SuperComplexNumber

A general implementation for super complex number, where $i^2 = -1$, $\theta^2=0$, $\mu^2=1$ (non-trivial)

There is some staff in quantum physics that allows you to have such object, that

$$ \theta^2=0 $$

meanwhile $\theta \ne 0$

And also there is a way to define such object that

$$ \mu^2=1 $$

meanwhile $\mu \ne 1$

So here I am presenting the mind blowing implementation for arithemtic for that stuff.

Thanks to some linear algebra machinery, I was able to construct such set, that this super complex number perfectly maps one two 2 by 2 matrix to exactly one super complex number, so these set of $real, i, \theta, \mu$ is basis for two 2 by 2 matricies, and I was able to create analytic continuation of any complex-valued function to this super complex numbers.

Look bro

var real = SuperComplex.Real;
var i = SuperComplex.Imaginary;
var th = SuperComplex.Theta;
var mu = SuperComplex.Mu;

SuperComplex v1 = 3 * real + 4 * i + 2 * th + 4 * mu;
SuperComplex v2 = 6 * real + 1.5 * i - 6 * th - mu;

System.Console.WriteLine(v1);
System.Console.WriteLine(v2);
System.Console.WriteLine(1/v2+2*v2*v2);
System.Console.WriteLine(v2.Apply(x=>1/x+2*x*x));

It outputs

(3 + 4i + 2θ + 4μ)
(6 + 1.5i - 6θ - 1μ)
(146.29999999999998 + 34.800000000000004i - 139.20000000000002θ - 23.200000000000003μ)
(146.29999999999995 + 34.799999999999955i - 139.20000000000002θ - 23.200000000000003μ)

Here in v2.Apply I only giving complex valued function, but it gives same output as value computed on super complex plane!

Isn't it insane?

Yeah, so you can take sin, cos, any function that is defined on complex plane is defined on super complex plane as well!

Code is pretty basic except for Apply method, which uses technique from this video to expand matrix-view of super complex numbers to work on any function.

See https://www.youtube.com/watch?v=-1loSrioE4Y&t=527s

Kemsekov/SuperComplexNumber

SuperComplexNumber