$\cdots 2222222222222222222222.0 \div \cdots 1111111111111111111111.0$

Yes, there is a theory of such numbers; these numbers are called 10-adic numbers, denoted $\mathbb{Z}_{10}$, and in the $10$-adic numbers the following manipulations are valid: we have

$$\dots 111 = \sum_{n=0}^{\infty} 10^n = \frac{1}{1 - 10} = - \frac{1}{9}$$

and

$$\dots 222 = \sum_{n=0}^{\infty} 2 \cdot 10^n = \frac{2}{1 - 10} = - \frac{2}{9}$$

and dividing them gives $2$ as expected. A related funny identity is

$$\dots 999 = \sum_{n=0}^{\infty} 9 \cdot 10^n = \frac{9}{1 - 10} = -1$$

the idea being that if you add $\dots 999$ and $1$ then you get $\dots 000$! The key feature of the $10$-adic numbers that makes all of this work is that large powers of $10$ are regarded as "small," and in particular there is a topology on the $10$-adic numbers with respect to which the series above converge.

The $10$-adic numbers have funny properties, though, the main one being that it's not true that if $ab = 0$ then either $a = 0$ or $b = 0$ (although the divisions we did above turn out to be fine). Here's a recent math.SE question where this sort of thing came up.

Via the Chinese remainder theorem $10$-adic numbers can be understood as pairs consisting of a $2$-adic number and a $5$-adic number, which are defined in the same way except that we work in base $2$ and base $5$ respectively. Because these bases are prime the resulting numbers turn out to be better behaved and mathematicians work almost exclusively with these.

A nice exercise is to show that for every prime $p$ other than $2$ or $5$ there is a $10$-adic number that deserves to be called $\frac{1}{p}$ in the sense that when you multiply by $p$ you get $1$; for example,

$$\frac{1}{7} = \dots \overline{857142}857143.$$

A harder exercise is to show that there exists a $10$-adic number that deserves to be called $\sqrt{41}$ in the sense that when you square it you get $41$ (actually there are four of them, rather than the expected two), and an even harder exercise is to determine exactly which square roots exist.

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