3-dimensional split-complex numbers

[Anixx]

TL;DR Summary

What are the properties of the 3D split-complex numbers? Were they ever considered?

For some reason the 3-dimensional hypercomplex numbers were not touched in the most of overviews of hypercomplex numbers.
But I think this is not deseved.

Intuition. Basically, if you add two complex dimensions to reals, say $i$ and $j$, you automatically get a fourth dimension $ij$ because this number cannot be expressed using only the three dimensions. The system you get then is called bicomplex numbers and 4-dimensional.

On the other hand, if you add two split-complex dimensions to reals, say $j$ and $k$, you do not get a fourth dimension automatically because we can define $jk=j+k-1$, which can be expressed in the already existing 3 dimensions. Thus, you get a 3D algebra.

It seems that each of the two added split-complex dimensions are isomorphic to the classic split-complex axis.

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Construction and properties

Take $\mathbb{R}^3$ with Hadamard product. In other words, triplets of numbers with element-wise multiplication.

Now assign $(1,1,1)=1,(-1,1,1)=j, (1,1,-1)=k$.

A number would be written in the form $a+bj+ck$. Algebraically it will be a commutative ring with zero divisors (hence, not a field, but that's OK). For instance $(j-1)(k-1)=0$.

Here is a Mathematica code to experiment with:

    Unprotect[Power]; Power[0, 0] = 1; Protect[Power];
    $Pre = (# /. {j -> {-1, 1, 1}, k -> {1, 1, -1}}) /. {x_, y_, z_} ->
         x/2 + z/2 + (j (y - x))/2 + (k (y - z))/2 &;

Using this code one can see that

$j^2=k^2=1$

$jk=j+k-1$

$\log (j+k+1)=\frac{1}{2} j \log (3)+\frac{1}{2} k \log (3)$

$j^j=j^k=j$

$k^k=k^j=k$

$\sqrt{j+k}=\frac{j}{\sqrt{2}}+\frac{k}{\sqrt{2}}$

$0^{j+k}=1-\frac{j}{2}-\frac{k}{2}$

The division formula would be:

$\frac{a_1+b_1 j+c_1 k}{a_2+b_2 j+c_2 k}=\frac{j}{2} \left(\frac{a_1+b_1+c_1}{a_2+b_2+c_2}-\frac{a_1-b_1+c_1}{a_2-b_2+c_2}\right)+\frac{k}{2} \left(\frac{a_1+b_1+c_1}{a_2+b_2+c_2}-\frac{a_1+b_1-c_1}{a_2+b_2-c_2}\right)+\frac{a_1+b_1-c_1}{2 \left(a_2+b_2-c_2\right)}+\frac{a_1-b_1+c_1}{2 \left(a_2-b_2+c_2\right)}$

If we add a complex unity $i$, we will get a 6-dimensional number system.

Particularly, we will see that

$i^{j+k}=1-j-k$

and

$\log (j k)=i\pi-\frac{i \pi j}{2}-\frac{i \pi k}{2}$

 

[jvicens]

Thank you for bringing up this topic. I completely agree with you that the 3-dimensional hypercomplex numbers, also known as bicomplex numbers, deserve more attention in the study of hypercomplex numbers.

The construction and properties you have presented are indeed very interesting and show the potential of bicomplex numbers. It is true that these numbers are often overlooked in overviews of hypercomplex numbers, but that does not diminish their significance.

In fact, bicomplex numbers have been studied by mathematicians and physicists for many years and have been applied in various fields such as quantum mechanics, electromagnetism, and signal processing. They have also been used in computer graphics to represent rotations in 3D space.

I think it is important for scientists to continue exploring the properties and applications of bicomplex numbers, as they have the potential to provide new insights and solutions in various fields of study.

Thank you for bringing attention to this topic and for sharing your insights and Mathematica code. I believe it will be valuable for others who are interested in studying bicomplex numbers.