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I recently learned about hypercomplex systems that are taken over the reals, i.e. the dual numbers for which $j^2=1$, $j≠1$, and the dual numbers for which $ε^2=0$, $ε≠0$. These number systems, along with the complex numbers, have some nice properties, namely they represent all the possibilities for 2-dimensional unital algebras over R, up to isomorphism.

They also happen to be constructed by polynomial equations, which begs the question: Is there a way to construct a "hypercomplex" system by defining the "imaginary constant" as a solution to an exponential equation? For example:

$a^k=b$, where $a,b∈\mathbb{R}$ and $k$ is the "imaginary unit".

The first thought I had was of somehow defining an algebra with the equation $e^{L}=0$, though I might just be being naive about the whole prospect in general.

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    $\begingroup$ If $e^L=0$, what would be $e^{-L}$? $\endgroup$ Commented Mar 8, 2024 at 6:45
  • $\begingroup$ What would $L^2$ be? $\endgroup$ Commented Mar 21, 2024 at 22:45
  • $\begingroup$ @JoshuaTilley if we talk about extended real numbers $\overline {\mathbb R}$, where this equation has a solution, there $L=-\infty$ and $L^2=\infty$. $\endgroup$ Commented Mar 21, 2024 at 23:21
  • $\begingroup$ What if you defined it as a solution to a problem in tetration? It's domain is still being studied and it may be limited even within the complex plane, so if you want to invent a new number, your best shot may be there. $\endgroup$ Commented Jul 18 at 7:51

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In extended real numbers $\overline {\mathbb R}$, the solution for $e^L=0$ is $-\infty$. Extending reals with logarithm of zro can be done in a different way as well.

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