Reversing a number of infinitely many digits

Question

Lets say we have a function that gets as input a real number and returns its reverse

e.g. 123.12 -> 21.321

So what happens when the input is a number α that has infinitely many digits. Does then reverse(α) exist as a number? Is this number well defined?

Answer 1

If the input is real numbers, then no. There is no real number which has an integral part with infinitely many digits.

But it's worse, do you map $\frac12$ to $5$ or the $\ldots9994$?

Answer 2

There is an idea like that, called the $p$-adics. The $p$ is because it works better when the base is prime instead of ten.