Implementation of tessarines in Mathematica

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  • Thread starter Anixx
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    Mathematica
In summary, the speaker shared a method for easily implementing split-complex numbers and tessarines in Mathematica. They provided a code using the $Pre function and suggested trying various functions using the split-complex unity J. They also mentioned a similar code for implementing dual numbers using the dual unity Eps.
  • #1
Anixx
80
12
Hello, guys!

I just wanted to share that there is a way to easily implement split-complex numbers (and tessarines) in Mathematica.

Code:
$Pre = If[FreeQ[#, J], #, Module[{tmp},
       
     tmp = Evaluate[
        MatrixFunction[Function[J, #], {{0, 1}, {1, 0}}]] //
       FullSimplify;
           tmp /. {{a_, b_}, {b_, a_}} -> a + J b]] &;

After this code you can use split-complex unity J in any functions. I recommend to try
Code:
J^I
I^J
J^J
Log[J]
Sqrt[J]
Gamma[J + 2]
Zeta[J + 1]
PolyGamma[J + 2]
Exp[J]
(-1)^J
2^J

Have fun!
 
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  • #2
A similar code implementing dual numbers (dual unity Eps):

Code:
$Pre = If[FreeQ[#, Eps], #,
    Module[{tmp},
     tmp = Evaluate[
        MatrixFunction[Function[Eps, #], {{0, 0}, {1, 0}}]] //
       FullSimplify;
     tmp /. {{a_, b_}, {0, a_}} -> a + Eps b /. {{a_, 0}, {b_, a_}} ->
        a + Eps b]] &;
 

Related to Implementation of tessarines in Mathematica

1. What are tessarines and why are they important in Mathematica?

Tessarines are a complex number system that extends the traditional complex numbers. They consist of two components, a real part and a dual part, which allows for a wider range of mathematical operations and applications. In Mathematica, tessarines can be used to solve more complex equations and model physical systems.

2. How do I represent tessarines in Mathematica?

Tessarines can be represented in Mathematica using the built-in function Tessarine[x,y], where x and y are the real and dual parts, respectively. Alternatively, the shorthand notation x + y ϵ can also be used, where ϵ represents the dual part.

3. Are there any specific functions in Mathematica that are designed for tessarines?

Yes, there are several functions in Mathematica that are specifically designed for tessarines, such as TessarineNorm, TessarineConjugate, and TessarineInverse. These functions can be used to perform basic arithmetic operations and manipulate tessarines in Mathematica.

4. Can tessarines be used in 3D graphics in Mathematica?

Yes, tessarines can be used in 3D graphics in Mathematica by representing them as points in 3D space. The real and dual parts of the tessarines can correspond to the x and y coordinates, while the z-coordinate can be set to 0. This allows for the visualization of tessarines in 3D space.

5. Are there any limitations to using tessarines in Mathematica?

While tessarines offer a wider range of mathematical operations compared to traditional complex numbers, there are still some limitations in using them in Mathematica. For example, some built-in functions may not work with tessarines, and certain mathematical operations may not yield meaningful results. It is important to understand the properties and limitations of tessarines before using them in Mathematica.

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