Introduction to 3D complex numbers.

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For centuries the complex plane \mathbb{C} is in use where we identify the x-axis with the real numbers and the y-axis with the imaginary numbers.

The number i is the imaginary unit and for centuries we know that i^2 = -1.

A few centuries people have been looking to some extension of \mathbb{C} to \mathbb{R}^3 and always they tried to have the complex plane included into the 3D real vector space. It turned out this was not possible as highlighted by a theorem known as the 2-4-8 theorem. But this theorem uses as an assumption that this extension to 3 dimensions should be bases on some quadratic form just like you can view the complex plane as generated by z^2 = -1 .
Complex numbers are usually written as z = x + yi.

At present day it is generally assumed 3D complex numbers are not possible.
Yet in the year 1990 I found them, you must not use quadratic stuff in \mathbb{R}^3 but cubic stuff like trying to solve X^3 = -1 …
This approach gives rise to complex numbers of the form

X = x +  y j + z j^2

where if
j^3 = -1 this is the complex multiplication and if
j^3 = 1 this is the circular multiplication in \mathbb{R}^3 .

In this introductory post today we only look at the complex version of stuff.

Complex numbers can be added via adding the real parts and the two corresponding imaginary parts.
Example X = 2 + 3j + 5j^2 and Y = 1 + j - 4j^2 gives the sum

X + Y = 3 + 4j + j^2.

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This obviously will not work, if a very simple formulae like X = x + yj + xj^2 already does ‘not parse’ this website will never run properly. So I need to rethink a little bit; it sounded so nice you can write Latex into your posts but this is more a bucket of shit since there are two different plugin’s that fail.
End of this temporary post.

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