Timeline for Is there a way to introduce quaternions and octonions in a similar way to how we are typically introduced to complex numbers?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 3, 2023 at 1:27 | answer | added | krm2233 | timeline score: 1 | |
| Sep 17, 2017 at 23:36 | answer | added | Kimball | timeline score: 1 | |
| Sep 17, 2017 at 15:00 | comment | added | anon | The motivation for constructing the quaternions is more geometric than purely algebraic. Hamilton wanted 3D rotations the way complex numbers yield 2D rotations, so wanted a 3D number system equipped with a multiplicative norm. However, the norm didn't work out to be multiplicative... until he realized that $ij$ (the product of $i$ and the new $j$) must be linearly independent, i.e. jut out in a 4th dimension. Then it becomes more evident the relations they must satisfy in order for multiplicativity to be achieved, after which we get 3D and 4D rotations via quaternions. | |
| Sep 16, 2017 at 23:27 | comment | added | user451844 | youtube.com/watch?v=3BR8tK-LuB0 might be a good start ? | |
| Sep 16, 2017 at 23:25 | comment | added | J. M. ain't a mathematician | "cannot solve it without" - that's a bit strict, I think. One could certainly choose to work instead with certain $4\times 4$ matrices with properties that allow that system to be isomorphic to the quaternions. The quaternions just happen to be a convenient and compact representation. And as already mentioned, the dual numbers are a very useful system. Have a look at the "split-complex numbers" as well. | |
| Sep 16, 2017 at 23:19 | comment | added | Tyberius | There actually is an interesting number system obtain by defining a new number $\epsilon^2=0$. These are called the [Dual Numbers])(en.wikipedia.org/wiki/Dual_number) | |
| Sep 16, 2017 at 23:14 | review | First posts | |||
| Sep 16, 2017 at 23:17 | |||||
| Sep 16, 2017 at 23:09 | history | asked | ajd138 | CC BY-SA 3.0 |