Number Systems Invented to Solve the Hardest Problem - History of Rings | Ring Theory E0
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- čas přidán 1. 06. 2024
- In this video, we explore the history of number systems that were invented in order to solve the Fermat’s Last Theorem, and we will see how concepts that were invented to tackle the problem such as rings and ideals gave rise to other number systems.
Links:
3Blue1Brown’s Video on Pythagorean Triples:
• All possible pythagore...
Chapters:
00:00 Intro
03:08 Number Theory
08:08 Classifying Number Systems
10:05 Adjoining to a Ring
12:35 Number Systems Invented to Solve FLT
18:15 Kummer's Ideal Numbers
20:03 Dedekind's Ideals
23:13 Modular Arithmetic and Quotient Ring
25:16 2-Dimensional Numbers
31:45 Higher Dimensional Numbers
37:30 Outro
Music🎵:
Forest of the Elves - Jonathan Segev • Forest of the Elves | ...
Midnight Tale - Kevin MacLeod • Midnight Tale
Arab Egypt Music - ALD • Arab Egypt MUSIC - Fr...
Summoner's Rift (Late Game) - Riot Games • Summoner's Rift - Late...
Confusion in my mind - BGM President • [브금대통령] (방황/혼란/Emotion...
Corrections:
36:09 Planar-quaternions, not dual-quaternions
36:17 M_2(R) is isomorphic to the split-quaternions
![[柴犬ASMR]曼玉Manyu&小白Bai 毛发护理Spa asmr](http://i.ytimg.com/vi/0TsXQ7z2Dh4/mqdefault.jpg)
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. :)
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* .
It's an interesting Field of study. 😄
One ring to rule then all!
yuri van gelder
I got a little lost on some parts, but it was definitely worth to continue watching! This was so interesting!
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
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
Thanks @@LillianRyanUhl you're absolutely correct
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.
I am in awe. To be exposed to the greatest minds in math is a transcendental experience.
Sometimes it's an algebraic experience 😉
That's a rational reaction.
@@DejiAdegbite A Natural thing to me
This is gold, I've got nothing else to add.
Must be a complex emotion to explain
p-adics?!??? Also A_inf, B_dR, B_crys, B_st, Galois deformation rings, Hecke rings, and so much more!! FLT really is astounding.
Beautifully done video. More, please, when you can. 🎉😊
💯
This video is so good. It feels like it was made to perfectly cater to my interests and current level of knowledge. I’m so glad that I found your channel. Thank you.
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
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.
You are right, its called planar-quaternions, not dual-quaternions
adding that to the corrections
Amazing analysis.
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.
A much needed refresher on rings, with additional paths for further education. Bravo
as someone who was struggling through some other videos about the quaternions, I am sufficiently glad this video is only 5 days old...
having said that, great video!
Sweet! This stuff is gold! Love the animation and explanations!! Well done ^^
Love how the music makes me feel like a Viking mathematical pioneer.
league of legends music lmai
This was amazing. This video made me understand concepts that I have heard before but never quite understood.
There were still some things I had trouble wrapping my head around espacially towards the end but overall this was a great experience.
Thank you!
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!!
Top tier math content right here
This is for sure a high-quality video! Congratulations! I subscribed right away, and I hope to see more high-quality content like this one!
learned a lot thanks, great explanations
Amazing exposition. Thank you so much!
Wow. Much of my abstract algebra class taken decades ago came together in becoming almost a coherent whole in my head. Much flashbacking. Thanks!
The summoners rift soundtrack just makes it even better
Such a nice ring theory primer!! 👏🧡
This is excellent. I learned a lot in a short space of time.
Thank you
The Art of Teaching applauds you!
The league of legends theme music at 10:05 LMAO. Great video btw
Great video! Many thanks!
Awesome video. More please!
rng - ring without identity
rig - ring without negatives
i love mathematician naming conventions
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.
@@Grassmplbro, stop, chill. And only then comment.
You are not making sense.
@@fullfungo
rig - ring without negatives.
A "ring" has additive inverse, so a "rig" in general is NOT a ring.
@@Grassmplclearly they meant something like “take the definition of ‘a ring’, and remove the requirement that [...]”.
@@drdca8263 I know that. But according to English grammar they didn't say it correctly.
amazing video, i could hardly understand anything in any conrete way but i felt an intuitive sense of some things and somehow the way you communicated the ideas felt super interesting
Great video, worth watching twice
i wish this video existed 8 years ago
good job man
this video is great youre doing gods work brotha
Awesome video! ❤
I am so happy to have stumbled upon your channel 😊
Thankyou, that is a very good educational video.
Need to watch it several times though, but that's good. 👍
Very good video that makes difficult math concepts simple.
I live for these kinds of videos
very intense and amazing!
Worth watching couple times.
This video is so great!!
This was so good!!
Please MAKE ALL the videos that you said you'll make later in this video ✨✨
Wow, great visuals
Woww great video, I forgot how much I loved ring theory
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.
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:(
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
Holy shit dude this video is awesome. Congratulations on your incredible work. You instigated my curiosity about number theory. Thanks a lot.
11:51 shots fired, shots fired!
36:18 Split quaternions and 2x2 real matrices are isomorphic to each other.
You are right, I'll add that to the corrections
this begs the question, is there any such thing as a "Rg" 9:23
Grateful I'm not going to have to study all that for a test at the end of the week! Well done
What I did today: got lost on CZcams and now I'm scared of numbers
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.
i like the shout out for 3b1b, however mathologer made really nice video about same subject
wow
mind really really blown
thanks
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)
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.
Great video. I think it would be even better split into smaller parts.
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.
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.
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.
I have never before seen (or been aware that I have seen) the units defined as neither composite nor prime. Thank you!
By the way, the “stupid” American system was actually British
I need episode 2!!!
I've been played league of legends all day, but I finally built up the willpower to close it and start on my homework
I pull up youtube to find something to watch while I do it
"Perfect, this is even a math video so I can't get even get distracted from math while watching it"
Then I heard the LoL music, and felt an urge to play just one more. The rift calls for me.
Wow I thought I was the only one that noticed.
Very nice video, thank you for your efforts. Which part of the video talks about the donut numbers shown in the thumbnail?
Bravo!
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 >
Thanks
Thank you!! You are my first super thanks!!
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?
4:57 ay i saw what you did.
Thanks!
Thank you so much!!
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
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.
@@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...
@@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
@@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.
@@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 ^^'
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.
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.
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 !
They are the same.
In fact, it was Euler who invented the symbol i because he got too lazy to write out sqrt(-1)
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.
Much in the video was on ideals. My interest, however, is on the complements of ideals, filters, because I've found a connection between them and cognition. I wish there were any videos on filters.
I've an MSC in chemistry, but these videos make me want to go bsck to university and learn maths again...
Cool stuff.
What software do you use for the visuals?
Mostly done in Manim, open source python library invented by 3Blue1Brown
good. make more videos exactly with mathematic terms
Incredible video! Do you have some kind of link to Gauss's proof of Fermat's theorem for n=3?
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
@@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!
이 영상을 너무 빨리 봐서 다음 영상을 기다리는 것이 고통이다
great content! although the background music sounds weirdly familiar, is it from some videogame?
"Summoner's Rift - Late Game" from League of Legends
@@HaramGuys i knew it! i was waiting in soloq watching this video, i thought my game might've bugged lol
0:17 In ~1700BC the Babylonians already had a positional numbering system in base 60
Pacman (original, anyway) is a cylinder, not a torus. You warp the sides, but not top/bottom.
Asteroids' world is a torus.
This video is really good. Can you share the source code for it?
Please do more history videos
34:25 Another term that can be used is "skew field".
4:59 you just had to use those numbers 💯
11:46 Also rhythm in Western/modern music notation.
Baez is right; the octonions really are the crazy uncle no one wishes to acknowledge.
They are useful enough to be considered honorary rings, just like how quaternions are considered to be honorary fields
I also like to imagine prime numbers
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.
Awesome video! One point that could be improved is your usage of the plural ("complex numbers" instead of "complex number") and "the" ("using the chinese remainder theorem" instead of "using chinese remainder theorem")
Loved the video (aaand the american unit system opinion😂)
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!
in the beginning of your video you talk about India and China. you do it for every country?
on wikipedia i’ve seen the word “ideal” for “rng” (ring with out unit) is it really the same ? or there is a mistake i don’t see ?
You didn't mention that the dimensions of fields over Q can vary really much, compared to those over R. And this in fact is more intuitive, since it is much easier to reason and calculate over Q (and its algebraic extensions) than over R. All those radicals in our notation suggest field extensions from Q in fact, and even more so do Bring radicals and more complicated things suggest such extensions from Q.
Working directly over R, it would be more natural to use no radicals at all except for square roots of negative numbers, expressing everything else in terms of Archimedically converging limits. So in practice, we work more over Q than over R, when calculating (algebraic) solutions. Or rather, we work over some finite dimensional subfield of the infinite dimensional algebraic closure of Q, plus extending this field with certain transcendental constants that may be defined through limits or otherwise through exponential and trigonometric functions.
Also, since you mentioned non-commutative rings (over R, not over Q unfortunately, which is much more interesting to study!), it would have been nice if you had mentioned non-associative rings with the Moufang property (which may or may not be the same as alternative rings). In such case, the octonions alternative field O, would have fit nicely besides the skew field H, the field C, and the ordered field R. Skew field being a different name for division ring, and alternative field being a different name for division algebra of course.
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?
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.
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.)