The set of Complex numbers can be used in lots of domains like geometry, vectorial calculations, solving equation with no real solution etc. But what are the uses of split-complex number that can't be done with complex numbers? I think you could do the same works in geometry or vectorial calculation in a "split-complex" plane but what advantages gives us to know that j is a solution of the equation $x^2=1$?
What I've thought so far is that using complex numbers and split-complex numbers together, we can have numbers of the form $a+bi+cj$ so everything that can be done in the complex-plan could be extended to 3 dimensional space by adding a split-complex part.
Best Answer
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 $B((x_1,x_2),(y_1,y_2))=x_1y_1-x_2y_2$.