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I wonder, what are some real or potential and purely theoretic difficulties you may face if you decided to implement the split-complex unity $j$ such that $j^2=1$ along with already existing complex unity $i$, with similarly extensive rule-set?

I mean, implementing $j$ immediately adds support for split-complex numbers and tessarines (which are isomorphic to all bicomplex numbers variants).

One would be able to write a tessarine as $(a+bi)+(c+di)j$ or $(a+bj)+(c+dj)i$, the both being standard forms.

On the surface, implementation seems not to be difficult as tessarines are both commutative and associative, which would allow to keep the usual rewriting rules.

Yet, they have non-zero elements by which one cannot divide. This may bring some difficulties. But since there is already zero, by which dividing is impossible, maybe this is possible to overcome.

Maybe some additional infinity-like symbols would be needed (but maybe not).

Maybe Solve and Reduce would need to produce more solutions.

What the developers say?

A separate but related question: would implementing dual numbers unity $\varepsilon$ be easier or more difficult?

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  • $\begingroup$ Maybe the same methods indicated for dual numbers when you asked about those 8-9 years ago? See also this prior MSE thread $\endgroup$ Mar 4, 2021 at 18:05
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    $\begingroup$ @DanielLichtblau I am not asking as a user, I am asking the developers' perspective. Implementing tessarines as matrices is easy. But not as universal. $\endgroup$
    – Anixx
    Mar 4, 2021 at 18:06
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    $\begingroup$ That strikes me as lying outside the scope of this forum. In any case, if these were implemented they would not likely go beyond the type of functionality found in the Quaternions.m package or the more recent Wolfram Function Repository version here $\endgroup$ Mar 4, 2021 at 18:10
  • $\begingroup$ @DanielLichtblau quaternions need special treatment, like the operator Quaternion and non-commutative multiplication (**) because they are non-commutative. In contrast, tessarines and split-complex numbers would need only a constant symbol added and can be re-written with usual rules of rewriting. $\endgroup$
    – Anixx
    Mar 4, 2021 at 18:13
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    $\begingroup$ True. One could for example represent them as elements in the (commutative) algebra C[x]/<x^2-1>. Extending this beyond basic algebra is a different matter entirely. $\endgroup$ Mar 4, 2021 at 18:33

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