Reddit mentions of p-adic Numbers: An Introduction (Universitext)

There's so much I want to say, but I have to go to bed. For now let me leave you with these:

This is a great book. It's probably the most accessible book on this subject that you'll find.

For a quicker read that still gives some motivation for these things, there's this paper written by a (then) undergraduate.

Finally, while I don't find the visuals extremely enlightening, this has pretty much everything you could ever want to know on that subject.

I like this book http://www.amazon.com/p-adic-Numbers-An-Introduction-Universitext/dp/3540629114

You should look at all the books on p-adic numbers in your library and find one you like.

We used this https://www.amazon.com/p-adic-Numbers-Fernando-Quadros-Gouvea/dp/3540629114/ref=mp_s_a_1_1 in a seminar.

Hi, a similar question was asked a couple days ago. I recommend reading GOD_Over_Djinn's excellent explanation here: http://www.reddit.com/r/math/comments/1h2i9v/playing_around_with_an_idea_related_to_prime/caqgyd5 or my own comment here: http://www.reddit.com/r/math/comments/1h2i9v/playing_around_with_an_idea_related_to_prime/caqgh42. The best way to learn about p-adic numbers is of course to read a book about them instead of just looking at wikipedia or reading what random people on the internet have to say. I cannot recommend enough Robert's "A Course in p-adic Analysis" if you have a basic knowledge of topology and analysis http://www.amazon.com/Course-p-adic-Analysis-Graduate-Mathematics/dp/0387986693. If you're more interested in p-adic zeta functions etc. look at Koblitz's "p-adic Numbers, p-adic Analysis and Zeta Functions" http://www.amazon.com/Numbers-Analysis-Zeta-Functions-Graduate-Mathematics/dp/1461270146/ref=sr_1_8?s=books&ie=UTF8&qid=1372366949&sr=1-8&keywords=p-adic+analysis. Although I haven't personally read it this book here also seems to be a more elementary introduction: http://www.amazon.com/p-adic-Numbers-Fernando-Quadros-Gouvea/dp/3540629114/ref=sr_1_3?s=books&ie=UTF8&qid=1372367005&sr=1-3&keywords=p-adic+analysis. The first 2 I know you can find pdfs of online. I don't know about the third. Alternatively, p-adic numbers are covered in a less technical sense in Bartel's notes on number theory here: http://homepages.warwick.ac.uk/~maslan/numthry.php. I haven't looked at them yet but I can say that his notes on representation theory are very good.

Any other construction I can think of aside from what I linked requires group theory or topology so its kinda hard unless you have a background in these subjects.

Edit: Having skimmed through Bartel's notes: they are an excellent introduction to p-adic numbers and he thoroughly covers them and their applications. I do recommend it.

Neal KoblitzIntroduction to Elliptic Curves and Modular Forms is fairly short as far as math books go, though not as short as the others here.

Gouvea's Intro to the p-adics is also not quite as short as what others have suggested, but it felt short to me when I was going through it.

> This seems very confusing to me, as it is defining p-adic expansion of numbers in terms of p-adic numbers ...

This is just a hand-wavy, intuitive explanation of what p-adic numbers look like. The fact is that once you formalize everything about the [p-adic valuation](http://en.wikipedia.org/wiki/P-adic_valuation) and the p-adic numbers, it turns out that every p-adic number has the series expansion that you mentioned.

> For instance, why, in the p-adic world, are positive powers of p small, and negative powers large? It seems like a prime number to a large power would be large, no?

When dealing with p-adic numbers, you have to forget all your intuition about the usual notions of absolute values and ordering of the real numbers, since they don't apply. Everything in the p-adic world is based on the p-adic valuations, which give their own topologies and notions of size. The p-adic topologies are very different from the topology on R. For example, any point within an open ball in the p-adic numbers can be considered that ball's center. Quirky things like this make it initially hard to grasp the concepts of p-adic numbers and their associated arithmetic, but once you practice working with them enough, they start to make sense.

> How does the limit of the sequence that they're talking about equal 1/3?

This again has to do with the fact that convergence in the p-adic topology is different from convergence in the usual Euclidean topology.

Some good resources for learning more about p-adic numbers are the following:

1. Gouvêa, Fernando Quadros, p-adic Numbers: An Introduction (Amazon, SpringerLink)
   
2. Koblitz, Neal, p-adic Numbers, p-adic Analysis, and Zeta-Functions (Amazon, SpringerLink)
   
3. Robert, Alain M, A Course in p-adic Analysis (Amazon, SpringerLink)
   
4. Serre, Jean-Pierre, A Course in Arithmetic (Amazon, SpringerLink)
   
   For me personally, learning general valuation theory was very useful for understanding p-adic numbers.