Introduction to p-adic numbers

Question

I am a freshman and for a final project of a subject I have to give an introduction to p-adic numbers, I look for some sources (books, videos, articles) to be able to do my work, the only bases I have is a very basic "number theory" (construction of $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, through equivalence relations and the construction of $\mathbb{R}$ by Dedekind cuts. And dense sets in $\mathbb{R}$ and $\mathbb{R}^n$), also basic things of modular arithmetic and diophantic equations and more things that are seen in the first year (calculus I, basic set theory etc.)

Could you give me some recommendations?

Thanks in advance.

Answer 1

If you're interested in video format content, Richard Borcherds (a prominent algebraist, if you're not familiar) has a three part video series on the $p$-adic numbers which is aimed at advanced high school students (https://www.youtube.com/watch?v=VTtBDSWR1Ac).

Answer 2

It's going to be pretty difficult to find a source appropriate for your level, so I really just recommend checking out lots of graduate textbooks on the subject and trying to take away the key ideas and calculations. One good text that I can think of is "p-adic Numbers, p-adic Analysis, and Zeta-Functions" by Neal Koblitz.

Answer 3

I first learned about $p$-adic numbers from (the first chapter of) W. Schikhof's Ultrametric Calculus, I was in my second semester with only very basic undergraduate knowledge in calculus and linear algebra back then, and I think I was lucky there. If you like R. Borcherds' videos in Noah Solomon's answer, this is doubly recommended because just like that, it starts with infinite digit representations. (Later one kinds of needs to forget that, but I still think it is a very good way to first learn about these beauties.)

On a personal note, I think the moment $p$-adics got me hooked was when I realised that something I had encounterd in high school suddenly made rigorous sense. Namely, little Torsten had realised that if you look at factorials of higher and higher numbers ($10!, 15!, 17!, 27!, ...$) they always end in longer and longer strings of zeroes. More precisely, he convinced himself that once some factorial $n!$ ends in a certain number of zeroes, then the factorials $m!$ of all $m \ge n$ must end in at least as many zeroes, and eventually more. The smartass that he was, he thought of that as

$$\infty ! = 0$$

and was very pleased with himself. Then he went to university, learned rigorous $\epsilon$-$\delta$ calculus and laughed about his younger self. Then he learned about $p$-adics, and after a while ... well, long story short, here he is on a math forum, writing

$$\text{$p$-adic} \lim_{n\to \infty} n! = 0 \quad \text{ (for any prime $p$)}$$

and again is kind of pleased with himself.