TITLE: Generalisation of integers for infinite length?
QUESTION [31 upvotes]: As most people know, an integer can only be finite length when expressed in the form of a series of digits in some base. However, real numbers in general can be infinite length, so long as there is a "first digit".
That is, if we express an "infinite integer" in the form
$$
\sum_{n=0}^\infty a_n 10^n
$$
where $a_n\in\mathbb{Z}_{10}$, then addition and multiplication should both be well-defined on this sequence. Indeed, I suspect that such a system may actually form a field, as additive and multiplicative inverses should exist (consider that $\bar{9}+1=0$ where $\bar{9}$ is the number for which $a_n=9$ for all $n$, and $\overline{285714}3\times7=1$, where $\overline{285714}3$ has $a_0=3$ and then $a_{6n+1}=4$, $a_{6n+2}=1$, and so on).
Has such a set with addition and multiplication been investigated? Does it have a name? Does it actually form a field, or does it fail one of the required properties of a field? Is the system functionally the same irrespective of chosen base, or do different bases change some properties of the system?

REPLY [6 votes]: Another fun way that the rings differ depending on the base used is that in base 5, $-1$ (i.e. $\bar{4}$) has a square root while $2$ doesn't, but in base 7 it's the other way around (and $-1$ is of course $\bar{6}$ in that base).
I should note that I haven't proved those, but my messing around with the idea pointed strongly that way. If you like programming it's a fun exercise to write up a class that implements these numbers. I did it in Python, as its yield/generator system made it easy to model these numbers as a potentially infinite list that produced the digits from units place up.