#p-adic numbers

[186-3]

What's your favorite real and solid number?

you fool, you asked me a math question

but you lucked out, because the vast majority of real numbers are gross and slimy. anything that is not a rational number must be banished into the pit. thus, you have limited me to talking about countably many numbers, instead of the uncountably infinite mess that is the real numbers.

so now, we talk about solid numbers. according to the first google result, a solid number is a number which can be expressed as the product of three natural numbers (positive integers). but that can be literally any positive integer, so i’m just going to assume you asked me for my favourite integer.

this is a question that has plagued me for much of my life. as a math major, your favourite number is your respite. it is the number to which you turn when you’re at your lowest, the number which props you up when you’re going down, the number that invites you into the warmth of its home when the rest of the world wants you out on the streets. you are soul-bonded with it. it is yours, and you are its.

and that is why i cannot reveal my favourite number. it is mine, and mine alone to revel in. to share it would be to give you the key to my life. it is too much power for one person to hold over another.

as you can probably guess from the schpiel at the start of this post, my favourite number is an integer, or as you said, a real and solid number. thus, in order to conceal my greatest weakness, i shall tell you my favourite number which is not an integer. but it is not a real number either. it’s not even a complex number. this number is the square root of -1 in Q5, which is approximately …31212 in base 5. it’s cool. trust me.

[numb3rth30ry]

Bruhat-Tits Trees.

Stop snickering. Mathematics is serious. 

jk/also,

Mathematics is beautiful. <3

[isomorphismes]

Fractions and whole numbers break down into repeated × and ÷ by prime numbers. 2's exponent in this prime decomposition, is the number’s 2-adic valuation.

Paul VanKoughnett

[numb3rth30ry]

Sierpinski in the Sky with Diamonds.

Peter Scholze is a Fields Medalist. His work centers around the study of perfectoid spaces, his novel and breakthrough discovery.

Please know: I do not understand this in the slightest. I just know that p-adic numbers are astonishingly beautiful. 

Mathematics is beautiful. <3

[forgottenbones]

1 Billion is Tiny in an Alternate Universe: Introduction to p-adic Numbers

[phenoct]

This is a short summary of what I recently learned about p-adics. Throughout this piece, $p$ is a prime.

[twocubes]

(This is a question about this post I wrote a while ago)

Well, the thing I did is indeed wrong in the real numbers. But! It turns out that the rational numbers don't only live in the real numbers, they also live in the p-adic numbers for any given prime p, which are numbers that emerge from a different notion of convergence on the rational numbers than the one that gives rise to the reals.

Informally, a sequence of rational numbers "converges p-adically" if for any n, there is some point onwards from which pⁿ divides all the differences between the terms. This means, among other things, that we can let numbers go on forever "in the wrong direction".

So, while normally you might write an expression like 1.121212...₃ and it would make sense as a real number in base 3 (the limit of the sequence of rational numbers 1₃, 1.1₃, 1.12₃, 1.121₃...), in the 3-adic numbers you might write ...222121.1₃ and it would be a perfectly valid 3-adic number (the "p-adic limit" of the sequence of rational numbers 0.1₃, 1.1₃, 21.1₃, 121.1₃,... ).

The weird thing then is that things can cancel "in the wrong direction". For example since 2₃ + 1 = 10₃, 223 + 1 = 100₃, 222₃ + 1 = 1000₃ and so on and so forth, we have ...2222₃ + 1 = ...0000₃ = 0, and so ...2222₃ = -1. (This implies that ...1111₃ = -1/2).

The full story, without all the hand-waving, is kind of long and incredibly fascinating, and I'm not remotely qualified to tell it. (In fact, I'm kind of worried that I might have done something wrong already, since I haven't actually officially learned any of this yet. I haven't even finished the book I'm about to recommend!) So I'll just drop a link to the p-adic numbers article on wikipedia and recommend "p-adic Numbers, An Introduction" by Fernando Quadros Gouvea (from which I lifted the trick in that post anyways :¬P) because it is a nice book.

The thing to understand is that strictly speaking, when writing a limit each of whose terms is rational, it is ambiguous which notion of convergence we should be using. It turns out that while the series 1+3+3²+3³+... = ...1111₃ diverges in the reals, it converges to -1/2 in the 3-adic numbers, and so we can use that fact to solve the problem from before. (Although, of course, it's kind of silly to do so this way...)

The usual development goes from ℕ to ℚ to ℝ to ℂ. Here, we have glimpsed a whole bunch of number systems that contain ℚ but aren't in ℝ. So, it turns out, there was a fork in the road there. This is what happens when you go the other way .‿.

[twocubes]

How to solve x = 1 + 3x

Write it down x = 1 + 3x

Replace x by 1 + 3x on the right x = 1 + 3(1 + 3x) = 1 + 3 + 3²x

Do this again and again forever

x = 1 + 3 + 3²(1 + 3x) = 1 + 3 + 3² + 3³x

x = 1 + 3 + 3² + 3³(1 + 3x) = 1 + 3 + 3² + 3³ + 3⁴x

…

x = 1 + 3 + 3² + 3³ + 3⁴ + 3⁵ + 3⁶ + 3⁷ + … Now, at this point your teacher will say that you have done something wrong; after all, the sequence 1, 3, 3², 3³, … obviously diverges to infinity, so its sum must diverge to infinity even more, right? But we don’t actually care what they say.

This is a geometric series with ratio 3. As you recall, the sum of an infinite geometric series with common ratio r is 1/(1-r). We apply this formula to get the solution: x = 1/(1 - 3) = -½ If we check, we do have 1 + 3x = 1 + 3(-½) = -½ = x, as required. ■