What are the uses of split-complex numbers?

Solution 1:

I think the known uses of split-complex numbers are probably going to be addressed by the Wiki page which MJD linked in the comments above, and other "fan pages" on the internet. So, I wanted to address this question in the post:

But what are the uses of split-complex number that can't be done with complex numbers?

In Clifford algebra (or geometric algebra, as called by a small segment of the population that uses them) these two algebras are used to encode the geometry of $\Bbb R$ under two different geometries.

The long story short is that a bilinear form gives rise to a geometry on a vector space. The "signature" of a real bilinear form determines its basic character, and since there are lots of forms with different signatures, you get different geometries.

The complex numbers study $\Bbb R$ with a bilinear form $B(x,y)=-xy$.

For the split-complex numbers, the bilinear form on $\Bbb R$ is just $B(x,y)=xy$.

The quaternions study $\Bbb R\oplus \Bbb R$ with the bilinear form $B((x_1,x_2),(y_1,y_2))=x_1y_1-x_2y_2$.