[julia-users] Re: Trivia question: Quaternions (ℍ), Octonions (𝕆), Sedenions (𝕊) etc. numbers are not supported, but are the in libraries or needed? And complex, vs. in others, say MATLAB..

[Steven Sagaert]

quaternions might be useful for 3D rotations but higher order constructs 
like octionions, etc will not be very useful for numerical computing. They 
might be useful to pure mathematicians (or as an alternative math formalism 
for some (speculative) QFT stuff) but not in "applied" math, and pure 
mathematicians use Mathematica IF they use a computer at all (most of these 
esoteric creatures still stick to pencil & paper :) ).

On Friday, August 28, 2015 at 11:50:45 AM UTC+2, Páll Haraldsson wrote:
>
> Mostly about math (that I do not know to well, this advanced, above 
> complex numbers), just seems relevant to generic programming..:
>
> In e.g.:
> https://en.wikipedia.org/wiki/Complex_number
>
> in the template at the bottom:
>
> Real numbers and extensions:
>
> Real numbers (ℝ) Complex numbers (ℂ) Quaternions (ℍ) Octonions (𝕆) 
> Sedenions (𝕊) Cayley–Dickson construction Dual numbers Split-complex 
> numbers Bicomplex numbers Hypercomplex numbers Superreal numbers Irrational 
> numbers Transcendental numbers Hyperreal numbers Levi-Civita field Surreal 
> numbers
>
>
> http://math.stackexchange.com/questions/86434/what-lies-beyond-the-sedenions
>
>
> I see that dual numbers are supported already with a library, and know 
> what Julia does support.
>
>
> Since Julia IS a scientific language, I checked:
>
> julia> 1+0im < 2+0im
> ERROR: `isless` has no method matching isless(::Complex{Int64}, 
> ::Complex{Int64})
>  in < at operators.jl:32
>
> that is the right thing. I wander what MATLAB does, since all numbers(?) 
> are stored as complex. 0im special cased?
>
>
> real numbers have properties that floating point already doesn't support, 
> so loosing those properties with higher order numbers seems not to be an 
> issue, but is it for the even higher order numbers and those properties? I 
> assume not, as there are no operators/functions and even if there where 
> operators (really just functions), then they would just stop working.
>
> In a language where everything is typed (not generic), this might not be a 
> problem, but leads to run-time "ERROR"s/exceptions. Should you use ::Real 
> at every point to not get those with Complex? What about other possible 
> numbers?
>
>
> Thanks in advance,
> -- 
> Palli.
>
>
>