@ValeryOchkov wrote:
In my students time I have had a dream about 3D "complex numbers": CN := 1 + 1i + 1j
Hi Valery. You can't, because you lost then usual algebra properties, the soul of the "number" concept: those things that can be added, multiply, divided, etc.
From the wikipedia article: https://en.wikipedia.org/wiki/Hypercomplex_number
"The Cayley–Dickson construction used involutions to generate complex numbers, quaternions, and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem says finite-dimensional real composition algebras are the reals ℝ, the complexes ℂ, the quaternions ℍ, and the octonions 𝕆, and the Frobenius theorem says the only real associative division algebras are ℝ, ℂ, and ℍ. In 1958 J. Frank Adams published a further generalization in terms of Hopf invariants on H-spaces which still limits the dimension to 1, 2, 4, or 8.^{[2]"}
^{Best regards.}
^{Alvaro.}
Hi. Maybe some complicate citation. What those theorems says is that if you define "numbers" like a+bi, a+bi+cj, a+bi+cj+dk, etc, you have algebra properties only for the cases 1,2,4 and sometimes 8. That's reals, complexes, quaternions and octonions. So, those are the only "true" numbers, because they have something like an algebra (for instance, a finite-dimensional real composition algebras) For the case 3 you can define z = a+bi+cj but can't call that thing as "number".
@AlvaroDíaz wrote:
Hi. Maybe some complicate citation. What those theorems says is that if you define "numbers" like a+bi, a+bi+cj, a+bi+cj+dk, etc, you have algebra properties only for the cases 1,2,4 and sometimes 8. That's reals, complexes, quaternions and octonions. So, those are the only "true" numbers, because they have something like an algebra (for instance, a finite-dimensional real composition algebras) For the case 3 you can define z = a+bi+cj but can't call that thing as "number".
I think about the using complex number Mathcad and Math in general functions for the solving nor only 2D problems (your for example) bot 2D problems too...
Oh, I see. But I can't get an algorithm which take some visible less time than your original vector-component solution, like in the two dimensions case.
Best regards.
Alvaro.