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Does anyone have a recommendation for a good book on split-complex numbers?

If it also covers dual numbers or the relation between split-complex numbers and special relativity or Minkowski 4-space or some analysis of split-complex numbers then all the better, but that's just gravy. I really just want a good reference for the geometry of the plane as expressed via split-complex numbers.

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The Wikipedia article on split complex numbers is enthusiastically maintained by an aficionado, and you can already find dozens of references there.

You can try to analyze the split complexes as working with the hyperbolic plane the way the complex numbers work with the Euclidean plane. There are differences of course. For instance (if you are using the imaginary unit $j$ where $j^2=1$) multiplying by $j$ reflects the plane over the line $x=y$. This differs from the complexes since they all perform orientation preserving transformations of the plane.

After you read about these a bit, the next logical step would be to look at geometric algebra which brings all the ideas under one tent.

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