Number Systems Invented to Solve the Hardest Problem - History of Rings | Ring Theory E0

[@DeclanMBrennan]

What a tour de force. I learnt a fantastic amount here in a very enjoyable way without being mired in detail. In this field, you truly are the *Lord of the Rings* .

[@DejiAdegbite]

It's an interesting Field of study. 😄

[@quiversky4292]

One ring to rule then all!

[@andrewzhang8512]

yuri van gelder

[@anstow]

Really nicely presented. At 37:11 Wedderburn and Artin showed that any non-commutative algebra over the reals is a product of *matrices* over R, C and H. Thanks for the wonderful refresher

[@LillianRyanUhl]

Those algebras have nilpotents; the only sorts of those algebras without nilpotents are those such that the ideal generated by each primitive idempotent is actually a division ring, meaning that that simple ideal is isomorphic to ℝ, ℂ, or ℍ

[@anstow]

Thanks @@LillianRyanUhl you're absolutely correct

[@georhodiumgeo9827]

Sir, this is 3b1b caliber work with maybe even deeper content. I can't believe I just found your channel. I know there are other number systems but to have a complete guide with the context for why they were made and a quick explanation is mind bending. I needed this video so bad I can't even describe how I even feel about it. Thank you.

[@oncedidactic]

As someone who never saw enough pure math to string together a full picture of these concepts and their origins, this is absolute gold. Will be very happy if there is more. :)

[@stevestarcke]

I am in awe. To be exposed to the greatest minds in math is a transcendental experience.

[@sgut1947]

Sometimes it's an algebraic experience 😉

[@DejiAdegbite]

That's a rational reaction.

[@notyourfox]

@@DejiAdegbite A Natural thing to me

[@jackwarren2849]

This is gold, I've got nothing else to add.

[@akhandanand_tripathi]

Must be a complex emotion to explain

[@nice3294]

Amazing video, somehow you managed to cover so much ground in this video while having it remain intuitive and understandable. I never realised how interesting rings and fields were

[@theflaggeddragon9472]

p-adics?!??? Also A_inf, B_dR, B_crys, B_st, Galois deformation rings, Hecke rings, and so much more!! FLT really is astounding.

[@jaafars.mahdawi6911]

I'm amazed at the scope you were able to cover in less than 40 minutes. Brilliant work really (or should i say, complexly :p). Keep it up.

[@andriypredmyrskyy7791]

Love how the music makes me feel like a Viking mathematical pioneer.

[@proced2344]

league of legends music lmai

[@tazking93]

A much needed refresher on rings, with additional paths for further education. Bravo

[@angeldude101]

The second anti-commutative 4D algebra with x² = 0 and y² = -1 is not the dual-quaternions as you said, but rather the planar-quaternions. The dual-quaternions are an 8D algebra and contains the planar-quaternions, containing an extra anti-commuting term squaring to -1. These along with several other algebras can be generated as Clifford algebras, denoted as Cl(p, q, r), where p is the number of orthogonal elements squaring the +1, q the number of such elements squaring to -1, and r the number squaring to 0. The planar-quaternions are Cl(0,1,1) and the dual-quaternions are Cl(0,2,1). As a bonus, the quaternions are Cl(0,2,0), ℂomplex numbers Cl(0,1,0), dual numbers Cl(0,0,1), hyperbolic numbers (the more descriptive name for the split-complex numbers) Cl(1,0,0), and the ℝeals are also included as Cl(0,0,0). These algebras are often very useful for describing geometric transformations in space, which is why they're often called geometric algebras. ℂomplex numbers are well known for describing 2D rotations, and the quaternions for 3D rotations. Geometric algebras extend these to higher dimensional rotations, as well as a few other things. Your third example, which is Cl(1,1,0), is often used as a simplified version of Cl(1,3,0), used for modelling a 2D slice of the 4D spacetime of Special Relativity. I loved seeing the binary rationals, not because I'm already a fan (this is actually the first time I've heard about them formally), but because I happen to be enjoy programming and computing, so I instantly recognized it as ideal fixed point and floating point numbers. It also made me consider how ℤ[1/10] would be the ring of all decimal expansions. (I'd assume finite, because otherwise it'd be indistinguishable from the ℝeals.) I was hoping for a little more time spent on modular integers, but they'll probably come up when you make the video on p-adics, because the p-adic integers with n digits of precision is equivalent to ℤ mod p^n. Again, my interest in computing makes me naturally more interested in the 2-adics specifically, and things like ℤ mod 256, ℤ mod 65536, ℤ mod 2^32, etc, since they're exactly the rings that 8-bit, 16-bit, and 32-bit integers represent. Integer "overflow" is usually treated as an error by most programmers, but it's just a natural part of doing modular arithmetic that should be completely expected.

[@EpsilonDeltaMain]

You are right, its called planar-quaternions, not dual-quaternions adding that to the corrections

[@stevestarcke]

Amazing analysis.

[@d.h.y]

Such a wonderful video. Please keep at it! I feel like I've just realized for what purpose those thick algebra books are so meticulously categorized!!

[@richardcheney6964]

i wish this video existed 8 years ago good job man

[@Zarol-h6z]

This video is such a good video, which helps me understand motivation and some math concepts. It helps me have a basic view of algebraic number theory. I highly appreciate it and will recommend it to my friends!

[@lucasbollmannbaum]

The summoners rift soundtrack just makes it even better

[@PRIYANSH_SUTHAR]

The Art of Teaching applauds you!

[@PerryWagle]

Wow. Much of my abstract algebra class taken decades ago came together in becoming almost a coherent whole in my head. Much flashbacking. Thanks!

[@joemattiaventurelli1030]

You got me mind-blown. Thank you, a beautiful video.

[@CatherineKimport]

I lost track of the number of times I started to get interested in something and then you said you were planning a later video to cover it in detail, guess I better subscribe lol

[@abdulllllahhh]

This video is so fucking good, I just recently got into number theory as a high school student, and for my 12th grade IB math IA I wrote about everything from this video.

[@farmertice7064]

Very good video that makes difficult math concepts simple.

[@Pdjwvdugowqnxbgze]

After seeing this video I want to take algebraic number theory next semester, but unfortunately there won't be enough time left for another course:(

[@ernestoherreralegorreta137]

Amazing exposition. Thank you so much!

[@maxhofman6879]

Woww great video, I forgot how much I loved ring theory

[@JR13751]

36:18 Split quaternions and 2x2 real matrices are isomorphic to each other.

[@EpsilonDeltaMain]

You are right, I'll add that to the corrections

[@ShenghuiYang]

Worth watching couple times.

[@АндрейВоинков-е9п]

Great video, worth watching twice

[@vorpal22]

My two favorite classes in grad studies were abstract algebra (where we did a lot of studying of rings, obviously) and my course in finite fields.

[@05degrees]

Such a nice ring theory primer!! 👏🧡

[@nicolasreinaldet732]

Me on the first half: Uhummm all makes sense. Me on the second half: Wtf, I will need to watch this again and read a book about it.

[@PlasmaFuzer]

Awesome video. More please!

[@OhInMyHouse]

Holy shit dude this video is awesome. Congratulations on your incredible work. You instigated my curiosity about number theory. Thanks a lot.

[@morgan0]

an example of division by zero being allowed with infinity as an actual number is IEEE 754 floating point arithmetic. infinity is just a bin from one number to infinity. and the way it represents numbers more like bins of numbers rather than discrete points is interesting as well (inf is a clear example of it, but also different scales has them at different sizes)

[@therealax6]

While this is true, floating-point arithmetic doesn't form a ring by any means. Take the smallest possible positive float, let's call it q (q = 2^(-159) for floats and q = 2^(-1074) for doubles). Now consider the product q * (0.5 * 2), which is clearly q * 1 = q. But on the other hand, what about (q * 0.5) * 2? Well, q * 0.5 is irrepresentable, so it either has to be rounded up to q or down to 0, But if you round up to q, then the overall result is q * 2 = 2q; if you round down to 0, the overall result is 0. Either way multiplication is not associative.

[@aleratz]

11:51 shots fired, shots fired!

[@petersieck7986]

Grateful I'm not going to have to study all that for a test at the end of the week! Well done

[@authenticallysuperficial9874]

Wow, great visuals

[@JohnSmall314]

The 'American' number system, initially based on the UK's 'Imperial' system makes use of the fact that powers of 2 are 'practical numbers', they have useful divisors. In the days before calculators and digital scales then measuring things is most convenient if you use 'practical numbers'. Hence the Babylonian system using base 60, and the old British system of Pounds, shillings and pence, with 12 pence in shilling, and 20 shillings in a pound. If you're weighing out money using scales those units are exceptionally useful. Likewise the crazy 12 inches in a foot, if you have to divide up lengths by 2 or 3 or 4 or 6, 12 has lots of divisors.

[@misterguts]

Yeah, I did kinda notice that, as smart as EpsilonDelta was in his presentation, he couldn't bring himself to leave out a little bit of random ignorant assholery.

[@therealax6]

There's nothing practical about powers of 2, as the only divisors of powers of 2 are, well, powers of 2. All of which also divide the corresponding powers of 10! The other numbers you mention, like 12 or 60, are highly-composite numbers. (A highly-composite number is a positive integer with more divisors than any other positive integer smaller than itself.) Those were chosen for the reason you mention. Powers of 2, not really - I can only assume that the choice to use powers of 2 often comes from the fact that most people can split things into halves visually with reasonable accuracy, but not into fractions with higher denominators.

[@Γιώργος-ε6τ]

Very beautiful maths involved, I appreciate it

[@tomkerruish2982]

Baez is right; the octonions really are the crazy uncle no one wishes to acknowledge.

[@HaramGuys]

They are useful enough to be considered honorary rings, just like how quaternions are considered to be honorary fields

[@gergokovacsjazzpiano8165]

I've an MSC in chemistry, but these videos make me want to go bsck to university and learn maths again...

[@ValkyRiver]

4:54 there is a related prime fact about the positive rational numbers, where every positive rational number has a unique prime factorization if one allows negative exponents. E.g. 6/5 = 2^1 * 3^1 * 5^-1 This is used in microtonal music for intervals in Just Intonation, and the derived notation is known as the “monzo”. E.g. 6/5 in monzo notation is | 1 1 -1 >

[@horrorspirit]

rng - ring without identity rig - ring without negatives i love mathematician naming conventions

[@Grassmpl]

Wrong. Those aren't "rings" (although we sometimes define ring to not include 1) USA - country in Texas Do you agree? No? My point exactly.

[@fullfungo]

@@Grassmplbro, stop, chill. And only then comment. You are not making sense.

[@Grassmpl]

@@fullfungo rig - ring without negatives. A "ring" has additive inverse, so a "rig" in general is NOT a ring.

[@drdca8263]

@@Grassmplclearly they meant something like “take the definition of ‘a ring’, and remove the requirement that [...]”.

[@Grassmpl]

@@drdca8263 I know that. But according to English grammar they didn't say it correctly.

[@moe.s6638]

I am so happy to have stumbled upon your channel 😊

[@charlievane]

Thanks

[@EpsilonDeltaMain]

Thank you!! You are my first super thanks!!

[@mehdimabed4125]

What a video !! The clearest I've ever seen of this kind of subject (and I've seen many !) In fact, I've always wondered if one could find a number system well suited for describing the maths of relativity ; I know that split-complex numbers handle Minkowsky 1+1 space-time, but does anyone know if such a number system exists for 2+1 or 3+1 (harder to visualize) space-time ? None of of the one presented in this video seems to fit, but I don't loose hope !! Thanks for the amazing lesson

[@HaramGuys]

you are looking for Clifford Algebra. In particular, Cl1,3(R) aka the spacetime algebra. I personally find it easier to understand it in the language of covariant/contravariant vectors and inner products. Hamilton initially invented the quaternions to represent a vector in 3D space, and Maxwell's equation was originally written in quaternions. But we now use the language of vectors and tensors instead.

[@mehdimabed4125]

@@HaramGuys I've read/seen through some stuff about Clifford algebra and geometric algebra ("sudgylacmoe" is a golden gem if you don't know it), but it never translates to a number system ; it never creates a new number or set of numbers like the quarternions do, and I wonder if such a number system could be made for describing this spacetime algebra...

[@mastershooter64]

@@mehdimabed4125 Can you really make a distinction between "set of numbers" and "algebras" tbh the world "number" doesn't have a precise mathematical definition whereas I can tell you what an algebra is. reals, complex numbers, quaternions are just examples of particular algebras, specifically fields and rings which have some structure unique to them. Like that the clifford algebras are an algebra which you get as the quotient of a tensor algebra. Actually clifford algebras generalize quaternions and octonions to any number of dimensions! so like you use quaternions for 3D rotations, you can use clifford algebras for rotations in n-dimensions

[@angeldude101]

@@mehdimabed4125 By "number" do you mean "division algebra"? Then there are none past the quaternions. Do you mean "field"? Then there are none past the ℂomplex numbers. By "number" do you mean "algebra"? Then Clifford algebras can provide that for any number of dimensions and several kinds of geometries, including Minkowski spacetime. One of the 4D anti-commutative algebras given in the video was actually Cl(1,1), which is often used as 1+1D spacetime to demonstrate the effects of relativity in a 2D picture, and is a sub-algebra of Cl(1,3), which is the full 4D spacetime physicists are usually interested in.

[@mehdimabed4125]

@@angeldude101 Thnaks for the answer ! By "number", I think I mean something like : a set of symbols that I can concatenate with other symbols (the operators, like "+" for example) in order to go from a symbols (number) to another... I'm pretty sure this definition is no rigorus at all, but by wrtitting it I realized that infact, the basis element e_i of Clifford algebras fit this definition :) But the problem is that everything seems so hard in these algebra (exterior product, quadratic form,...) ; for quaternions for example, we just have 3 rules (i^2 = j^2 = k^2 = -1), and everything follows as usual (despite we loose commutativity of course, but it is easily shown from basic aritmetic with quaternions). For example, I don't understand how to reconstruct complex numbers in Clifford algebra language. Apparently, in Clifford algebra e_i*e_j = -e_j*e_i when e_i =/= e_j , but to me, the basis elements of complex numbers are noted "1" and "i", and 1*i =/= -i*1 .... I've read in Wikipedia that "Hamilton's quaternions are constructed as the even subalgebra of the Clifford algebra" ; why quaternions aren't just Cl(3,0) ?? Maybe what I'm looking for is a 3d/4d algebra with "simple enough" aritmetic ? I don't really know ^^'

[@cykkm]

I'm dumbstruck! Please, please, don't stop! You make connections between high-level mathematical concepts so… palpable. It's easy to fill in the blanks when you understand how pieces snap together. I for one, could never grok the motivation behind ideals.

[@rewrittenperspective547]

This was so good!! Please MAKE ALL the videos that you said you'll make later in this video ✨✨

[@wompastompa3692]

Pacman (original, anyway) is a cylinder, not a torus. You warp the sides, but not top/bottom.

[@foo0815]

Asteroids' world is a torus.

[@pneumaniac14]

this video is great youre doing gods work brotha

[@김완기-f5e]

이 영상을 너무 빨리 봐서 다음 영상을 기다리는 것이 고통이다

[@PhiloPhysics]

That moment when the background music is literally league of legends…

[@luizmenezes9971]

I used dual quaternions to store and manipulate the kinematic state of one object in one dimension. And a vector of dual quaternions to do so in 3 dimensions. Dual quaternions are a lot more useful than people give them credit for.

[@theskilllesss8106]

I really need to get a grasp on this concept, what is the difference between sqrt(-5) and i sqrt(5) ? Is it written this way to induce the decomposition but to say not to cover the complex plane ? But I don't see how... Isn't it just a notation thingy ? By the way I'm studying Maths in French so some notations or rather the way you name things really differs to the point that translating "directly" from English to French isn't right, I might have overlooked something really obvious and if so I'm really sorry I did ! In all cases it was a really cool video, I hope you'll continue !

[@EpsilonDeltaMain]

They are the same. In fact, it was Euler who invented the symbol i because he got too lazy to write out sqrt(-1)

[@gcewing]

I think writing it that way makes it clear that we're injecting just a single object into the field, not i and sqrt(5) separately.

[@riccardo.toscano]

I also like to imagine prime numbers

[@Peccomment]

Thanks a lot, a jewel!

[@EpsilonDeltaMain]

Thank you for your support

[@Just_a_user3]

Very nice video, thank you for your efforts. Which part of the video talks about the donut numbers shown in the thumbnail?

[@aweebthatlovesmath4220]

I need episode 2!!!

[@wyboo2019]

24:20 for anybody wondering more about why zero divisors are an issue, one intuitive reason is because it removes one of our main methods of equation solving. for example, if you were trying to solve x^2=x, you'd bring both to one side and factor to get x(x-1)=0. now, normally two numbers multiplying to get 0 means that one of them is 0, so you can break this into two cases: one where x=0 and one where x-1=0, and then you have your solutions but when you have nonzero numbers that multiply together to get 0, you lose this method of equation solving, because you can no longer assume that one of x or x-1 equals 0, because they could just be a pair of zero divisors it's the same reason we study p-adics for primes p instead of just n-adics for any natural number n, because composite n leads to zero divisors in our n-adic system on the other hand, the idea that two numbers can multiply to give 0 is super intriguing and definitely worth investigating. what other consequences of 0 divisors are there, and how can we work around them if possible?

[@stanleydodds9]

The way I think of it is that being in an integral domain is equivalent to always being able to cancel multiplication and get a true implication. So if I have an equation ax = ay, and there are no zero divisors, it is true to say that x = y. So essentially, it's the precursor to being able to do division; division is injective, but only where it's possible. In a field, we have the stronger result that division is always possible and is injective, except by 0.

[@holothuroid9111]

Great video. I think it would be even better split into smaller parts.

[@pacificll8762]

This video is so great!!

[@jeromejean-charles6163]

Great indeed. To me this is very much in the spirit of the "naturalist" approach to mathematics advocated by John Conway. It helped me gluing/ unifying several of my mental pictures in algebra. I would like to know if in fact you appreciate John Conway?

[@korigamik]

This video is really good. Can you share the source code for it?

[@seneca983]

34:25 Another term that can be used is "skew field".

[@TheZectorian]

I love how league music is playing in the background

[@weeb3277]

in the beginning of your video you talk about India and China. you do it for every country?

[@bromax8686]

i did not expect to hear league music when watching a maths video

[@MrBeiragua]

I would say that the connection between natural numbers and language is not that old. For thousands of years counting was made with gestures (counting fingers) and language was used only for general description, like the words thouSAND and HUNDred come from the same root *kent, that meant "a lot".

[@angelchavez4824]

Please do more history videos

[@wiri2391]

I think what makes finding the solutions to the quadratic equation with the restriction to positive integers much more difficult is exactly that: the restriction. While taking the square of an integer is perfectly fine in its scope of „counting“ (just by multiplication), its inverse is beyond the scope of the number system as taking the square root can result in irrational numbers. Therefore, finding solutions to the Pythagorean equation naturally belongs into the realm of irrationals, wouldn’t you agree? We observe similar effects in other fields, e.g., in geometry where analyzing surfaces in 3D is much harder and nuanced as analyzing volumes, because volumes naturally belong in 3D while surfaces are restricted objects embedded into 3D. Similarly, 2D regions in 2D are easier to handle than 1D objects (lines) in 2D. What are your thoughts on that? Anyway, very nice video!

[@thomasschoenborne5771]

Thanks!

[@EpsilonDeltaMain]

Thank you so much!!

[@JamesLewis2]

Although I realize that non-associative algebras are not particularly important in number theory, the usual result from Frobenius is that there is a fourth finite-dimensional real division algebra, the eight-dimensional octonions O. (Curiously, the higher-dimensional Cayley-Dickson algebras have zero-divisors but also multiplicative inverses for all non-zero elements, which only makes sense because they're non-associative; the octonions are alternative, but beyond that, the algebras are just power-associative and are of little use, with only the sedenions S and trigintaduonions even having commonly used names.)

[@xniyana9956]

Wait a second, is that League of Legends music I'm hearing in the background of this video?

[@tiltltt]

great content! although the background music sounds weirdly familiar, is it from some videogame?

[@HaramGuys]

"Summoner's Rift - Late Game" from League of Legends

[@tiltltt]

@@HaramGuys i knew it! i was waiting in soloq watching this video, i thought my game might've bugged lol

[@purwic]

good. make more videos exactly with mathematic terms

[@1.4142]

In China and other eastern countries, a stock going up is shown as red, and green is down.

[@HaramGuys]

I know that in china, red is considered a very positive color, but they still use the term 赤字 for negative balance right? I believe it is western borrowed meaning

[@1.4142]

@@HaramGuys The dictionary says this minister 苏绰 invented 朱出墨入 in 545 bc to keep track of revenue and deficits. But this other guy on the internet says this might be a misreading, and cites 《周书·苏绰传》, which uses 朱出墨入 and meaning the colors of the documents you submit to the government being black and the ones the government issues being red. So it predates western use either way.

[@wcsxwcsx]

You know, red is an auspicious color in China, which is why it is used to represent good, positive things.

[@fyu1945]

Incredible video! Do you have some kind of link to Gauss's proof of Fermat's theorem for n=3?

[@EpsilonDeltaMain]

Here is a much more streamlined version the proof, Gauss's proof but in a more modern language thats easier to understand: qr.ae/pKXrJ3 Stuff I have talked about in this video, such as units, ideals, modulo, UFD etc, all shows up in to proof

[@fyu1945]

@@EpsilonDeltaMain Thank you I'm gonna look into it. I actually had most of what you talked about last semester and your video served for me as a great summary of what I've learned. Instant sub!

[@popescucristian8978]

4:59 you just had to use those numbers 💯

[@Mr.Nichan]

11:46 Also rhythm in Western/modern music notation.

[@deltalima6703]

Epsilon delta is a nice name. I think so, at least. ;)

[@moralboundaries1]

Needs more of an interesting narrative and engaging animations imo. But it's interesting.

[@Kurtlane]

Wow! Excellent stuff, as far as I could understand (about 1/10). Everyone else's explanations left me totally stuck. Please, does anyone know a book (a textbook would be best) that introduces one to these things. Doesn't have to cover everything here. Thanks.

[@gamespotlive3673]

Guys there is no league of legends music they are lying to you.

[@mekaindo]

o

[@minecraftermad]

I don't like the infinity from 0 division because it loses information. Id personally call it a new number with the divided numbers value as metadata. So you can still compare sizes etc and even get back from the conceptual infinite size

[@PC_Simo]

1:11 Umm… Natural numbers include 0. Whole numbers are natural numbers with negative whole numbers. *EDIT:* I guess you corrected that, as soon, as I continued the video 😅.

[@Stdvwr]

I have a higher math education and I barely understand this video. But the comments are all positive, so I wonder if I forgot how to take new information of if the commentors have most of these concepts already in their heads

[@sheepcommander_]

FLOATING POINT NUMBER JUMPSCARE

[@MykhailoIvancha]

Did anyone create a system that has a shape of a hyperbolic plane? Would be cool

[@maxvangulik1988]

23:49 gravity coil

[@_Noopy_]

please make a video for the gauss's proof of a^3 + b^3 =/= c^3 ..... please!

[@oberonpanopticon]

This is a bit over my head, I couldn’t really follow along with anything that came after “since the beginning of human language…” Still a good video though! Probably! I mean it could be complete gibberish for all I know but the other comments seem to agree that it’s good!

[@crimsonmegumin]

same :(

[@WindLighter]

The problem with constructing higher dimentional fields (3, 4,5, etc dimentional) is that those fields need numbers to consist of n+1 components for n dimentions. It's funny however that for n=1 we get reals and for n=2 we get complex numbers but in unusial notations.

[@thefance4708]

i thought the rule was 2^(n-1) though? e.g. n=4 is octonions, which has 2^(4-1) = 8 degrees of freedom, no?

[@WindLighter]

@@thefance4708 it's the different approach. I'm still exploring how numbers in my notation behave as the rules was constructed out of assumption that number signs are a separate entities from number magnitudes. If you imagine number as a path along a closed set of vectors in an n-dimentional space and treat them as an orderd list of scale factors for each of the vectors (zeros included) there is a set of rules that makes that a field. For example, positive numbers in that notation are a lists of two values where first value is larger than the second and negatives are lists with the second value larger then the first. The tricky part is how to map those numbers from n to m dimentions though. Some draft is in this document: docs.google.com/document/d/1P8E7HOe1KcdT68j75nqSLoquEH0AFSXHdcNlIbQLF5k/edit?usp=sharing

[@evandrofilipe1526]

Casual geometric algebra w

[@lost4468yt]

What I did today: got lost on KZbin and now I'm scared of numbers

[@djtomoy]

I just imagined a number, septney phine

[@richardneifeld7797]

Overwhelmingly informative. Thank you. May I suggest less content in each video. 5-10 mins each?

[@mariotabali2603]

I hit ‘like’ because there is no ultra like

[@CrimsonKnightmare1]

please put a trigger warning at 10:05 I was BOMBARDED with LEAGUE OF LEGENDS music, it triggered my ptsd *shudders*

[@mrocto329]

ff15 open mid rep top feeding

[@coarse_snad]

Maybe I'm not good enough at math yet, but I found the explanations towards the middle of the video to be hard to understand. I've rewatched from 22:45 like 10 times now, and I feel like if completely missed what "principled" means in this context. First half of the video was fantastic though!

[@EpsilonDeltaMain]

I feel like I really brought that fact out of context. The full proof of FLT for regular primes is several pages long and I didn't want to go through it in detail, and just make a historical introduction of why these concepts were introduced in the first place. here is the full proof in the link: kconrad.math.uconn.edu/blurbs/gradnumthy/fltreg.pdf and it relies on one important theorem Kummer's Lemma: planetmath.org/kummerslemma which relies on the fact that p is regular, and the fact that I^p is principal implies that I is principal is key in proving the Kummer's Lemma

[@coarse_snad]

Yeah, those proofs seem scary.

[@edcorns3964]

C(1) = R C(2) = C C(3) = H C(4) : a[0] + a[1]*i[1] + a[2]*i[2] + a[3]*i[3] + a[4]*i[4] + a[5]*i[5] + a[6]*i[6] ... C(N : N>1) : a[0] + Sum([n=1..N-over-2], a[n]*i[n]) N-over-2 = N!/[(N-2)!*2!] = N*(N-1)/2 Explanations: * The number of imaginary components i[n] in N-dimensional space is equal to the number of (2D) planes in that space. * Each imaginary component i[n] defines one rotation (counterclockwise by convention), in one plane, by 90 degrees. * C(N) are N-dimensional complex numbers (for any number of dimensions N>0), where the problem (which can be brute-forced) is determining all the valid permutations of i[j]*i[k]=i[m] (non-commutative multiplication), where 'valid' refers to correct results of rotations in all the planes in N-dimensional space.

[@Hans-s8v]

Am I crazy, why is Leauge of legends music blaring in the backdrop