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

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.

• Maybe you can think of Hyperreal numbers? See en.wikipedia.org/wiki/Hyperreal_number for a (brief) introduction. – AlexR Aug 21 '13 at 17:56
• Maybe start with something simple: how would you convert such a number from decimal to binary for example? – ithisa Sep 12 '13 at 23:03

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.