Intuition for the p-adic metric
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- เผยแพร่เมื่อ 9 เม.ย. 2021
- There's intuition behind the p-adic metric??? No way!
Second video with more on distance metrics here: • Ostrowski's Theorem (p...
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Yes, the BEST way to start a Sunday morning!
1y later, watching on a Sunday. Still true
it’s 6 am on a thursday for me.
im trying to do research with p-adic numbers, this is super helpful
This was fascinating to watch and easy to understand, I love your videos!
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
Dude, your content is amazing, I'm looking forward to your next video
Yes! Thank you for this! I'm glad somebody's finally doing some good videos on the p-adics!
Thanks! I’m still very new at these p-adics, your intro is gentle enough I think I can follow you!
I'm so glad to hear that! That's always the goal :)
im looking at the beautiful foliage beyond the glasses
Thank you! I've been waiting all my life for good p-adic explainer videos!
Amazing! Fascinating p-adic numbers
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.
Your videos are awesome. Thank you!
Thanks for making these! Never heard of p-adic numbers before your videos. You're right that intuition is a luxury in math!
Brilliant!
Great job! I felt like a 71 year old 2nd grade student learning arithmetic for the first time. I call it Mad Hatter arithmetic!
This is great and so down to earth! Exciting thank you! Looking forward to other lessons :)
Nice thumbnail :D Happy there's someone like you doing these videos, I like the style.
this is a great video thanks for helping me understand
Woah! This channel is awesome! Just subbed
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.
That sounds pretty close to what I've come up with, which is literally to just round the p-adic numbers to a given amount of bits and see if they match. The direction I came from was very different though, namely I came to it through studying computer integers. Binary numbers of 8, 16, 32, 64, etc bits. When converting from a wider integer to a lower one, all that gets left is the lowest bits that fit in the new type. While this is usually seen as a dramatic change in value when thought of as ℝeal integers, in the context of 2-adic integers, such a truncation is honestly little more than just rounding the number to a certain number of significant digits. The only thing is that the meaning of "most-significant" and "least-significant" are swapped.
Good stuff thanks
One of the digits in 3-adic number could be negative, so corresponding to reading variables from the right to the left
I can barely add and I hate math but this is pretty beautiful
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."
Thank you so much
Thank you very much I enjoyed watching the video
This video was very interesting! I loved your explanations. I really hope you keep making videos.
Questions:
Am I correct in saying that the unit circle at the origin in the p-adic numbers is the set of all rationals that don’t have a multiple of p in the denominator?
Does the unit circle also somehow include all of the infinite numbers that don’t “have a multiple of p in the “denominator””? If so, what does that mean?
I am loving your videos. I am curious, what program do you use for your videos? It’s wonderful.
Thanks man! I do my animations using a program called Manim (it’s made by 3b1b if you’ve seen some of his stuff). For videos like this I just screen record the tablet though :)
0:04 "how ig"
You should explain the image in the thumbnail of the video. I looked at the article linked in the description of the image in the article about p-adic numbers on Wikipedia, and I didn't understand it.
That’s actually been the plan for my next video on p-adics! It’s just taken me a long time getting around to it :)
I wonder how many secrets of quantum and physics are hidden in this numerical encodement
I don’t know why I’m somehow obsessed by p-adic numbers. I would like to understand what it’s used for. For me they don’t represent things in our world (like all the other numbers). It’s just like a game. The next step may be to pile upp digits like a deck of cards and try to define addition, multiplication and so on. That will make no more sense than p-adic numbers.
The major use of p-adic numbers is indirect: it's in analytic number theory. Number theory describes the properties of numbers (and more, but let's keep it low key). Not all properties are easily proven by purely algebraic methods. For example, Lambert's proof for the irrationality of pi employs analytic tools like derivatives and integrals. It's highly constructive, not at all elegant IMO. Still, that proof takes place in the realm of the real numbers. Some theorems in number theory are more easily proven in the field of p-adic numbers - while still applicable to the subset of rational or whole numbers. This is not surprising, since p-adic numbers are constructed on the characteristic of divisibility.
The p-adic numbers that are not rational, the infinitely ongoing series to the left, are just as "unreal" as the "real numbers" which have infinite progression to the right. What makes the "real numbers" more "real" or more realistic, or more intuitive, is that the metric is more intuitive. We can '"see" the real numbers filling the gap between the rational numbers. The p-adic metric is not intuitive: we don't think of 4 and 5 to be further apart than 4 and 8, that is 2-adically. We don't "see" 16 falling in the middle of 4 and 8, let alone how an infinite progression to the left would fill the gap in between the powers of 2.
I made my masters thesis on the p-adic gamma function - and I still have a hard time grasping p-adics intuitively. Which is why I return to videos like these :)
what kind of camera did you use to record your video (nice work).
Question - we know the 'right' hand digits of a 'infinite' number, and we can say there's a difference in what we can see - but how do we know in the hundredth trillionth term off to the left there is a bigger number than differs?
Noob question. Is there a possibility of the number that you what to measure be a prime?
So is size all about the number of zeroes in the right most part? So the non-zero digits don't matter for the size?
Does ostrowskis thm care that it's 1/x would any decreasing positive function be fine?
Trying to create a new distance measure based on surreal birth ordering. The nth surreal number can be ordered by birth between 0 and 2 with:
(2n+1)/2^floor(log₂x)-2
Resulting in a linear ordering of the surreal numbers by value.
All variable being positive integers.
Distances obey:
D(a,b)
The "distance" between two integers, under this order, will be as close as their left matching binary bit pattern... regardless of their size.
The distance value of the nth number being the nth odd over it's binary size.
Enumerating the nodes of a binary tree from left to right going from top to bottom. As we write on a page. If the node numbers were transposed onto a horizontal axis, then the numbering order would match this.
ok google, set reminder to watch this video 2 more times.... X_X you do explain it quite well btw
Wouldn't it be more correct to tslk about "p-adic distance" instead of "p-adic numbers"?
Why p-addic numbers works only for p being prime?
I think they only form a field if p is prime, but you can probably still define multiplication and addition and subtraction for them for any integer p > 1 ?
think of it like rooms. a 2-adic system is a system where all the numbers fit into 2 rooms with 2 rooms inside them with 2 rooms inside them and so on. if you propose a 4- adic system, that is just a 2-adic system without the 2 biggest rooms. it's like simplifying fractions, if a p-adic is not a prime it's not the simplest form of that p-adic.
so what is D(4-4) = D(0)?
Precisely. One requirement of a distance metric is that the distance from one member of a group to itself is 0 (I talk about this in my video on Ostrowski’s Theorem).
If I had you as a tutor or friend in undergraduate, I'd have become a math major. Because explanations don't come any clearer than yours.
3:03 is confusing because you're supposed to be contrasting the two systems but you misspoke
You'd only get better at this. Make more.
Yo mathematicians getting hot af what the hell is going on here
Ayo 🤨📸
are you a programmer?
I need the number of your dentist