Climbing past the complex numbers.
Vložit
- čas přidán 21. 09. 2023
- Head to squarespace.com/michaelpenn to save 10% off your first purchase of a website or domain using code michaelpenn
🌟Support the channel🌟
Patreon: / michaelpennmath
Channel Membership: / @michaelpennmath
Merch: teespring.com/stores/michael-...
My amazon shop: www.amazon.com/shop/michaelpenn
🟢 Discord: / discord
🌟my other channels🌟
mathmajor: / @mathmajor
pennpav podcast: / @thepennpavpodcast7878
🌟My Links🌟
Personal Website: www.michael-penn.net
Instagram: / melp2718
Twitter: / michaelpennmath
Randolph College Math: www.randolphcollege.edu/mathem...
Research Gate profile: www.researchgate.net/profile/...
Google Scholar profile: scholar.google.com/citations?...
🌟How I make Thumbnails🌟
Canva: partner.canva.com/c/3036853/6...
Color Pallet: coolors.co/?ref=61d217df7d705...
🌟Suggest a problem🌟
forms.gle/ea7Pw7HcKePGB4my5
I once came across a physics paper that employed the Trigintaduonions (T). Thirty-two dimensional numbers!
I really need the link. I want to see its application...
Need the link
This was it. Not sure what's harder to read, the math or the broken English!
arxiv.org/PS_cache/arxiv/pdf/0704/0704.0136v2.pdf
Geometric Algebra for n-dimensional space has 2^n coefficients in its objects; so it gets quoted as 2^n "dimensions". 5D space for conformal algebra is common, which is 2^5 dimensional. The 2^n comes from n directions in space included or not; because directions in space square to real numbers -1, 0, or 1.
Crazy. Wonder how long time it would take to understand än article like that.
I'd love to one day learn enough to understand a word of what this video is teaching.
Same 😂
Check out the channel dialect. You will get an intro to vector calculus. They also demonstrate elementary matrix algebra. They just started a series on Christoeffel tensors so if you can get Riemannian geometry, you're well on your way to getting quaternions, it's just adding more to matrix operations.
@@philipm3173 is there a channel to explain anything of what you just said?
@@sazam974 3Blue1Brown
@@sazam974 they have a 16 part course called the essence of linear algebra which introduces vectors and linear transforms.
"Coming up" with quaternions for myself during a boring university lecture is still one of my proudest moments.
howw
I did the same in 9th grade
Like come up with the multiplication rules? Or something else?
@@schizoframia4874probably the multiplication rules
Was there a bridge nearby ?
I would love to see a video on the splithypercomplex numbers!
Zorn matrices might be nice. Yes, that Zorn.
So splithy
split hyper complex?
@@pugza1s731 yes
Yes, especially a way to systematically construct splithypercomplex numbers over any field, including those of characteristic 2. This most likely means not all independent generators can be anticommuting but instead may be "skew commuting" rather.
Why was Hamilton considered such a jokester?
Because he always said i j k.
The first telephone company in Hamilton Ontario was started by a physicist. He built a young ladies dormatory on the 2nd floor of the exchange because he knew Hamiltonian operators do not commute.
Seeing how we start losing common features like having no zero divisors or communitivity as we apply this construction, I'd be curious if we lose anything else after the sedenions, or if they have the same basic properties after that.
After the sedenions, your balls fall off
Lmao
I kinda hope it all unravels into complete anarchy as you move up through the dimensions, and then suddenly assumes strict rules again.
Repeat ad infinitum.
That would be awesome :)
Yes, this is a very deep question.
Is there an infinite family of (increasingly abstruse) algebraic properties, which are incrementally lost as the ladder is climbed?
Or, do the (somehow meaningful) algebraic properties completely run out at some point, and as abstract algebras, the higher-dimensional conjugation algebras are all the same? (But then, they are continuous algebras parameterized by 2^n copies of the reals... which in itself is an algebraic property. They are not isomorphic...)
If you're hoping for things to come back being like the wonderful unity of the complex algebra, sorry, that won't happen.
Any algebras that does just what the complex numbers do, is itself the complex numbers. Etc.
Fortunately, different algebras are different, and life is richer!
These things came to life as abstractions, but people have applied them to real-life problems.
For example, multiplication by quaternions preserves geometry in 4 dimensions, and thereby, motion and scaling of solid 3-dimensional objects in 1 time dimension.
Their non-commutativity reflects the non-commutativity of 3-D rotations.
And there's a nice trivia for the "motivation" of this construction. If we would like to preserve the norm multiplication rule, |X Y| = |X||Y|, we have to stick to the 2^n dimensions.
So long as |a| is quadratic in the components of a. Otherwise matrices provide a counter example.
doesnt this have to do with the topological characteristics of spaces? I've been getting into topology and there are some theorems dealing with parity of dimensions and how they don't allow for certain constructions
On sedenions there are Zero divisors
@@benjaminojeda8094 Yes, and they are not a composition algebra, that is they fail the important rule:
forall X and Y: |X Y| = |X||Y|.
Where || is the quadratic norm form in question (related to a symmetric bilinear form sometimes called an orthogonal form).
@@Utesfan100 I am pretty sure that, as long as we investigate *composition algebras* rather than merely the special case of *normed division algebras*, there are 3^n dimensional algebra analogues with cubic norms, to these 2^n dimensional algebras with quadratic norms.
In the general case these norms are not necessarily positive definite but merely nondegenerate indefinite.
Since quaternions have very interesting properties when it comes to describing rotations in 3D space, I'd love to see a video about practical (or not so practical) applications of these higher dimensional algebras. Also, what about algebras, that don't obey this 2^n dimension rule? Great video! 🎉
Google "cohl furey"
Such as those used in matter theories.
😢😢😢😢😢
They don't have applications.
There are attempts at applying Octonions to physics.
Probably not the beat idea, but there are some interesting results there.
The most famous onsight in history was Hamilton's onsight of the quaternions
Ummm...I think Newton's insight about gravity and Einstein's insights about relativity (among others) are *just a tiny bit* more famous than that. ;-)
@@w.randyhoffman1204we are talking about history of mathematics here :)
@@mathophile716Newton’s insight into Calculus then.
If I see you spell it like that, it's on sight.
@@praharmitra Let's not debate what's greater, Newton, Hamilton, doesn't matter. All true geniuses.
This is fantastic. I’ve been looking forward to this video for a while so thank you, Professor!
THIS IS GREAT! Thank you Michael!
Really cool! And yes, interested in the split (and the dual) variants!
I love complex numbers. Subscribed! Any video on this topic is appreciated.
it would be nice to see a video about the split octonians
I would like to see this too
Wow!! Simply, one of the best explanation.
Amazing construction! I already know about the first five of these algebras, but I've never seen this way to get from each one to the next, and I never even knew there were infinitely many of them! Great, educational video!
Your best video in a while. You always make good videos but this one was particularly great
Congratulations, the explanation was clear and usefull. Thanks.
Would love to see some examples of zero divisors in the sedonians
The basic idea is that a^-1 * (a*b) ≠ b ≠ (b*a) * a^-1 generally. That is multiplication by the reciprocal of a is not the same as division by a, which means these reciprocals are not true inverses.
You need to pick four fully independent sedenions a, b, c and d (none of them expressible through multiplying and/or adding the others).
If you pick them to be orthogonal i think you may form (a+b) * (c+d) or something and show that is then zero.
I don't remember exactly though, it was some years since i read about it.
Iirc there is some claim about the zero divisors of the sedenions being closely connected to the Lie group G_2.
I love the cayley dickson construction!
woah this is really interesting, im glad you made this video
Thanks for the video.
Good one mate, and the suggestions in the other comments already cover what I'm further interested in in number sets. I learned these in university, but we didn't go into the regularities or construction rules of them.
This did it for me. I just joined your Patreon. Sigh, I work full time writing code for folks, so not always possessing enough free time, but I like to try. :)
By far, one of my favorite videos...
great video!
I think it would be interesting (although maybe not practical 🤔) to do a follow up video on semi-algebras and the fact that if you tensor any of these algebras with C then you're going to get a matrix algerba over C. I think this is really interesting as it effectively shows that all these algebras are somehow just matrix algebras.
Matrix algebras ate associative which isn't the case for octonions.
Interesting, the conjugation on the pairs (a, b) is reminiscent of the twist structure (a, b)* = (b, a) but using two different negations instead. So if you think of b as being the complement of a everywhere
a video on the split octonians would be neat
well done. really cool 😁
This is fascinating.
Can you do geometric algebra next? The dimensions scale up forever there too but much more nicely, it seems
Yes please
It just seems like all extensions beyond complex numbers are lacking. Real numbers are wonderful, but the extension to complex just feels like perfection. Everything beyond feels like you lose more in elegance and properties than you gain in extra dimensions.
Mathematics doesn't have to feel elegant to be useful though ;)
Of course, we could all just agree that these definitions don't make sense. Though we then might lose some useful applications...
I think, grahams number could be seen as not really elegant, but that doesn't really matter, right?
tell that to roboticists extensively using quaternions
@@Nettlebed7 or game developing where it feels like half of the times I look up something I need to understand quaternions to understand how it works
has anyone done some theory on the proprieties of infinite dimensional number? like, aleph_0-nions or something. How would they work? _can_ they work? would they have any useful proprieties?
it seems like such a wild concept that it can't be usefull but then again p-adics are a thing
The Title or Description should mention that this is covering the Cayley Dickenson Construction as this is one of the better and more complete lectures on the topic.
Has anyone done papers/research on infinite dimensional numbers?
FWIW, the word is "sedenion," not "sedonion." It comes from the Latin _sedecenarius_ meaning "sixteen-fold." So the word should really be "sedecenion," but I guess that was too long.
Sedenants are 4D graphical regions.
Are there any theorems like the convolution theorem that exclusively permit big algorithmic speedups (e.g. O(n^2) -> O(n log n) for multiplying large numbers, enabled by complex numbers and Fourier transform) using quaternions or octonions?
What properties do we lose going from the sedenions to the 32-dimensional algebra?
anything of interest is lost
Last remaining shreds of sanity
That from these honored dead, they take increased devotion to the task for which they gave the last full measure of devotion. 💀
I am going to pronounce that "oct-onions" and you can't stop me.
especially consiering he calls them "sed-OH-nee-ans" instead of "sed-EN-ee-ons"
@@cd-zw2tt What do you call people from Sedona, Arizona?
@@TimothyReeves "John." But he's the only one I know in Sedona. 🙂
@@cd-zw2ttHmm? Who said onions?
I would love to see some content where you could motivate an isomorphism from this construction to the language of geometric algebra. It seems to be related
Well octonions and above are non-associative, but Clifford algebras are _always_ associative, so the isomorphism stops at the quaternions.
Maybe it is more of a coincidence that the more "primitive" structures like complex numbers and quaternions show up in unrelated contructions. And yes, Octonians and above don't fit in any geometric algebra, because of associativity, as was said above.
What would the cartesian product be? Addition of elements with different grades? The isomorphic embeddings of R, C and H may just not be related to each other in a way that resembles the Cayley-Dickson construction, because it uses tools not available within one Clifford algebra.
Or maybe you would need a very large one to have blades of grades that don't interfere with each other that they become "free" = independent like the components of a Cartesian product, but then you don't actually benefit from Cliffordness.
Can you post the "inverse Cayley´-Dickson construction". The construction to go from S to R?
aa* is another way of writing a and not a --- indeed x*(x+1)=1 (adding one is conjugation) in the finite field Z2 is an irreducible polynomial and the "splitting field" is F4, where something can be both true and false without the logic dissolving into triviality ... so there is a sequence of logics, 2 valued, 4 valued, 3 valued ... and then stop because 3 is enough. #RM3
Id like to see what insights this gives us into Abstract Algebra, if we keep climbing tonhigher and hugher dinensions!
none at all
WOW...just wow!
Thanks for the great video
I find it easier to see those algebras as the even subalgebra of a n dimensional geometric algebra with unit metric
2D gives the complex numbers, 3D the quaternions, ...
The conjugate is then simply the reverse and all properties simply derives from the geometric product (no need to memorize the multiplication tables)
Also it is then easy to go to split algebras by changing the metric , for example dual quaternions is the even subalgebra of a 4D geometric algebra with a 0,1,1,1 metric
All geometric algebras are associative, the octionions aren’t! That means the octionions and beyond are not the even subalgebras of some geometric algebra.
Depending on your perspective this is either disappointing (a nice pattern doesn’t hold) or exciting (the Cayley-Dickinson Construction is a different way to generate algebras that happens to line up with the Geometric-Algebra-derived ones in low dimensions!)
@@kikivoorburg Thanks for your remark! The next question is where does the equivalence breaks between the 2 algebras ( octonions and the even subalgebra of a 4D geometric algebra with unit metric) . I guess those 2 algebras still share a lot.
Is it possible to do 3D region mappings as in the 2D region mappings in C (complex field)?
2:40 Michael is a time traveller!?
It all makes sense now...
Yes, please, more :)
I don't suppose you would know a nice mnemonics or shorthand to remember this last diagram? It's ... complex
Imagine one.
When you kept pronouncing "sedenions" as "sedonians", I kept thinking about Sedona, AZ.
I find the rules for going from one algebra to the next fascinating. The video states that it was produced by looking at R->C->H, but is this the only set of rules that can do this? And is it minimal or maximal? Can you remove or add additional rules? I'm guessing you can't just remove them, but what about removing and adding a different rule, or reframing the whole picture.
Nope, it isn't! There's another process that generates the same 3 algebras at the start, but never loses its associativity, and that is Clifford algebras.
The Reals are Cl(0,0); Complex numbers Cl(0,1); Quaternions Cl(0,2). Past that it diverges from the Cayley Dickson construction with Cl(0,3), Cl(0,4), and so on. Like the Cayley Dickson construction, each is 2 times larger than the prior, and in general Cl(0,n) is 2^n dimensional (or rather Cl(p,q,r) is 2^(p+q+r) dimensional).
Most explanations of Clifford algebras won't actually define the algebras in this way though, instead generating them from Cl(2), Cl(3), and so on, and then using the even subalgebras to extract the Complex numbers and Quaternions respectively. This method more cleanly shows that the systems in question are specifically the algebras of rotations in 2 and 3D respectively.
Some while ago i tried working out something similar to Clifford algebra, but using the Moufang identities and scalar squaring, anti-commutation and anti-association relations rather than associativity and just scalar squaring and anti-commutation relations.
This would produce O from H, but then produce something entirely else from O.
Never got the calmness of mind to complete my reasoning though, life is a bitch sometimes.
Didn't expect a surprise Fano Plane at the end there... are there more shapes as we go to other sets???
Where do the split-complex and dual numbers fit into this scheme?
I don't know if it's a relevant question but : I understand that the octonions can be seen as an algebra with a construction process that derives from the Fano plane.
Is there a way to perceive what an equivalent way of constructing an algebra would lead to when starting from a non-desarguesian projective plane, e.g. the Hall plane or its dual ? Something like the equivalent of a skew field but it would be an algebra? I thought intuitively about a malcev algebra but have absolutely no clue how to prove it.
And I forgot to tell, but of course thank you for the great quality of your online content. Helps amateurs such as me a lot!
Which book is good to read about this fascinating subject?
I'd recommend Chapter 33 in "The book of involutions", a book by Alexander Merkurjev, Jean-Pierre Tignol, and Max-Albert Knus. But I'm a graduate student mastering algebra, so this might not suit your preferences. In that case I'd recommend "On Quaternions and Octonions". A book by Derek A. Smith and John Horton Conway.
I like to see the complex system plus abut instead of moving forward of it just do the colpexer system of that system
If rotations in N dimensional space can be described by, for the lack of better phrasing, 2^(N-1)-ions and fractal geometry allows for fractional dimensions, that leads me to wonder if we can talk meaningfully about numbers like objects between the complex and the reals or the complex and the quaternions, and, if so, what sorts of properties would those numbers or number like objects would have, assuming they exist?
I was wondering the same thing, although I hadn't considered the fractal nature.
Perhaps they would be half-associative 😂😂
We should invent a system.
At this point I think we might want to ask what limits there are, if any, to the amount of different types of numbers and the possible properties of said numbers that can be logically constructed within the rules of mathematics. To what extent can the set of ALL numbers be comprehended at all?
That is not how it works. There is a correspondency between rotations of vectors, and of spinors though, but you need to study something called Bott periodicity and Clifford periodicity to understand it properly.
For real vectors of dimension n, the corresponding spinors can be real, complex or quaternion, and they can be either single spinor or two half spinors.
Each complex spinor has a conjugate spinor, these function somewhat similar to half spinors.
SO R¹ ≈ Spin R¹
SO R² ≈ Spin C¹
SO R³ ≈ Spin H¹
SO R⁴ ≈ Spin H¹±
SO R⁵ ≈ Spin H²
SO R⁶ ≈ Spin C⁴
SO R⁷ ≈ Spin R⁸
SO R⁸ ≈ Spin R⁸±
SO R⁹ ≈ Spin R¹⁶
SO R¹⁰ ≈ Spin C¹⁶
SO R¹¹ ≈ Spin H¹⁶
SO R¹² ≈ Spin H¹⁶±
SO R¹³ ≈ Spin H³²
SO R¹⁴ ≈ Spin C⁶⁴
SO R¹⁵ ≈ Spin R¹²⁸
SO R¹⁶ ≈ Spin R¹²⁸±
So for example quaternions are related to 3D and 4D rotations. While octonions (with various internal structures) are related to 5D, 6D, 7D and 8D rotations.
9D rotations are special since they introduce tensoring of previous spinors with 16D spinors, which continue by periodicity onwards.
I know that there's an identity for writing (a^2+b^2)(c^2+d^2) as a sum of squares, and similarly for (a^2+b^2+c^2+d^2)(e^2+f^2+g^2+h^2), the first coming from norms in C and the second from norms in H. Is there an analogous formula in 8,16,... dimensions?
By the way, I'm pretty sure that diagram at the end is basically projective 2-space over F_2. We can treat e_1 as the vector (0,0,1), e_2 as (1,0,1), and so on. Then that explains the looping on the collinear points.
I think the second identity you mentioned is this one: en.wikipedia.org/wiki/Euler%27s_four-square_identity
@@jakobthomsen1595 Thanks for the article. It looks as if it's only possible up to 8 variables if you want linear expressions in the squares, but there are analogues for any power of 2 if you allow rational functions.
Thanx to octonions, there is such a formula in 8 dimensions, yet there is none in 16 dimensions or higher.
So do we keep losing structure past the sedonians, or do all the ones after it have the same basic properties?
Also, we can add virtual numbers, or numbers with negative absolute values, to get even more complex numbers.
Such numbers are called split complex numbers, or rather they are part of those numbers.
Yeah, and there could be a virtual unit, v, so that abs(a*v)=-a.
|iv|=-i
also, there could be
|(-a)v|=a
Interesting, isn't it?
The general form could be:
a+bv
Higher orders could entail:
a_0+a1v1+a2v2+...+a2^n-1v^2^n-1
Just my speculation.
@@Gordy-io8sb Try conic complex numbers, also called tessarines and bicomplex numbers. They are a commutative algebra over the complex numbers, indeed they are a composition algebra with complex quadratic norm.
All complex numbers are represented as norms (or absolute values if you like) of numbers in this algebra.
Is it possible to do mappings similar to the 2D mappings in complex.
This is the epitome of
“Elementary students when their math has letters”
“Higher math students when their math has numbers”
What goes wrong if you try to use the same construction with two different algebras, for instance if you take R x C?
@6:43 you refer to the results as a "non-negative integer" when I think you meant to say "non-negative real", and likewise shortly following.
I've heard it say that each step up looses a degree of freedom or something like that and they become increasingly more limited in use.
I'd be more interested in someone discovering a successful system for say tricomplex numbers or some. Complex is two. Quaternians is 4 and 4 dimensional. Good for 3d rotations. But amongst that ladder we haven't found one that does the in-betweens.
There's no purely 3dimensional system it goes from 2d to 4d. And I believe one could be discovered but it's rules might be unique and outside of the ladder.
It's arbitrary but arguably all systems in mathematics are. As long as it works and it's useful it really doesn't matter how different it is from those in that ladder. That's what I mean by arbitrary.
Mathematics is infinite and the number of discoverable calculatable systems are infinite. As well as the uncomputable systems.
Geometric algebra explains all of this.
You can define one by saying how many and what type of basis vectors you want to have.
In general G(x, y, z) is saying there are x vectors that square to 1, y vectors that square to -1, and z vectors squaring to 0.
G(2, 0, 0) is basically 2d space and isomorphic to the complex numbers.
To get something for 3 dimensions simply:
G(3, 0, 0) -> 3D space.
The rules for working with geometric algebra are very simple and gives as a greater understanding of the objects we use.
Find out more: czcams.com/video/60z_hpEAtD8/video.htmlsi=87Zw1fA8KWX3Bedu
G(4, 0, 0) -> Quaternions
G(8, 0, 0) -> Octonions
G(2^n, 0, 0) -> 2^n-"nions"
It’s worth noting you _can_ make 3D number systems, they just don’t act like complex numbers.
If you demand that the elements of your algebra i and j square to -1 like with the complex numbers, Quaternions, etc. It is (as far as I’m aware) provably impossible to get 3D numbers to work.
This, however, does work:
1^2 = 1
i^2 = j
j^2 = i
i^3 = j^3 = -1
Giving the set {1, i, j} which is 3D and closed under multiplication! It does have zero-divisors though.
Look up the video “Let’s invent the Triplex numbers”, that’s where I got this example from. It’s well worth a watch!
I think you could define any dimensionality just by having a unit q where
q^n = +- 1
This algebra is closed under multiplication and has n dimensions:
{1, q, q^2, q^3, … , q^(n-1)}
Edit: also, multiplying by i or j in the Triplex numbers both corresponds to some rotation about the diagonal axis (passing through 0 and 1+i+j) so I’m not sure you could do other 3D rotations with it.
@@evandrofilipe1526 the relationship for the Octonions doesn’t actually hold, because all geometric algebras are association and the Octonions aren’t! The Cayley-Dickinson Construction splits off from GA after the Quaternions
Geometric algebra:
unites them all
Thanks. Gotta split!
Is there a Fano-like diagram for quaternions?
I see quaternions come up in 3-d rotations, but what is the "killer app" of the octonions?
Unit quaternions (isomorphic to SU(2)) generate rotations since they're a double cover of SO(3). Their relationship with the groups SO(3) and SU(2) are among the reasons of many other applications of quaternions to physics. In a similar way, octonions are deeply related to physics and the standard model; for more info I recommend taking a look at:
czcams.com/video/ng1bMsSokgw/video.html
Quaternions are used for both 3d and 4d rotations, while octonions may be used (in less obvious ways) for 5d, 6d, 7d and 8d rotations.
I didn't see yet the video but i just saw the last diagramm and i don't think that two arrows has to ends on e5 beacause it doesn't make the diagramm invariant under rotations. To be symmetrical, you have to reverse the arrow between e5 and e7.
What sort of idiot gets the diagram that charts the system wrong?
So wait I can start my own domain on squarespace? Is it an associative and commutative domain?
Can you explain why there are also 480 possible definitions for octonion multiplication?
That diagram at the end looked a lot like the fano plane?? Is there some connection between these algebras and projective geometry???
Yes. There is a relationship between these algebras and projective spacea over the field GF(2) = Z/2Z. This is because every generator unit squares to a scalar.
@14:30-ish there's an interesting claim about phi(C). The fact that phi(C) behaves like R' does not rule out the supposition that C has some je nais se quoi not preserved under phi.
It is interesting: are there possible systems with arbitrary number of imaginary units, say 5, 6, 7 and so on? If no then why? What are the applications of such hypercomplex systems? We know vast usage of real and complex numbers, using quaternions for 3D rotations, how other systems are used?
The answer I find the most satisfying is from geometric algebra (though it’s far from rigorous):
Basically, any time you introduce an element in a geometric algebra, you end up ‘generating’ other elements from the multiplication.
Let’s try to construct an algebra with just a single “-1” squaring unit we’ll call “i”. Then we have:
i^2 = -1
Which ‘generates’ the scalar values. So actually we can’t have a geometric algebra with only “i”, we need {1, i}.
Let’s try another -1 squaring element “j”, giving {1, i, j}. Watch what happens:
1^2 = 1
i^2 = -1
j^2 = -1
ij = ?
What does “ij” equal? Well, in geometric algebra, different basis elements anti-commute. Hence:
ij = -ji
This lets us figure out what it squares to:
(ij)^2 = ij ij = -i jj i = ii = -1
So by introducing “j” we also ‘generate’ another -1 squaring element which we can label k.
So we have {1, i, j, k} where ij = - ji = k and by extension ijk = (ij)^2 = -1. This is the definition of the Quaternions!
Notice that we were able to have the 2D complex numbers and the 4D Quaternions, but not 3D-complex like numbers since it sort of “wants” to generate another element.
Sadly the explanation doesn’t extend to Octonions since they’re non-associative and all geometric algebras are associative, but I find it a nice way to understand the case for Quaternions more intuitively at least
This felt like playing around with notation yielded a more general ways to describe a set of similar yet previously un unified systems. Like something Conway would do, with the transcendental numbers. It’s cool to see.
Although I didn’t understand why it was fine to just assume that certain properties of the constructions held - like associativity or commutativity - without first proving them. That probably just made this video shorter and more engaging which I guess I can get behind :) …
For associativity and commutativity at least, I think the assumption of their holding in the reals is reasonable since that proof is probably a bit out of scope of the video. He proved commutativity in the complex numbers, and that's the last algebra where it holds. He did skip associativity in the complex, but I think the proof of that would look the same as the proof of it in the quaternions, which he did show
Ok this is a wild question but could we use numbers of n dimension where n approaches infinity? Negative dimensions? Dimension approaching zero?
How is addition for A' defined? Isn't this important?
Is quaternions as H the upper half plane?
the buildup to groups of lie type is appropriate.
2:24 this is the time police, did you see your future self coming from the 21 century!
we want cayley-dickson construction for split complex and dual numbers
Know can go left rather right and make hyper real numbers.
Does watching Michael show associativity for the quaternions count as doing it once in my life? 🙂
That depends on exactly how compulsive you want to be / have to be, or alternatively, it depends on whether you have a life outside mathematics.
(But maybe I'm being unfair to real mathematicians. I just don't have it in me to be one, and my first sentence was "sour grapes").
the classic undergraduate question
Why does the cross product 'only' work in three and seven dimensions?
Because no higher C-D algebras than quaternions and octonions are composition algebras. Cross product works trivially in one and zero dimensions also.
Yes, split Quaternions :D
Has anyone invented an easier way to label the octonions? It seems bulky and obtuse. Like, if Ea * Eb = Ec, there should be a simple function relaring a, b, and c. But I could be totally wrong - maybe that isnt possible.
The answer for the value of x when x²=-1 is any number on an infinite dimensional unit sphere in a complex plane
It seems that the octonions are somewhat arbitrary in the order they are mapped to e1 to e7. Is there some reason this standard definition is so seemingly random in its mapping? If I were starting from scratch I'd probably choose (0,1), (i,0),(0,-i),(j,0),(0,-j),(k,0),(0,-k) or some similar "natural" progression. Obviously this would alter the details of the multiplication table but the essence would be the same. I'm therefore curious why the standard definition seems so arbitrary.
It's arbitrary because it's a definition that was developed historically and then used with that definition after that. I see this type of misconception on CZcams Math videos, it's just a convention.
@@holliswilliams7717 So, like the tragedy of electrons being negative.
can’t wait for the dārskubï-helvetica numbers
Can we use these things to do higher-dimensional rotation?
Not really. Higher dimensional rotations are generally much better described using Clifford algebras. In general, the algebra of rotations in n dimensions is Cl⁺(n), or the even subalgebra of the Clifford algebra over R^n. While this aligns with the Complex numbers and quaternions for Cl⁺(2) (aka Cl(0,1)) and Cl⁺(3) (aka Cl(0,2)), they diverge beyond that as Cl⁺(4) only has 6 elements that square to -1 and 2 elements that square to +1, along with being associative but, I think, having zero divisors.
If you've read about 4D rotations, it shouldn't actually be that hard to see why things diverge since it's the first time you encounter "double rotations" or rotations that can only be described as 2 rotations around separate axes of rotation, rather than just a single rotation around only 1 rotation axis. This is also why I omitted Cl⁺(1) = Cl(0), since in general, an additional acid of rotation gets added every 2 dimensions. 0 and 1D have 0; 2 and 3D have 1, 4 and 5D have 2, etc.
The extra element that squares to +1 instead of -1 is exactly used to represent these double rotations, whereas there are only 6 distinct primitive rotation axes. (1 not-rotation, 6 single rotations, and 1 double rotation.)
@@angeldude101 Also, a torus is a 4D shape that has 2 radii, a doughnut is a 3D cross-section of a torus, that's why it has no beginning, middle, or end.
@@AlbertTheGamer-gk7sn Uh... What‽ A torus is a 2D manifold in 3D space. There's absolutely nothing 4D about it unless you go out of your way to rotate it into the fourth dimension. Spinning a torus around its two separate axis (the linear one through its hole and the circular one through its interior) can act like a 4D double rotation in some ways, but it's just as easily described as resembling 3D screw motion, or, well... Rotation around a circular axis (which is exactly what we initially defined it as).
What's the advantage of using these algebras over Geometric Algebra?
aka Jacob's Ladder
So the complex plane is really just an extension of the number line into 2d. And quaternions are the 4d version?
There is also a real-imaginary-virtual number system that is 3D, with virtual numbers that have negative absolute values.
@@AlbertTheGamer-gk7sn I always viewed negative numbers as a kind of hack. Like "let's just add one bit of data to numbers and see what happens".
But it seems they are quite intrinsic to all of math. I guess i've changed my mind on how i feel about negatives now.
How do you pronounce "sedenions"? I'm just a math layperson. It was misspelled in the video, so maybe that's where the misunderstanding lies.
22:30 why did we do water?
At 6:44 probably you meant "non-negative real numbers" instead of "non-negative integers"
Real numbers are not a field. They are a projective scheme. There is no multiplicative inverse for some a that is zero, so they try to make an exception, but it doesn’t really work.
why do dimensions of these algebras appear to be powers of 2?