P-adics

When I was young, learning math(s), I would often wonder what more "advanced" mathematics (like my older brother and sister were taking) could be about.  At a young age, I remember actually coming to the conclusion that you learn to add, subtract, and multiply, with really big numbers!  I had no clue there could be so much more to it.

Maybe because of those early imaginings, some years ago I started playing around with a mathematical idea that's not dissimilar to doing basic operations with big numbers.  (Maths are great for thinking about when you're drifting off to sleep, since they can consume your whole brain and there's no room for the troubles of the day.)  I got pretty far with it, I think, and told a few people that I thought I discovered something, before coming to the conclusion that nah, it was all bunkum.  Here's a social media post I had started composing to talk about it (never posted), right after I gave up on it:

Disclaimer: I’m not a mathematician, but I’ve been playing with some back-of-the-envelope sort of stuff (actually done mostly while lying in bed at night, drifting off to sleep) and have made a tiny bit of progress on some (I’m sure) unprofound mathematics that probably wouldn’t have been a challenge for most of you.  But it’s been a lot of fun!  So I thought I’d write it up here.  And it all started with a misconception…

I began considering digit sequences where you know the low-order digits but not the high order digits -- for example, “...20445089241” is just some number that ends with those digits, but the “...” means that we don’t know how the number begins.  The misconception was, I began to consider these numbers as infinite -- that is, the “...” goes on forever.  So these numbers are all effectively infinity, but they just… end differently.  Knowing the final digits was enough to give them different properties -- for instance, divisibility.  For example, a number could be infinite but still may or may not be divisible by 5, depending on whether the last digit was a 0 or a 5.

(I’m sure you can all see the fallacy here, that any such infinite digit sequence is equal to any other, and there’s no divisibility difference between them, but pondering the possibility was enough to keep me going.)

So, treating them as a class of numbers, I began to ask the obvious next questions about operations and closures.  Adding two such numbers is trivial -- e.g. “...1230734 + ...6534221 = ...7764955”.  Subtracting two such numbers was doable, but not knowing which was “larger” meant it yielded two answers (A-B and -(B-A)).  Multiplying was trickier, but I managed to figure out the algorithm for generating the result starting with the least significant digit.  Division was a bit trickier still but I was still able to figure it out eventually (requiring an assumption that it divides evenly -- you don’t really know unless you know the higher order digits).  From the division algorithm I devised a way to figure out the square root of one of these numbers (again starting from least significant digits).

Then I had the odd thought, since these numbers were infinite, but I could still apply math operations to them, then could I find a number A that is its own square root (i.e. A*A=A, excluding 0 and 1)?  Turns out I could!  For instance, “...141376 * ...141376 = ...141376”.  This got me excited.  But a later Google search revealed that I had just rediscovered automorphic numbers!

It wasn’t until I started trying to devise a way to convert these numbers into different bases that it fully dawned on me that my premise was flawed -- all these infinite numbers were equivalent and didn’t have any unique properties; they’re just different representations of the same thing.  But all the methods for basic operations still work for finite numbers where you just don’t know the high-order digits of the number, so that’s pretty cool, to me at least.  Thanks for reading!

It was years ago that I basically stopped thinking about the idea.  But then recently, I saw a passing mention that p-adic numbers were numbers that go off infinitely to the left, and my ears pricked up.  So I went off and watched this video and it immediately clicked.  It really feels like I was close to (RE)discovering them!  I was wrong in dismissing them as "just" infinite numbers, and I shouldn't have just given up.  There were still a few places where I would have had to make a logical leap that I don't think I could have.

It turns out p-adic numbers were discovered in the late 18th century, and have been expanded by some top minds ever since, so I won't ever feel bad that I didn't flush out the whole system.  On the contrary, I feel quite exhilarated at the discovery and will do a bit of further play with them.  And I'll be putting p-adic numbers into my personal keep chest of things I (re)discovered, along with chart parsing and cyclical cellular automata.  Even if I blundered on to familiar territory, it still makes part of wish I had pursued mathematics more academically -- I would have had no chance at greatness I think, but there's millions of mathematicians who pursue it out of love and curiosity, and just to be part of the search for fundamental truths, which I think is noble.