3-dimensional split-complex numbers

• I
• Anixx
In summary: Your Name]In summary, the conversation discusses the importance of bicomplex numbers and how they are often overlooked in overviews of hypercomplex numbers. The properties and construction of bicomplex numbers are presented, including their potential applications in various fields of study. The conversation also mentions the Mathematica code that can be used to experiment with these numbers. Overall, the conversation highlights the significance of bicomplex numbers and encourages further exploration and study of their properties and applications.
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.

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:

Code:
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}##

WWGD and Maarten Havinga

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.

What are 3-dimensional split-complex numbers?

3-dimensional split-complex numbers are a type of number system that extends the traditional complex numbers by adding a third dimension. They are also known as "hyperbolic numbers" and are represented as (a, b, c) where a, b, and c are real numbers.

How do 3-dimensional split-complex numbers differ from traditional complex numbers?

Unlike traditional complex numbers, 3-dimensional split-complex numbers have a non-commutative multiplication rule, meaning that the order in which the numbers are multiplied matters. They also have a different conjugation rule, where the conjugate of a number (a, b, c) is (a, -b, -c).

What are some applications of 3-dimensional split-complex numbers?

3-dimensional split-complex numbers have applications in physics, particularly in the study of special relativity and Minkowski spacetime. They are also used in computer graphics and robotics for their ability to represent rotations in three dimensions.

Can 3-dimensional split-complex numbers be visualized?

Yes, 3-dimensional split-complex numbers can be visualized using a 3D coordinate system. The real part of the number is represented on the x-axis, the split-real part on the y-axis, and the split-imaginary part on the z-axis. This allows for a visual representation of the multiplication and conjugation rules.

Are there any other types of split-complex numbers?

Yes, there are other types of split-complex numbers, including 2-dimensional split-complex numbers and 4-dimensional split-complex numbers. Each type has its own unique properties and applications.

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