What are good some good analysis texts on p-expansions for real numbers?

Question

I'm self-studying real analysis, And I want some good references to study the p-expansions of real numbers.

Answer 1

Real numbers do not have $p$-adic expansions. If you’re asking for $p$-adic expansions of rational numbers, that’s fine, but the real number $\pi$, for instance, is meaningless in any $p$-adic context.

Even the innocuous number $e$, for instance, can not be given any sensible definition $p$-adically. The series for $\exp(x)$ is not convergent at $1$, so you may ask, “Well, why not the value such that $\log(e)=1$, please?” But that’s no good. The series $\log(1+x)=x-x^2/2+x^3-3-\cdots$ has a rather large domain of convergence, but you easily see that setting it equal to $1$ gives you infinitely many roots, and among these, no single one can be discerned as preferable to all the others.

So my recommendatrion to you is to follow the advice of commenter @DaveL.Renfro and look into the wonderful book by Gouvêa.