So for example

\begin{align*}&923;...567;...312 \\ &=f(2,3)f(2,2)f(2,1);...f(1,3)f(1,2)f(1,1);...f(0,3)f(0,2)f(0,1).\end{align*}

There are plenty of things to iron out like whether or not function that don't have a left most digit should be considered "numbers" like should ...444 be a number or ...413 ($\pi$ backwards) and is there a reasonable way to compare them. Or what would 1;...999+1 be? Also a good concept of distance seems like it would be difficult to set up.

I personally have some doubts that this sort of number system has any sort of use, but it was fun to play around with so I figured I would ask.

Answer 1

Conway's invention of the surreal numbers may be relevant to what you are thinking about. The idea was developed by him and others but as far as I know it is not a particularly useful concept. It is extremely cool though.

Answer 2

George Bergman describes a construction (parts of which he leaves as an excercise) of a number system where the digits can extend infinitely to the right and left of the decimal point in his book An Invitation to General Algebra and Universal Constructions. The book is available for free online as a pdf here. The construction starts in chapter 7, about halfway down page 240. This construction uses some category theory (inverse limits and the like).

He describes it as a simultaneous generalization of the $p$-adics and the reals base $p$, and has an exercise to prove that both the above fields embed as dense subsets of this construction.