# Introduction to 3D complex numbers.

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.