Can you model p-adic numbers computationally?

Question

Please forgive if the question is not perfectly formulated. The general notion is, is there some way to model - build a representation of - the p-adic numbers, in computer code? For example, could we generate at least a finite number of them, somehow? Is it possible in any way to index them?

Answer 1

Yes. $p-$adic numbers can be seen as a series $$x=\sum _{k=0}^{\infty}x_i\cdot p^i,$$ where $x_i\in \{ 0,1,\cdots ,p-1 \}.$

So, you can think of a number as a sequence of finite sequence in $\mathbb{Z}/p\mathbb{Z}.$

For example, if $x=10$ and $p=2$ base $2$ 0101 gives the number. Check that $-1 =(p-1)+(p-1)\cdot p+\cdots =\frac{p-1}{1-p}$ at least formally.

Answer 2

In the spirit of Turing's paper ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGS PROBLEM we represent real numbers (infinite entities in general) as limits of sequences of rationals. We restrict the sequences of rationals to sequences defined by a recursive function and have a modulus of convergence defined bu a recursive function. The same idea holds for $p$-adics as they are limits of sequences of rationals. We have of course complexity consideration in the above spirit. The computable $p$-adic numbers and the complexity thereof was the subject of my Ph.D. dissertation.