/sci/ - Science & Math » Thread #13028287

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Anonymous Sun 25 Apr 06:36:39 2021 No.13028287 View Reply Original Report

Quoted By: >>13028291 >>13028295 >>13028369 >>13028808 >>13028817 >>13029657 >>13029738 >>13031617

What is the quickest way to understand quaternions

Anonymous

Anonymous Sun 25 Apr 2021 06:38:07 No.13028291 Report

Quoted By:

>>13028287
i don't know

Anonymous

Anonymous Sun 25 Apr 2021 06:40:29 No.13028295 Report

Quoted By:

>>13028287
pair of vectors with defined multiplication of vector

Anonymous

Anonymous Sun 25 Apr 2021 06:41:20 No.13028297 Report

Quoted By:

The quickest way is to have had a nice mother who took the time to teach you basic mathematics as soon as you were old enough to learn so that math became as natural to you as walking or speaking your native tongue. Once you've have that experience, understanding higher concepts becomes lightning quick.

Anonymous

Anonymous Sun 25 Apr 2021 07:10:33 No.13028369 Report

Quoted By: >>13029677

>>13028287
Look up 3b1b's vid on it

Anonymous

Anonymous Sun 25 Apr 2021 10:45:27 No.13028808 Report

Quoted By:

>>13028287
First learn about field extensions.

Anonymous

Anonymous Sun 25 Apr 2021 10:48:05 No.13028817 Report

Quoted By: >>13029674

>>13028287
Find one and experiment with it. Oh they don't exist in the real world? Then they should not be studied in mathematics either.

Anonymous

Anonymous Sun 25 Apr 2021 10:51:34 No.13028824 Report

Quoted By:

A quaternion can be described as a 2-tuple of a scalar and a 3-vector: $a + bi + cj + dk = (a, v)$
Then addition is normal elementwise addition and multiplication is given by:
$ (x, u)(y, v) = (xy - u \cdot v, xv + yu + u \times v) $

Anonymous

Anonymous Sun 25 Apr 2021 16:44:24 No.13029657 Report

Quoted By: >>13029664 >>13031623

>>13028287
a+bi+cj+dk = (a+bi) + (c+di)j = z+wj
A pair of complex numbers with a funky multiplication.
If you take two of these and multiply them together, you find it is possible to represent them in a matrix with no reference to j or k.
$z+wj \sim \begin{bmatrix}z& w\\ - \bar w & \bar z\end{bmatrix}$

Anonymous

Anonymous Sun 25 Apr 2021 16:46:16 No.13029664 Report

Quoted By: >>13029676

>>13029657
Now expand w and z to 2x2 blocks

Anonymous

Anonymous Sun 25 Apr 2021 16:48:12 No.13029674 Report

Quoted By: >>13029700 >>13029728

>>13028817
Electromagnetism is most elegantly described via the calculus of quaternions.

Anonymous

Anonymous Sun 25 Apr 2021 16:48:54 No.13029676 Report

Quoted By: >>13029680

>>13029664
You get no imaginary numbers at all

Anonymous

Anonymous Sun 25 Apr 2021 16:49:04 No.13029677 Report

Quoted By:

>>13028369
it's one of his worst videos, extremely unhelpful

Anonymous

Anonymous Sun 25 Apr 2021 16:50:07 No.13029680 Report

Quoted By: >>13029707

>>13029676
Yeah, that's the point.

Anonymous

Anonymous Sun 25 Apr 2021 16:56:59 No.13029700 Report

Quoted By: >>13029713

>>13029674
no it isnt lmao, modern vector calvulus was invented specifically because people (Heaviside) were fucking tired of cumbersome quarternion equations in EM

Anonymous

Anonymous Sun 25 Apr 2021 16:58:02 No.13029707 Report

Quoted By: >>13031623

>>13029680
$a+bi+cj+dk \sim \begin{bmatrix}a& b& c& d \\ -b& a& -d& c\\ -c& d& a& -b\\ -d& -c& b& a \end{bmatrix}$

Anonymous

Anonymous Sun 25 Apr 2021 17:00:44 No.13029713 Report

Quoted By:

>>13029700
Then they ironically came back as pauli matrices.
They were probably the right choice from the start.

Anonymous

Anonymous Sun 25 Apr 2021 17:05:04 No.13029728 Report

Quoted By:

>>13029674
Really, is there a good textbook that takes this approach?

Anonymous

Anonymous Sun 25 Apr 2021 17:09:04 No.13029738 Report

Quoted By:

>>13028287
you need to be comfortable with complex numbers, including the idea of them itself. quaternions are ""the same thing just bit more complicated"".

Anonymous

Anonymous Mon 26 Apr 2021 01:01:19 No.13031617 Report

Quoted By:

>>13028287
visualize the I j k components as axis of rotation

Anonymous

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Anonymous Mon 26 Apr 2021 01:04:29 No.13031623 Report

Quoted By:

>>13029707 >>13029657
How about Octonions and Sedenions?

https://en.wikipedia.org/wiki/Cayley–Dickson_construction

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