#p-adic

[numb3rth30ry]

You still taking to go orders?

Cool lemme get uhhh (a)&(b) combo?

yeah imma need like, 1/2 uhh... that derivative, uhh 3-piece of that Cantor 3-adic McFractalFood meal

and a side of mint bbq pepsi abstraction

[curiosamathematica]

An Infinite Universe of Number Systems

The p-adics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory.

[tactile-vedic-math]

Mathematicians welcome computer-assisted proof in ‘grand unification’ theory

Proof-assistant software handles an abstract concept at the cutting edge of research, revealing a bigger role for software in mathematics.

[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]

Bruhat-Tits Trees.

Stop snickering. Mathematics is serious. 

jk/also,

Mathematics is beautiful. <3

[momoguido]

Tonight for bedtime watching, I watched in bemused disbelief as two vocaloids gave a very gentle, child-like introduction to p-adic number systems.

https://youtu.be/HlA5NV-Md_w?si=8HF-8aGtvwohClKB

[mantacid]

I learned about p-adic numbers recently.

Imagine if you will, a base 3 number (meaning the number has ones, threes, nines, … places instead of ones, tens, hundreds…) that’s all 2s but extending to the left of the decimal point.

x = …2222222222222

Adding 1 to this — making sure to carry if the sum in any place value gives three or more (because it’s in base 3 not base 10) — gives…

x + 1 = y

… 1 1 1 1 1 1 1 1

… 2 2 2 2 2 2 2 2 2

+ 1

———————————————————————

… 0 0 0 0 0 0 0 0 0

y = 0

… zero. Just zero.

So we have a massive number, basically infinity, that when one is added to it, becomes zero. This means that that number is actually -1 (since if x+1= 0, subtracting 1 from both sides gives x = -1)

I feel like this has something to do with telescoping sums, but it may take me awhile to put that intuition into more formal language.

[isomorphismes]

The 3-adic integers, with selected corresponding characters on their Pontryagin dual group