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.

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 :(

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

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

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

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

"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.

Alon Amit himself! Never expected to see you on KZfaq. 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.

Thanks! Glad this helped give you a way to picture them!

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. 👍

3b1b mentions it himself in a very old video. It was titled something like "inventing math".

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

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!

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.

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.

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.

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?

Thanks for letting me know about this! I will have to check it out.

Thanks, Luca!

Th 10-Adic number of views have decreased since the first view 😥

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.

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!

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.

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?

*infinite* signed *decimal* integers (10's compliment)

@@blockmath_2048 *complement

Thanks and welcome!

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.

I also fell in love with physicses and chemistries and engineerings.

Thanks! That’s so great that you worked out the exercise!

Thank you! I’m glad it resonated!

Thanks, Armin!

Thank you! You definitely will!

Great to hear!