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* .
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
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.
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
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.
Iam an engineer and math is my love and hobby and because of that, i love these kinds of videos. Thank you very much for this very interesting and entertaining video 😊 greetings from germany
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!
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!
Something interesting: Those three number systems, the Complex numbers, Split-Complex numbers, and Dual numbers are analogous to the only observer independent coordinate transformations, all of which show up in physics. Complex numbers --> spatial rotations Dual Numbers --> Newtonian (Galilean) transformation Split-Complex numbers --> Lorentz transformation (special relativity) It's so unbelievably mysterious that these things come out of pure mathematics and yet are deeply ingrained in the machinery of reality itself.
This video feels very nostalgic for me. My undergrad abstract algebra class used a textbook that was literally called, "learning algebra from early attempts to prove Fermat's last theorem"
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
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!
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.
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
this video is incredible! I finished an introductory abstract algebra course recently, and this video really contextualises and clarifies the motivation behind concepts which seemed quite arbitrary and confusing to me at first. (Why we should care about structures like Z[i] and Z[w], reasons for distinguishing between primes and irreducibles, quotient rings, an intuition for why f(x)/p(x) is a field precisely when p(x) is irreducible)
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 >
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.
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 ^^'
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.
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'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.
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?
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 !
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.
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.
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.
gotta push back on the US measuring system. I worked home construction there and Canada, and I can say that doing everything by halves makes simple measurements very easy. we didn't need a calculator to build shelves!
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. :)
im from bangladesh im from bangladesh
@@BirdsAreVeryCoolstinker, noone asked
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
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 got a little lost on some parts, but it was definitely worth to continue watching! This was so interesting!
one of the best videos i have ever seen on youtube in my 10+ years of watching youtube
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
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.
Iam an engineer and math is my love and hobby and because of that, i love these kinds of videos.
Thank you very much for this very interesting and entertaining video 😊
greetings from germany
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.
Thanks a lot, a jewel!
Thank you for your support
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
Wow. Much of my abstract algebra class taken decades ago came together in becoming almost a coherent whole in my head. Much flashbacking. Thanks!
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!!
A much needed refresher on rings, with additional paths for further education. Bravo
Sweet! This stuff is gold! Love the animation and explanations!! Well done ^^
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!
Beautifully done video. More, please, when you can. 🎉😊
💯
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!
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!
i wish this video existed 8 years ago
good job man
The summoners rift soundtrack just makes it even better
Something interesting: Those three number systems, the Complex numbers, Split-Complex numbers, and Dual numbers are analogous to the only observer independent coordinate transformations, all of which show up in physics.
Complex numbers --> spatial rotations
Dual Numbers --> Newtonian (Galilean) transformation
Split-Complex numbers --> Lorentz transformation (special relativity)
It's so unbelievably mysterious that these things come out of pure mathematics and yet are deeply ingrained in the machinery of reality itself.
Thanks
Thank you!! You are my first super thanks!!
p-adics?!??? Also A_inf, B_dR, B_crys, B_st, Galois deformation rings, Hecke rings, and so much more!! FLT really is astounding.
The league of legends theme music at 10:05 LMAO. Great video btw
That cought me off by surprise lol
This video feels very nostalgic for me. My undergrad abstract algebra class used a textbook that was literally called, "learning algebra from early attempts to prove Fermat's last theorem"
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
Perfect introduction to Algebra as a whole. I wish I will make a math video as good as this: content-like and structure-like
This is excellent. I learned a lot in a short space of time.
Thank you
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!
Thanks!
Thank you so much!!
learned a lot thanks, great explanations
Love how the music makes me feel like a Viking mathematical pioneer.
league of legends music lmai
Top tier math content right here
Very good video that makes difficult math concepts simple.
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.
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
Holy shit dude this video is awesome. Congratulations on your incredible work. You instigated my curiosity about number theory. Thanks a lot.
You got me mind-blown. Thank you, a beautiful video.
Such a nice ring theory primer!! 👏🧡
The Art of Teaching applauds you!
Amazing exposition. Thank you so much!
this video is incredible! I finished an introductory abstract algebra course recently, and this video really contextualises and clarifies the motivation behind concepts which seemed quite arbitrary and confusing to me at first. (Why we should care about structures like Z[i] and Z[w], reasons for distinguishing between primes and irreducibles, quotient rings, an intuition for why f(x)/p(x) is a field precisely when p(x) is irreducible)
I wish I had such great videos when I was studying this topic. It helps me to understand the topic way better than the professors.
Thankyou, that is a very good educational video.
Need to watch it several times though, but that's good. 👍
This was so good!!
Please MAKE ALL the videos that you said you'll make later in this video ✨✨
I am so happy to have stumbled upon your channel 😊
I live for these kinds of videos
Great video, worth watching twice
Woww great video, I forgot how much I loved ring theory
Grateful I'm not going to have to study all that for a test at the end of the week! Well done
Why is the background music summoner’s rift lol
I literally just thought “why is league playing rn?”
Awesome video. More please!
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
Awesome video! ❤
Worth watching couple times.
0:17 In ~1700BC the Babylonians already had a positional numbering system in base 60
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 >
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.
this video is great youre doing gods work brotha
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.
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 ^^'
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:(
Very beautiful maths involved, I appreciate it
Great video! Many thanks!
Wow, great visuals
League of Legends music for a math video is something I hadn't seen before.
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.
very intense and amazing!
i like the shout out for 3b1b, however mathologer made really nice video about same subject
36:18 Split quaternions and 2x2 real matrices are isomorphic to each other.
You are right, I'll add that to the corrections
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.
11:51 shots fired, shots fired!
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.
This video is so great!!
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?
I'm surprised it took 12:42 till Euler was mentioned
Great video. I think it would be even better split into smaller parts.
20:03 hmm, seems like (r1), (r1, r2, ...) are least ideals, contains elements r1 or {r1, r2} etc. But is it true for any ring?
this begs the question, is there any such thing as a "Rg" 9:23
Multiples of 2 ≥0 ?
wow
mind really really blown
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?
I need episode 2!!!
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.
fantastic!
I've an MSC in chemistry, but these videos make me want to go bsck to university and learn maths again...
이 영상을 너무 빨리 봐서 다음 영상을 기다리는 것이 고통이다
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.
Euler and Gauss discover a bunch of cool s***, and then Kummer comes in and just starts making stuff up.
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
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
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.
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.
good. make more videos exactly with mathematic terms
Cool stuff.
What software do you use for the visuals?
Mostly done in Manim, open source python library invented by 3Blue1Brown
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.
gotta push back on the US measuring system. I worked home construction there and Canada, and I can say that doing everything by halves makes simple measurements very easy. we didn't need a calculator to build shelves!
Pacman (original, anyway) is a cylinder, not a torus. You warp the sides, but not top/bottom.
Asteroids' world is a torus.