Intuition for the p-adic metric

There's intuition behind the p-adic metric??? No way!
► Music Credits:
● Music By: 'Homie Cat'
● Track Name: 'Pure Heart'
License for commercial use: Creative Commons Attribution 4.0 Unported "Share Alike" (CC BY-SA 4.0) License.

[ComplexVariables]

Yes, the BEST way to start a Sunday morning!

[angeldude]

There's a very simple way to count the matching digits, at least for 2-adics: take the number's negative and AND it with the original number. Then you just use a lookup table or tree-like structure to narrow down which power of 2 the result is. Actually, you often won't even need to do this since most computers have a dedicated instruction just for finding the 2-adic valuation of an integer, often by the name of ctz (or "count trailing zeros"). Alternatively, don't even bother with the lookup table and just take the reciprocal immediately, if you can represent it anyways; no need to take the logarithm if you're just going to exponentiate it immediately afterwards.

Thinking in terms of computer integers actually helped me understand the p-adic metric, since with only 8 bits, 256 is indistinguishable from 0. In terms of modular arithmetic, you'd say the two numbers are congruent modulo 256. In terms of 2-adic arithmetic, you could say that the difference between 256 and 0 is a rounding error with only 8 known bits. If two numbers can round into each other, then they have to be pretty close. It does give the amusing consequence of flipping the concept of "most/least-significant digit/bit."

[edwarddahl]

Hi SuperScript, thanks for the video. I'd like to point out that there is a natural meaning to the distance between p-adic integers. Take p=2 and consider the sequence of rings which are given by modding Z by 2**n for different n. The first few rings in this sequence are {0}, {0, 1}, {0, 1, 2, 3}, ... Now note that each of these rings is just the set of endomorphisms of an abelian group. For example the ring consisting of {0, 1, 2, 3} is exactly the endomorphism ring of the abelian group consisting of the fractions {0, 1/4, 1/2, 3/4} where addition is modulo one. The important point is that two endomorphisms in this ring are "close together" if their action on more elements of this abelian group is identical. This corresponds exactly to the notion of closeness for p-adic integers. From this point of view, p-adic integers do not measure the size of a set. Instead, p-adic integers are labels of transformations of a set. And closeness for p-adic integers means that the two transformations agree at many points.

I'm a physicist but I became interested in p-adic numbers about 15 years ago. I am familiar with the usual pedagogical approach - defining them as completions of Q under this novel metric. But these approaches completely ignore a central fact about p-adics: they are closely related to endomorphism rings of what is called the Prufer group, also know as the quasicyclic group. And my opinion is that by understanding p-adics from this different perspective, the meaning behind the metric becomes quite clear. Likewise, the fact that p-adic numbers fit naturally on a tree is closely connected with their structure as transformations.

[ezhanyan]

i'm not a stem major or anything, and videos like this one here are a perfect way to spark my interest in a subject. this is great, you earned yourself a sub

[aaronh]

This was fascinating to watch and easy to understand, I love your videos!

[imauz]

im trying to do research with p-adic numbers, this is super helpful

[yslars]

Yes! Thank you for this! I'm glad somebody's finally doing some good videos on the p-adics!

[brazenzebra]

Great job! I felt like a 71 year old 2nd grade student learning arithmetic for the first time. I call it Mad Hatter arithmetic!

[snotgarden]

Thanks! I’m still very new at these p-adics, your intro is gentle enough I think I can follow you!

[Dr.Cassio_Esteves]

Dude, your content is amazing, I'm looking forward to your next video

[bendavis]

Thanks for making these! Never heard of p-adic numbers before your videos. You're right that intuition is a luxury in math!

[ScottBlomquist]

Thank you! I've been waiting all my life for good p-adic explainer videos!

[etta]

Nice thumbnail :D Happy there's someone like you doing these videos, I like the style.

[Dhrumeel]

I love the way you explain things! This is the first of your videos I've come across and I'll be watching all the other ones. I get the feeling that I'll not only learn math from you, but also pick up some tips on clear exposition.

[Epoch]

I can barely add and I hate math but this is pretty beautiful

[TheJoyLoveShow]

This is great and so down to earth! Exciting thank you! Looking forward to other lessons :)

[xeessdu]

im looking at the beautiful foliage beyond the glasses

[coocavender]

I wonder how many secrets of quantum and physics are hidden in this numerical encodement

[jacobpaniagua]

Woah! This channel is awesome! Just subbed

[alexanderten]

One of the digits in 3-adic number could be negative, so corresponding to reading variables from the right to the left