The complex number family. 

I got stuck at that too. Hopefully what you said is what he intended. I guess sqrt should be well defined given beta^2 is real.

I wouldn't call these weird... they are quite natural.

@@jsmdnq Some of them are natural... Others are rather irrational, and some don't even seem real

​@@Tim3.14 lol 😂

I haven't seen this stuff before, but something wasn't quite adding up around there.

i was looking for exactly this comment. Thanks

I usually just act like the split complex numbers are just the basis vectors of GA

There is also one author, M.E. Irizarry-Gelpí, who uses different terms: „perplex numbers” for numbers with j^2=1 and „nilplex numbers” for numbers with ε^2=0. The latter term is unique to this author. I like these terms more than “double numbers” and “dual numbers”, as those are confusing, whereas “nilplex” is immediately associated with ε^2=0.

Wait how? I'm not getting that.

​@@neopalm2050 it's not hard. I prooved about 3 weeks ago. Now I will see of dual numbers have some kind of CR equations, but just by it's definition it shouldn't.

ah yes, the duel numbers :D today: pi vs e! fight!

The “trouble” of which you speak is called “gimbal lock”. We encounter it in CG, too.

​@@MrLikon7 Last I heard, Euler brokered a 5-way peace agreement between _e,_ π, _i,_ 0, and 1.

As a fellow engineer, I feel that the tool-in-a-toolbox approach that we take is as it should be. I wouldn't characterize it as a 'problem.' There is way too much knowledge out there for any one person to learn in a lifetime. So we have to be very choosy about what we learn, and carefully manage how much time we spend learning. After all, we do need to leave time for ourselves to solve problems in the real world and earn a living! I watch these kinds of videos-math, modern physics, biology etc.-because, yes, I do find them very interesting. But I will not be using them in my work-unless I change my line of work 😜. In fact, I don't even use most of the things I learnt in engineering college-they taught us way too many things, hoping to cover all bases. On the other hand, I've had to unlearn and relearn a lot of other things as my work kept evolving. No doubt I could study these topics in detail if I wanted to-but that time would have to come from something else that I'm already doing. Priorities! Back in college, if I tried to major in all the courses that I was interested in, I'd never have graduated. As it is, 4 years was plenty! 🤣

@@MrLikon7 yes. Epsilon lost to itself. Its square is 0.

Lmao

It's that or "I" or "e"

that is a useful convention for the abstract general setting - after all we do not usually call the identity matrix 1

that made me so confused

​@@taeyeonlover it's confusing because Michael's notation is confusing; it is better to "write 1/sqrt(abs(beta²)) · beta" to emphasize that the denominator is a scalar multiplication, because if you confuse it for ordinary multiplication then it looks like you can cancel the numerator and denominator ( even more so for "j = beta/sqrt(beta²)" )

@@schweinmachtbree1013 ah no it was just that the sqrt missing threw me off

Since those three conic sections are obtained by intersecting a plane with a double-cone at different angles, I imagine the parameter you're looking for would relate to that angle.

The thought that occurred to me is that the hyperbolic trig functions relate to a hyberbola in the same way the standard trigs functions relate to a circle. I wondered if there's a complementary extension of the trig functions that relate to parallel lines. Of course, if there were, it would be completely useless!

@@elliottmanley5182 The circular and hyperbolic trig functions are ultimately just the even and odd parts of the imaginary and real exponential functions respectively. So finding a dual cosine and sine would just be to find the even and odd parts of e^αε = 1 + αε + (αε)²/2 + ... Normally you'd go further, but we already hit ε² so everything past it would be 0. So a cosd(α) = 1 and sind(α) = α. Significantly less interesting than sin/cos or sinh/cosh.

@@angeldude101 _less_ interesting? I'd say the fact that sind and cosd are algebraic instead of transcendental is pretty interesting. Especially from computational standpoint, because it means you can do arithmetic with them with fewer approximations.

Visualizing the basis (1,a) of R(a), where a^2 is in R, in a cartesian plane, we can indeed observe that the unit circle, that is all z in R(a) s.t. z•z*=1 (where the * denotes the conjugate) goes from a circle (a^2=-1) to a ellipse (-1

Thank you. That makes sense. I had to pause the video and look through the comments for an explanation when that bit didn't follow.

Hopefully, this won't be perceived as a rant by most. I think that the analogue of complex analysis over the double numbers is not as nice as their linear algebra. This is because linear algebra uses the conjugation operation (a + bj)^* = a - bj. Linear algebra over the double numbers depends strongly on this operation. Complex analysis studies the analytic functions, which do not include conjugation. I think that because of the isomorphism R(j) = RxR, a holomorphic function over the double numbers is just an arbitrary pair of differentiable real functions, so double-number analysis is effectively real analysis AFAICT.

Those are indeed all isomorphic to their respective 2D R-algebras

1 + -1 = 0, j² + i² = ε²

@@angeldude101 nice!

This raises the question of other quotient rings such as real polynomials/x³+x²+1. Maybe they end up just being complex numbers with a weird choice of I or maybe not

There's a fractal called the Burning Ship Fractal that you get if you mess up the Mandelbrot equations slightly.

@@thewhitefalcon8539 burning ship utilize standard complex numbers, but get absolute value (modul) from the complex part (is not that easy but this is the basic idea)

which area of math is this I would really love to know

I was wondering the same thing

@@Zeitgeist9000 I have searched for it and I got the result something about hypercomplex number in group representation theory

@@Zeitgeist9000 I just wanna ask you one thing that anytime you hear about any new math topic that has recently been discovered what would be your thoughts about it pls answer

like you wanna learn it or just "ahh heck with it I am not doing this"

I had this problem too, I still do to some extent, but it gets better. It also depends on how familiar is the subject you read.

It's definitely much easier to see (e1e2)(e2e3) = e1e2e2e3 = e1e3 = -e3e1 than to try and remember ij = k.

@@angeldude101 Agreed. GA to me has been a revelation. Covers large parts of Tensor calculus, Quaternions, Complex numbers, Dirac algebra .... Then you have the physical equations. ∇F=J/(ϵ0c)+MI One equation, the whole lot. On complex numbers, I think there is a case for teaching the GA approach first at a very basic level, to show that things can square to -1 in a real sense.

Quaternions are actually a subset of the geometric algebra (as are complex numbers and split complex numbers).

It's also a bit like the cross product.

@@Noam_.Menashe But the main difference is that the cross product is evil and needs to die in a fire because it's completely unintuitive both geometrically and algebraically. The wedge product by contrast is very intuitive both geometrically and algebraically.

Dual numbers can be used for automatic differentiation since the "imaginary" term acts like the derivative in many ways. While it's not specifically the dual numbers, a basis squaring to 0 is also useful in representing translations of objects, much like complex numbers are good at rotating, and split-complex numbers are good at hyperbolic boosts (more niche than the other two, but very useful in relativity).

Probably has applications for things involving perturbations. Maybe even stochastic calculus.

If so, how about taking a similar "family" approach and explaining how constraints of addition and multiplication drop away as you move from complex to quaternions to octonions to sedenions?

A bit more background reading today: I started to get a handle on Clifford algebras and then got lost completely when I got to Bott Periodicity.

@@elliottmanley5182 8N = N8

If you were to just square a split-complex number, then you'd get something like (x+yj)^2 = x^2 + 2xyj + y^2; not quite what we're looking for. The conjugate is needed to eliminate the middle term of the expansion and get a pure difference of squares.

Now I might need to go re-watch them as it's been a while. Those were good ones.

I just realized it's actually the reverse but still

Interesting if this mixture yields anything

Geometric algebra lets you specify the signature of the algebra as how many bases square to 1, -1, and 0. You can have G_1,1,1(R) which has {1, i, j, ε, ij, iε, jε, ijε}. In practice you don't need this since two positive bases already have an imaginary product, and it's possible to make a dual basis vector from a positive and a negative basis.

I believe that this 4d structure would just be the set of all 2x2 matrices. If you represent i,j, and epsilon as matrices it makes sense

You could also make them as 8x8 matrices (technically in 3 ways, but all will act the same). You can also make them into 4×4 matrices in three distinct ways, following each pair, which won't commute with the other one in the pair, but they should both commute with the one left out.

The hyperbola and straight lines don't converge to anything, so the closest thing they each have to pi would probably just be infinity.

Split quaternions? Only j^2 and k^2 are 1

I believe the `jk=i` form rules are generally derived from a rule `ijk = -1`.

Yup! e^ix = cos(x) + isin(x), e^jx = cosh(x) + jsinh(x). Dual numbers have an odd variant: e^iε = 1 + εx.

modulus is length. usually modulus squared is product of self and conjugate m(z)^2 = z * conj(z) and this number should be "simpler" than the object you started with (vector -> scalar) it should "project" down to the length scalars

It's a later insight. Minkowski represented special-relativistic spacetime by putting a factor of i from the complex numbers on the time coordinate, but defining the norm in a "Euclidean" way otherwise, without taking complex conjugates, so that a minus sign appeared in the calculation. In one space and one time dimension that's equivalent to the split-complex numbers, just written slightly differently.