1 Billion is Tiny in an Alternate Universe: Introduction to p-adic Numbers
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- čas přidán 1. 05. 2024
- The p-adic numbers are bizarre alternative number systems that are extremely useful in number theory. They arise by changing our notion of what it means for a number to be large. As a real number, 1 billion is huge. But as a 10-adic number, it is tiny! #SoME2
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Notes and references:
The last 30 digits of 2^1000000 and other large powers can be computed using modular arithmetic, by working modulo 10^30. In Mathematica, use the function PowerMod. In Python, use the third argument of pow. These functions implement the method of repeated squaring or one of its variants:
en.wikipedia.org/wiki/Exponen...
en.wikipedia.org/wiki/Modular...
Bézout's identity can be used to prove that the numbers from 2 to p-2 pair up perfectly, and the partner of a given number can be computed using the extended Euclidean algorithm:
en.wikipedia.org/wiki/Bézout%...
en.wikipedia.org/wiki/Extende...
The 2-adic limits arising from the (2^n)th Fibonacci numbers were established on page 216 of this paper:
Eric Rowland and Reem Yassawi, p-adic asymptotic properties of constant-recursive sequences, Indagationes Mathematicae 28 (2017) 205-220.
doi.org/10.1016/j.indag.2016....
Hensel's lemma gives conditions for Newton's method to work in the p-adic numbers:
en.wikipedia.org/wiki/Hensel%...
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0:00 Introduction
2:16 Properties of the real numbers
3:19 10-adic integers
6:55 Properties of the 10-adic integers
10:06 Division?
12:47 Limit points
13:50 5-adic limit
15:36 Fibonacci numbers
16:31 Square roots of -1
18:25 What are p-adics good for?
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Animated with Manim. www.manim.community
Music by Marc Rowland and Cody Leavitt.
Thanks to @catpfaff for helpful feedback on an earlier version.
Web site: ericrowland.github.io
Twitter: / ericrowland
Great!
Let me just add a remark. Eric already mentioned that the field of p-adic numbers provides a completion of the rational numbers (it contains the rational numbers, it extends the usual addition and multiplication on rational numers, and there are no holes : ) ). He also mentioned that this completion is based on a different version of an absolute value, the p-adic absolute value, where p is a prime. A theorem by Ostrowski from 1916 says that there are no other ways to assign absolute value on rational numbers, more precisely, every nontrivial absolute value defined on rational numbers is equivalent to either the usual one, leading to the notion of real numbers, or to a p-adic absolute value, for some prime p, leading to the notion of p-adic numbers. Thus, the p-adic numbers appear as natural objects, on par with the real numbers.
I knew you as a little boy, as happy as happy is, and no surprise that you are where you're at. Your entire family is so grateful and proud of you in all ways.
I think this ought to be pointed out every time the p-adic numbers are brought up. In my first semester of analysis we had a problem set where we were asked to complete the rationals with respect to the p-adic norm. We also did some stuff with ultrametric inequalities. At the time I just trudged through it, assuming that it was just a different, quirky system that our prof gave us because he was a number theorist. There was no mention at all of Ostrowski's theorem. Had I been shown that the p-adic numbers were so natural I think I would have taken a deeper look. Alas, I just moved on and now I'm more into algebra :)
Is it possible to extend the p-adic numbers for each value of p such that every finite algebraic expression involving a single variable has a solution?
With absolute value you mean norm right?
@@MrAlRats Perhaps you're referring to an algebraic closure.
An algebraic closure of a field F is defined to be a field K containing F satisfying the following:
1. Any element of K is a root of a single-variable polynomial equation with coefficients in F.
2. Any single-variable polynomial equation with coefficients in K (in particular polynomials with coefficients in F) has a solution in K. Any field satisfying this second condition is said to be algebraically closed.
Thus, in particular, K is a field containing F such that every single-variable polynomial has a root in K. In fact, one can show somewhat of a converse: if K is a field containing F such that every single-variable polynomial has a root in K, then K contains an algebraic closure of F.
One can show that given any field F, there exists an algebraic closure of F. Thus, if we denote Qp to be the field of p-adic numbers, then we can say that there exists an algebraic closure of Qp. Interesting enough, any algebraic closure of Qp has a natural norm on it, and the algebraic closure is not complete with respect to the norm. Thus, you may complete it again, and obtain a field denoted Cp. This field is complete, but perhaps it is no longer algebraically closed. However, a proof shows that it is actually algebraically closed.
When i was a fucking child i thought ….99999 should be a number. Then came grade school and it convinced me that I’m wrong. Now this?!?! Lol
I'm flabbergasted to find out this is your first video, since it's so high-quality -- was hoping to binge on more of your content. Thanks for making this one!
There was one channel that released one video that was an amazing maths video that explained things perfectly and with great visuals that went viral, and then they never made a video ever again.
I made one video with respect to another trending hashtag thing a while back (MegaFavNumbers or something), and I got reasonable feedback, but I have no desire to make any other videos. However, I did mine much more bare bones, with no animation. I feel if I spent the time learning how to make a video of this kind of quality I'd feel like I'd need to make more to actually have it make sense to spend the time to learn. It's possible that the animation bits are actually really easy if you get the right software, but I've never explored it.
@@123TeeMee what channel was it
@@badbad6763 this one
@@123TeeMee still waiting for hackerdashery to upload again :(
I choose to believe we've discovered the secret digits of infinity.
This is probably the best explanation I’ve seen done solely because of the visuals. So many online and printed sources go right over people’s heads because they just delve straight into the math without explaining why it’s even interesting or useful to represent numbers this way. I spent countless hours slowly chugging through the math and slowly coming to the realization that they’re much more closely related to modular arithmetic than representations of real numbers in some sense. Thanks so much for making this and truly getting to the heart of what this field of mathematics is about!
Truly remarkable presentation! Very well explained and easy to follow (coloring the numbers helps a lot), I love it.
Thank you so much!
@@ValkyRiver oh hi! didn't expect to find you here :D
I beg to differ. I don't know my way around padic numbers and from my point of view he might as well just be making random things up.
A number is close to zero because it has lots of zeroes in it. Sure.
The lim of some large exponent number is zero. Sure...
Multiplying the number by itself magically yields itself. Sure.
The way he presents things doesn't make sense and he doesn't explain how it would make sense.
Maybe it does if you already know what he's talking about.
Edit: sry for sounding negative. Let's be real, things are complex. Maybe sometimes even p-adic? I don't know
@@mkevilempire to me it makes some sense this way: real numbers can have lots digits after the dot, like pi=3.1415..., but when we 'revert the roles' of the digits before and after the dot, we can think about the numbers which infinitely grow up as if they have some limit in the same sense as 3.1, 3.14, 3.141, 3.1415... sequence has it's limit as pi
So taken literally, 1.3, 41.3, 141.3, 5141.3 'converges' to something like ...5141.3, and just as irrational numbers aren't the part of set of rational numbers 3.1, 3.14, ..., we might come up with a new sort of numbers (...5141.3?) which sort of should not exist because a number can't have infinite amount of digits before the dot, but it's less weird when you remember that
1) something like complex numbers exist for example (and how is 'i' even a number after all? We consider 'i' to be a number because now it's a part of new system we created and we think that this system works) and
2) we revert the meaning of a big and small number for p-adic so that adding digits to the left of the number doesn't matter that much and doesn't make a number grow 'exponentially' (after we changed the meaning of |x| anyway)
I do not agree, I do not understand shit the only thing I understood was that powers of 2 after a while have same last digits and then it all goes down and makes absolutely 0 sense
For some reason I just watched a 21 minute video about a field of mathematics I have never heard of. And I am not even a mathematician!
Keep up the good work, your visual style, animation and presentation style is amazing. Your voice is also very nice to hear.
smae
that's what #SoME2 (and the og #SoME) are all about!
Same
well in four-adic, your misspelling of "same" would be represented as " " I'm guessing. What a farce!
This is like creative interspecial gender identities for people who don't want to do real mathematics@@nou5440
This was an amazingly well done video, you should be proud. I wanna especially shout out the audio quality, your voice was crisp, clear and a consistent volume. Whatever amount of effort you put into your audio setup was well worth it
18:06 answer: sqrt(-1) exists in the p-adic numbers iff p = 1 (mod 4).
It's well-known that for p > 2, sqrt(-1) exists mod p iff p = 1 (mod 4). This is enough to show impossibility for p = 3 (mod 4).
Now an induction proof to show that it indeed exists for p = 1 (mod 4). Assume there exists x^2 = -1 (mod p^k) for some k (the above establishes the base case k=1), then we need to show that there is some digit d between 0 and p-1 such that (d*p^k + x)^2 = -1 (mod p^(k+1)). Well, from the first equation there exists m such that x^2+1 = m*p^k. Expanding out the second equation, we need d such that (2xd+m)p^k is divisible by p^(k+1), in other words 2xd+m = 0 (mod p). But this is a linear congruence mod p where the linear coefficient is nonzero, so it has exactly one solution (note: this is where we need p to be prime). This completes the induction.
Amazing video btw. Really beautifully explained, it feels like I learned something even having studied p-adics before.
Unfortunately this means that it's not possible with p = 2 either, which would've been useful for computing, since the 2's complement integers used are effectively truncated 2-adic integers.
I dont get what the induction shows?
I thought about this too, the 1 or 3 congruence-modulo 4. so my next question is where do the gaussian primes (referring to this fact that 5, though a prime, is still a product of two complex numbers 5=(2+i)(2-i) fit into this p-adic story?
These talks as always , go over my head , it would be very helping if you can describe about your math journey and the approach to get to that level. As an interested learner
When giving the representation of negative numbers, I immediately recognized two's complement. When programming, a negative integer n is represented as the bitwise negation plus 1. So -1 = ...11111111. The main difference is that true 2-adic numbers are unbounded, where as computer integers have a finite size. Regardless, I was able to truncate the sequence given in the video to 8 bits giving (-75)^2 = 181^2 = -7 = 249. Similarly, while rational numbers are usually represented in an equivalent to scientific notation, by using a p-adic method, you can represent in 8-bits 1/3 = 171 = -85, which in binary is written as 1010_1011. Unfortunately 1/10 doesn't seem representable, or at least as a 2-adic integer.
Yep, that's all correct, though arguably saying computer integers have a finite size in the 2-adic integers is a little bit ambiguous. I think the correct term would probably be precision. The 8-bit representation of a 2-adic number would be accurate to the first 8 digits, or accurate to a 2-adic distance of 2^-8. The fraction 1/10, like all rational numbers, is representable in the 2-adic numbers. It is not, however, representable in the 2-adic integers. This is because the denominator of 1/10 is coprime with 2. You could represent 2-adic numbers using an analogue to floating point numbers, however.
But 1/10 can't be precisely represented in binary either right?
@@aniruddhvasishta8334 With a 2-adic encoding, any fraction with an even denominator, such as 10, appears to be impossible. Any fraction with an odd denominator however appears to work perfectly fine.
1/10 is not a 2-adic integer, but it is a 2-adic number. Like how there are decimals (say, 0.2) which are not integers but are real, 1/10 is not a 2-adic integer but it is a 2-adic number. Its representation would be 1/5 shifted to the right by 1 (e.g. …100110.1)
so that's why some calculators have trouble at 0.1 + 0.2, good to know
Whenever I watch math videos, all I remember is a quote by a teacher and I can't remember where in life I heard it.
"Are we ever going to use this in the real world?"
"You won't, but some of the smart kids might."
Your channel is very promising! The inspiration of 3B1B is pretty clear, and this is not a problem by any means. I'm excited to see more of your videos
I love that with real numbers having a highly composite base is desirable, whereas with n-adic numbers, things only come to life when you use a prime base. It makes sense in a weird way though in this reverso-world of numbers
"reverso-world of numbers"
Oooh man! I love this as a name for the study of these weird fields whose models 'behave well' (norms/absolute values being consistent, etc)...
Sounds much better than: An introduction to Abstract Algebra, with applications in p-adic numbers.
very good note
Why is a highly composite base for real numbers desirable?
@@gregoryfenn1462 Mostly it helps with every day maths. It means that when you have a round number of something, there are more subdivisions of that set.
This is why a lot of things are sold in dozens, they can be split in 2, 3, 4 or 6.
prime bases can be useful in the Reals, too. it's mostly bases that are in between, like base-10, that are crappy.
consider for instance how base-7 expansions of fractions would look. they'd virtually all be like how 1/7 looks as a decimal expansion, except for where the denominator is a power of 7. this consistency is worth something. and it's actually that same type of consistency which makes a highly composite base desirable. in base-14,414,400, for instance, you can deal with fractions that have denominators that are powers of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, etc. and come out with a nice, short, expansion. that's some pretty lovely consistency. but still, a fraction with a denominator of 17 will have an expansion that's like 1/7 in decimal, and these are unavoidable regardless of the base. so in a sense, a prime base optimizes consistency by rendering the largest possible number of expansions awkward and infinite.
in this sense base-9 and base-11 would both be better choices than base-10, but for the opposite reason that base-12 and base-8 would be. and still, base-13 and base-7 would be better choices than base-10, and so would base-14 and base-6, or base-15 and base-5. base-10 is truly an astoundingly poor choice. which is probably why everyone who knew how to do math before the Bronze Age Collapse used something other than it, making it painfully obvious that the main reason we use it today is that the powerful nations that arose from the ashes of the Bronze Age Collapse valued wearing shoes more than counting, and thus couldn't easily use the base-20 number systems of pre-Iron Age Europe.
One funny thing that happens with the p-adic numbers is this: when we complete the rationals Q to the real numbers R in the usual way, we find that R is missing some solutions to polynomial equations such as x^2=-1, so we take the algebraic closure C (complex numbers) and we're done - C is both complete and algebraically closed. In the p-adic case, we move from Q to Q_p as explained in this video, and we can then take the algebraic closure of Q_p, but... this algebraic closure is, again, incomplete - it has holes! We complete that field again, and the result is luckily both algebraically closed and complete. But it took one extra step...
Alon Amit himself! Never expected to see you on CZcams.
I'd like to add that this final field is actually isomorphic to the complex numbers! (as a field) Though obviously it does not have the same topological structure.
@@sidechannel5510 if it has the characteristic of a duck, is algebraically closed like a duck, and has the cardinality of a duck…
But can't we construct C_p directly from Q_p in one step? Surely the number of steps depends on the route we take. Right?
@@sidechannel5510 Who is he?
@@MrAlRats I’m not sure what “step” might mean here. I spoke specifically of two kinds of steps: metric completion, and algebraic closure.
I'm sold - reals just lost their job, I'm switching to Competing Brand numbers today!
You managed to touch on so many topics I've been curious about in p-adic arithmetic. I really appreciate the worked examples and motivation. I'm eagerly awaiting a sequel.
Excellent video ! I have encountered p-adics several times, and could never find a satisfying way to visualize them. This video achieves just that, in my opinion. I would gladly watch more videos about some other applications or properties (their peculiar topology, for example).
Thanks! Glad this helped give you a way to picture them!
It's interesting how many programmers and coders this video resonates with
Signed integers are basically just truncated 2-adic integers with the same addition, subtraction, and multiplication as long as you disable overflow checking. 2-adic division on the other hand is pretty different from signed-integer division since the latter truncates to a normal integer. That said, a 2-adic reciprocal operation isn't too hard to do letting you do 2-adic division with normal signed integers and wrapping multiplication. In 8-bits, 181^2 = -75^2 = -7.
It's because it gives rock solid mathematical foundation to that practical trick of negative numbers representation. This very fact brings peace into my inner coder's mind ✌🤤👍
@@igorvoloshin3406 does that mean numbers in a computer have always been p-adic????
@@samuraijosh1595 it's a matter of understanding. When you know what is it, you'll see it where you didn't before. 👍
Remember in my limits and proofs class, I asked about having a number defined by a formula and having infinite left digits cause I thought we have infinite right ones. Was told it didn't converge so there was no number and it was a silly question. Only now that I know of P-adic numbers have I discovered it's wasn't a silly question.
Wow! What a blessing by the algorithm for a first video, the quality is astounding and I can’t wait to learn more from you! Best of luck with your journey
Really well presented and really interesting. I’ve never heard of p-adic numbers. Of the SoME2 videos I have seen so far, this is my favorite one
3b1b mentions it himself in a very old video. It was titled something like "inventing math".
I don’t have any background in mathematics so after first 5 minutes I was just trying to catch the overall feel of what you’re talking about (more or less successfully). But it was so much pleasure listening to someone that is so passionate about anything that I not only made it to the end of the video but I want more!! 😃
I knew about p-adic numbers before, but this is an awesome introduction!
I mostly knew about them as a neat trick, but your introduction motivates them really well!
Eric, thank you so much for this gem, it's a great visual exposition of seriously abstract math topic! This is the only video on your channel, and I *soooo* much hope it's only the first of very, very many!
Man you deserve way more subs, this is such a good and intuitive demonstration. Easily on par with "the greats" in "math youtube"
Thank you for the video! My professor is always joking about p-adic numbers on his lectures and it was nice to finally learn something comprehensive about them ;)
The application at the end is wild. It's very cool that you chose one that so clearly illustrates a result inconceivable in the real numbers and yet simple enough that it is completely clear given the background presented in the video
This is the single most helpful video on p-adic numbers I have ever seen. Bravo! Please make more!
This was an amazing video, well explained, and the end usecase is just purely beautiful. The style reminds me a lot to 3blue1borwn. Keep up the good work. I hope CZcams shows this video to waaaaay more people
Editing style, animations, the topic, this is very reminiscent of 3blue1brown, but keeps it unique! Nice job man! Love this video :)
Exactly! Big 3B1B vibes here.
It's 3B1B animator called manim
@@Henrix1998 oh really? That makes sense then :)
All these new channels are inspired by 3b1b. Before Him, no math channel presented this way. All hail the Great Teacher Who Taught All to Make Cool Maths Videos
I love the #SoME submissions so much. Though I've only seen a few, this one might be among the best so far.
okay your video was beyond fantastic, it was the clearest and most accessible explanation of p-adic numbers I've ever seen, BUT YOU DIDN'T INCLUDE THEIR METRIC SPACE PROPERTIES!! you can define a distance function on the p-adic numbers that is internally consistent and fits the standard definition for a metric on a set and that's part of why they're so cool!! overall great vid tho!!
it would be amazing to watch the video about thier metrics space properties
Yes that is definitely part of why they're so cool! I had to stop somewhere though! =)
@@EricRowland I guess it's forgivable... but only if you make another video about it 😉 great work again, friend, so glad you put it out there!
cant we define size in a "circular" instead of a "linear" fashion? meaning that numbers of the same size are on a ring around zero?
I sometimes visualize this way too, but there are flaws, as a normal circle still has a circular order (a line joined up with itself), but these circles wouldn't.
@@sgcoskey There is no such a thing as a cyclically ordered set. This is because this would contradict the irreflexive property. For instance, it is nonsensical to have a set {0, 1, 2} ordered by 0 < 1, 1 < 2, and 2 < 0, since transitivity implies 0 < 0, which violates irreflexivity. So, there necessarily cannot be such a thing as a "circular order," that is just a contradiction.
@@angelmendez-rivera351 What you say is very true! (There is such a thing as a circular or cyclic order, but it can't be defined this way due to the reasons you point out, and it isn't too relevant to the discussion for me to have brought it up, so, sorry!) What I mean is that the balls in the p-adics don't have the same structure as the balls in R^2 or any real space. One "feature" of the p-adics (as any ultrametric space) that people often point out is: for any ball B, every point x in B is at the center of B! (So "center" isn't a unique item.)
k mod n ? The modulo operation reminds me of your question.
@@zbnmth Yes, you can define a structure of numbers where, for example, 1 + 4 = 0. This is Z mod 5. The intuition is to order the elements as 0 < 1 < 2 < 3 < 4, but this actually fails, because an ordered field must satisfy the axioms that a < b implies a + c < b + c, and yet, 3 < 4 would imply 4 < 0, which means 4 < 4, which is false. So while the algebraic structure is valid, you cannot totally order the elements.
Never thought I'd learn so much about p-adic numbers from a CZcams video.
I really hope this channel blows up. Great content.
Amazing video. I am doing research in number theory and the p-adics pop up from time to time, but I have never seen such a intuitive explanation.
This is beautiful. I thought you had millions of subscribers and I was going to binge all of your videos. And then I realize this is your 1st video. Subscribed, and notifications turned on.
12:15 Of note is that this is hardly the only system with non-zero solutions to cx=0, where c is an arbitrary non-zero constant. Heck, such numbers even have an official name - zero divisors. The example that immediately comes to mind is the dot product on Rⁿ. If a and b are members of Rⁿ and ab = 0, this implies only that a and b are orthogonal to each other (i.e. that cosine of the angle between them is 0), not that one of them must be zero. That's partly why there's no standard version of division defined on Rⁿ. Though, interestingly, the complex numbers can be seen as simply R² together with the complex multiplication rule, and complex multiplication is of course invertible, so there is division there.
curious notion!
This is misconceived. The dot product is not a multiplication, since it is a map from R^n*R^n to R, not a map from R^n*R^n to R^n. A multiplication forms not an inner product space, but an algebra over a field. C is an algebra on R^2 over the field R, where multiplication is defined by (a, b)·(c, d) = (ac - bd, ad + bc). In other words, C is a 2-dimensional R-algebra. Now, if an algebra is unital and associative (which they need not be), then said algebra is actually a ring. So, algebras over vector spaces can sometimes form fields, since a field is a commutative division ring. A K-field, where K itself is a field, is a commutative division algebra, and a division algebra is necessarily both unital and associative.
Also, zero divisors are a general ring-theoretic concept. A left-zero divisor is some d such that there exists some x such that d·x = 0. A right-zero divisor is some d such that there exists some y such that y·d = 0. A two-sided zero divisor (or just zero divisor in short) is both a left-sided divisor and a right-sided divisor.
@@angelmendez-rivera351 That's just semantics though. Fact is, the dot *product* is commonly referred to as a form of vector multiplication, and the definition of words is determined by common usage -- i.e. words mean what we use them to mean. Sure, the dot product doesn't have all the same properties as multiplication on the reals and it may not satisfy the formal definition of multiplication used in abstract algebra, but all that means is that there's more than one notion of what multiplication is.
@@Lucky10279 *That's just semantics though.*
This is a silly objection. All of mathematics is semantics. Semantics are a key component of what makes mathematics what they are.
*Fact is, the dot product is commonly referred to as a form of vector multiplication,...*
...and people commonly incorrectly use the word "theory" to actually refer to a "hypothesis." This observation proves exactly nothing. We are discussing mathematics here. How "most people" use a word, correct or not (usually not correct), is utterly irrelevant.
*...and the definition of words is determined by common usage -- i.e. words mean what we use them to mean.*
In colloquial language? Yes. In academic matters of research? No.
*Sure, the dot product doesn't have all the same properties as multiplication on the reals and it may not satisfy the formal definition of multiplication used in abstract algebra, but all that means is that there's more than one notion of what multiplication is.*
It _could_ mean that, but no, it does not mean that, in this instance. It just means people have historically used language that is inadequate to talk about concepts that were not well-understood until much later on, and that our language today should change to accomodate this new understanding.
8:45 That reminds me of overflow with floating point arithmetic, as well of the two's complement representation of negative binary numbers.
Reminds me of this too!
I'm actually curious if this wasn't the precursor to 2's compliment representation...or rather, the advance that enabled its development.
Two's complement representation is just 2-adic representation rounded, like 3.141 is pi rounded. In other words 2adic is like two's complement with infinite amount of bits.
@@devgumdrop3700 Huh. Interesting.
I hope you keep up making these cuz the first video is remarkable... Can't wait for the next
i am blown away. your first video, the quality is stunning, the content itself is fascinating and delivered really well. and bam, 175 K views. great work!
I'm not very easy to impress, but this one video has genuinely made my jaw drop three separate times. Amazing content, and ammazing presentation too!
Dude, thank you SO MUCH! I'm a... recreational mathematics enthusiast, let's say.. and I've been trying to understand p-adic numbers on and off for a couple years now. The problem is that a lot of the resources on the topic are incredibly dry and dense, akin to a really bad muffin. Or something. While I like proofs and whatnot, when I first learn a new topic I'd rather initially try to understand it on a more intuitive level. I'm not sure that's the right word, but I'm talking about when all the puzzle pieces in your mind finally fit together and you can almost _"feel"_ why something is the way it is on a fundamental level (that makes sense, right?). Anyway, you finally helped make the concept click for me after a really long time so you clearly have a knack (sp?) for teaching!
Well you've got a new subscriber now. Thanks for the help man :)
Awesome explanation!
Looking forward to seeing any next of your videos.
Don't stop!
I HAVE BEEN looking for an explanation of the p-adic numbers THIS IS GREAT THX
A while back I was thinking about Cantor’s diagonalization proof for the uncountability of the real numbers, and I realized that the difference between the set of numbers whose digits only lie to the left of the radix point (integers) and the set of numbers whose digits only lie to the right of the radix point (all real numbers between 0 and 1 inclusive) is that numbers in the latter category are allowed to make use of all infinitely many digits at once, where the ones to the left always terminate after some positive power of 10. The p-adic numbers flip this system completely on its head! I never imagined something like this would make any amount of sense.
I wonder if there’s a system that allows for infinite positive *and* negative powers of the base at the same time? My guess is “no” just because it would be practically impossible to even start any numerical calculation, but maybe I’m wrong here.
If there is such a system, it is not like these. These are formed from the completion of the rationals, which basically means including to the rationals all the infinite sums of rationals where the terms get closer and closer to some number. This is given by the distance formula |x-a|, and it has been proven that even for the very basic properties of the absolute value function, the only metrics that satisfy it are the traditional one and the n-adic ones. This is, I guess, meant to be your beginning to an answer.
Having infinite poxitive and negative powers would make it …99999999.99999999…==0. Then you can divide that by 3 and end up with …33333333.33333333…, etc.
There could be a way if the digits repeat it at least 1 side.
Are there functions consisting of variables and p-adic numbers?
Greg Egan published a short story titled "3-adica" a few years ago, and it fascinated me. The story isn't entirely about p-adic numbers, but it does feature a virtual world which obeys the topology of 3-adic numbers along with some distance analogies that are equal parts interesting and confusing.
Watching your video really helped me fill in some of the gaps in my understanding. It might be time for me to revisit that story :)
Thanks for letting me know about this! I will have to check it out.
Great video! Really intriguing to see the properties of alternative algebras. Would love to see more videos like this. Subscribed!
Had no idea of these concepts, great material you have here. Easy to follow along and beautifully presented. Thanks
This is a truly amazing video. Thank you for sharing it with us! As someone said:"This needs way more views!" :D
Thanks, Luca!
Wow, that's impressive ! You'll surely get my vote in the peer review ! I didn't really heard a lot about p-adic numbers before, thanks for changing that ! :)
Easily the clearest explanation of p-adic numbers I've ever stumbled across. You have a new subscriber!
you mentioned that ...9999=-1. We can view the left-hand sum as 9+90+900+9000... To evaluate this sum, we can use the formula a/(1-r) with a=9, r=10, and we will get -1. You can use this method to calculate a lot of P-adic number's values
This needs way more views!
Th 10-Adic number of views have decreased since the first view 😥
This was excellent, thank you! I hope you find time to make more videos as informative as this one :)
This is awesome! Id love to see more from this channel!
If you saw the end of this video and thought "what? you can do calculus with these guys?" and that piqued your curiosity, I recommend the text "Ultrametric Calculus" by Schikhof. It's a pretty gentle ("advanced" undergraduate level) introduction to the calculus on p-adic numbers. It talks about the usual suspects of calculus, but also goes into comparisons of p-adic numbers with rational, real, and complex numbers, and shows how they relate to one another. It's a very rich subset of analysis and gives a completely different (and yet still quite familiar) flavor of calculus as you know it.
I am more intrigued by the claim at 20:30, that a sequence of rationals which would normally be recognised as a _failure_ of Newton's method is apparently a _success_ in the p-adic world. Can you recommend something about Newton's method (and other numerical methods) applied to p-adic numbers?
If you look up Hensel's lemma, that will give you conditions for Newton's method to work in the p-adic numbers. Questions about numerics turn into questions about congruence modulo powers of p.
@@user-qm4ev6jb7d A lot of the ideas behind numerical methods carry over wholesale to p-adics, because it's really just calculus on a metric space (ignoring the "minor" issue of being non-Archimedian). The difference is that the metric (and so the topology and the way convergence works) is different. In other words, if you understand the topology of p-adics and know how to do calculus on them, then you get numerical methods for free by swapping out the respective pieces.
In this particular example, we have a sequence of numbers resultant from Newton's method. Convergence of a sequence (in a sequentially complete metric space) is related to the distance between pairs of terms, d(a_n,a_m), getting arbitrarily small -- this is called the Cauchy criterion.
For reals, d(x,y) = |x-y| with the real absolute value, and our Newton sequence is all over the place with respect to this "usual" distance. However, when we swap out for the p-adic absolute value, we get that the Newton sequence is actually getting closer together with respect to the Cauchy criterion. This allows us to formally see its convergence.
Further, we can use the same techniques as we do for real numbers to get convergence rates and error estimates, thus giving us suitable cut-off points for running finite computational algorithms. It turns out that Newton's method typically has "+1" digit per iteration in p-adics, instead of the usual "x2" digits per iteration it has in the reals.
The relevant ideas are covered in the text that I recommended, including Hensel's lemma (see Eric Rowlnad's comment) but most of the details for specific methods you'll need to work out yourself; I'm not aware of any comprehensive resource(s) of numerical analysis over p-adics.
Question as a video maker: how do you get your manimations to line up so well with what you say: is it all done in editing, or do you somehow have the animations playing while you're recording your narration. If you create visuals first, how do you have exactly what you're going to say in your mind? This frustrates me to no end, that everyone seems know something I don't.
My approach was to start with the animation, then record audio, then alternate between adjusting the animation timing to fit the audio and adjusting the audio timing to fit the animation, until I got what I wanted, all the while realizing I needed more animation to match audio in certain places and more audio to match the animation in others. It was extremely time-consuming! I hope with experience it will get faster. Maybe others have better/different workflows?
Hi Eric, actually yes, I think you may find making one small tweak to your method to save you time in production.
What I’ve found is the most efficient way to create a smooth production is as follows:
1. Write the script.
2. Record the audio.
3. Animate to match the audio.
What makes a presentation “feel” smooth is the flow of the spoken word.
Figure out what you want to say, how you want to say it, and record that.
Now, doing so without eventually having to resort to back and forth tweaks does involve a bit of imagination. Essentially, during the writing stage (stage 1), visualize the eventual animation you will make to match your words. You’ll find your visualization of eventual animation will, at times, not be enough to match up with the words you’ve written at certain sections (and vice versa). But by working it out here at this stage, you save a ton of time over trying to work it out after stage 3.
If you have trouble visualizing animations you have not yet created (though I suspect you do not!) you could substitute visualization for storyboarding. This doesn’t have to be complicated, and could be as simple as thumbnail sketches of difference “scenes”.
I personally don’t use storyboarding as I don’t find it necessary because I have no trouble visualizing the eventual animation I’ll make while in the writing stage (or remembering it in the animation stage). Though, relying on visualization and memorization alone may not be the best approach all of the time. Storyboarding can be useful if one either cannot easily see the visuals in their mind’s eye, has a hard time remembering them, a production has a long runtime, or one is trying to collaborate with others on a production.
Really amazing video. Hope this helps you make more such presentations with less effort in the future.
@@storyrewrite Thanks, this is great advice! I will definitely try it out for the next video.
@@storyrewrite thank you very much, both of you!!! Hopefully my work will go more smoothly in the future 🙂
@@storyrewrite I don't do videos, but this is good advice for any kind of presentation. Thanks!
I never comment, but this video was absolutely incredible. Your visuals, voice, and intuitive teaching style were captivating; I'll have to look more into the p-adic numbers!
Hold up. Anyone else thinking 2-adic numbers might shed light on the Collatz conjecture? After all, the conjecture is centered around the question of "how divisible by 2 is 3x+1?"
ETA: some digging on Google reveals an excellently-written 2019 paper with lots of good references, by Oliver Rozier, that explores this idea and surveys other work.
0:21 "now let's add *A FEW* more rows"
How does adding 1 to …4444 give 0? Wouldn’t it just be …4445? This part I didn’t understand.
Edit: I now understand that it’s because it’s a base 5 number system
So the study of p-adic numbers is modular arithmetic's big brother.
The facts that ...9999999999 ≡ -1 (10-adic), and ...11111111111 ≡ -1 (2-adic) are very familiar to anyone who has looked at how computers get math done.
This was a great video, and I have the urge to further explore p-adic numbers, except I must sleep :(
This is fantastic! I've tried to understand the p-adics for years and never had this level of clarity before! I get it now! Thank you
The negative representation in 10-adic numbers reminds me of 2's complement from comp-sci
As someone who hasn’t done much advanced math after finishing high school, this video scratches an itch for learning new math. It would be helpful if you could also point to some recommended readings for those of us who’d like to learn more.
Keep up the good work regardless!
Thanks for your effort, please keep making more videos. We need more people like you.
Fascinating ! I hope to see more from you !
This is great! Very well presented. I suspect the ending with Newton's method will feel a bit rushed for most folks new to the subject, but nevertheless this is really well done.
1:25 An interesting question left out is why the last digits of 2^10^n "converge" in base 10 (or the last digits of 2^5^n converge in base 5 etc...).
I guess this is the most underrated channel ever! Kudos to you Rowland!
This is the weirdest, yet most beautiful thing I've seen in a while...
SoME2 is creating some great maths videos and I'm loving it
Which book would you recommend for learning p-adic numbers (or maybe 2-adic numbers if they are simpler) for beginners? And what are the required prerequisites?
Fernando Gouvêa's book "p-adic Numbers: An Introduction" is quite good. It assumes some background in elementary number theory, algebra, and analysis, because these are necessary to really develop the theory.
this video is great! I've always struggled to get any kind of intuitive sense for what the p-adic numbers actually "are." the colored visual of the numbers converging to the left really helped make it click!
2 minutes into the video, and I have to stop, and say, holy crap ! where in the heck did this guy come from ? ! This is a totally great and amazing video !! I can't wait for more ! Now, back to the video ...
Derek from Veritasium essentially plagiarized the first part of your video, just thought you should know man
not entirely, also it is just a good explpanation
yes, i immediately recognised
Rewatching, yeah it’s plagiarised
Would you, then, consider the first part of this video to be plagiarized from Richard E. Borcherd's first video on p-adic numbers?
I like how the negative 10-adic integers actually are represented like a signed _binary_ integer
*infinite* signed *decimal* integers (10's compliment)
@@blockmath_2048 *complement
Thanks so much for making this video! I took an abstract math class on calculus several years ago (and several times) and struggled a lot. The p-adic numbers were discussed and I did eventually pass the class but I had no idea that they sometimes contain the square roots of negative numbers. Actually I think I did learn that but I didn't understand the implications or what it really means. I wish I could have watched this video as I struggled with that class, thanks so much!
Unbelievable first video! You have a new subscriber, already looking forward for new content. Thanks and congrats!!!
Thanks and welcome!
Loved the video!
I have a question: do the 2-adic numbers contain the square root of 7? If they do, why can't you divide the square root of -7 by the square root of 7 to get the square root of -1?
Great question! In fact they don't contain square roots of 7! You can see why if you try to build one.
@@EricRowland does that mean that the 2-adics contain at most one of sqrt (-x) and sqrt(x)
@@WaluigiisthekingASmith Yes, exactly. If there were 2-adic numbers a,b such that a^2 = x and b^2 = -x, then (a/b)^2 = x/-x = -1. This would imply that a/b is a square root of -1, but the 2-adics don't contain square roots of -1.
Seeing this video made me remember why I fell in love with maths in the first place.
I also fell in love with physicses and chemistries and engineerings.
Beautiful!!!
I wish I had this video 6 months ago when I had to learn and write about p-adics for my undergrad capstone.
This video covered most of the questions I had on p-adics that I was barely able to find on math stackoverflow, especially the stuff on when the p-adics contain sqrt(-1).
Subscribed
A very well explained complicated concept, I love it. Wish to see more of it
Comtent on the level of 3blue1brown.
2:49 substitutions?nvm
oh my god
the smile on my face when the numbers lined up in the x²+7 thing at the end
absolutely amazing video; keep going!
This is fantastic! Your teaching style and the way you paced this lesson helped me grasp something I've struggled to fully understand before. I even immediately reached for my notebook to do the exercise, and I learned a lot by getting hands-on with p-adic numbers with a goal in mind. The pattern is also so satisfying that it feels obvious in retrospect. I love the way you ended the video, too! Just great all around. I've subscribed!
Thanks! That’s so great that you worked out the exercise!
So that’s what Donald trump meant when he said a small sum of 1 million
Hi, Eric. This is entirely brilliant! Over the years, I've revisited the subject, but it never really clicked. This is a truly lucid and accessible presentation. Thanks!!
Thank you! I’m glad it resonated!
What a beautiful way to visualize the p-adic's. I've never quite seen them that way, but it's really lovely.
You explained it better than anyone other person I have heard explain it. The colour coding is just magnificent!!
This was gorgeous. Thank you so much for sharing
Great stuff! It's amazing what CZcamsrs (like you!) have done for mathematics!
Thank you so much! Its been a while since I heard of p-adic numbers, but since then never found a intuitive explanation that newbie like me could understand! Your video really helped me.
Amazing!
“Now lets add a few more rows”
* adds thousands more *
Amazing video. First time seeing you on youtube and an instant subscribe. Will be sending this to my friends too
What a happy day, that I get to discover such a great new channel as this! I hope to see more of you 🥰
Thank you! You definitely will!
What a beautiful and exciting introduction to p-adic numbers! You even snuck in an example from your recent research with Reem Yassawi on "p-adic asymptotic properties of constant-recursive sequences" (doi:10.1016/j.indag.2016.11.019, arXiv:1602.00176).
I can't wait to see how you will follow this video up! You set an extremely high bar for yourself :)
Thanks, Armin!
This is INSANE!!! How have I not seen this before?? NICE WORK! (Earned my sub!)
The conclusion made the entire subject so much clearer for me, thank you!
Great to hear!