p-adic numbers are actually quite simple; so simple, I've actually seen children invent it on their own!! They're just like ordinary decimal notation numbers, e

p-adic numbers are actually quite simple; so simple, I've actually seen children invent it on their own!! They're just like ordinary decimal notation numbers, except instead of allowing these to extend infinitely to the right (as in 3.14159...), you allow them to extend infinitely to the left. So, in addition to familiar numbers like 45 or 67.8, you also have numbers like ...95141.3 or ...999 or what have you. You add and multiply these according to the same rules you're already familiar with. So, for example, if you add 1 to ...999, the last digit of the output is 0, and you get a carry. Making the second to last digit 0, with another carry. And so on and so on, making the result ...0000 over all. Thus, ...999 acts like -1. (If we were working in base two, this last example would be just like the "two's complement" you are perhaps familiar with from computer arithmetic!) In fact, most discussion of p-adic numbers isn't done in base ten. Instead, people typically focus on p-adics in a prime base (hence the p). Why? Because in a prime base, you will find that every nonzero p-adic number has a multiplicative inverse, which is very convenient (while in a composite base, you will find that sometimes nonzero numbers multiply to zero). But there's nothing actually stopping you from making use of the notion for non-prime base, should you be interested in doing so; the notion is still perfectly coherent. (That having been said, another reason mathematicians focus on p-adics in prime bases is that the Chinese Remainder Theorem essentially allows one to reduce the study of all other bases to this case.) There's a lot of beautiful further theory to explore here, but again, the basic idea is hopefully quite simple. Let me know if you have any questions and I'll be happy to try explaining further or more clearly.