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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.

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Sadly, your idea doesn't work; there are no three-dimensional numbers analogous to the two-dimensional complex numbers. You'll get into trouble when you try to decide what $i\cdot j$ is. William Hamilton famously struggled for many years to try to find a way to make it work before discovering the (four-dimensional) quaternions. –  MJD Apr 25 '13 at 18:58
    
If j is a solution to x²=1, then j is real, so what does cj really add as an extra dimension? –  imranfat Apr 25 '13 at 19:00
    
@imranfat No, here j stands for the elementary unit of split-complex numbers, j≠1 and j≠-1 so j is not real. –  moray95 Apr 25 '13 at 19:02
    
@MJD If you think that multiplying by i is a rotation of π/2, then $ji$ should be i or -1. Using matrix representations of i and j, ij= \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}. Which is of course hard to define as a number. –  moray95 Apr 25 '13 at 19:06
    
It seems like you want $i=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ and $j=\begin{pmatrix}0&1\\1&0\end{pmatrix}$. But if you do that, then $i+j=\begin{pmatrix}0&2\\0&0\end{pmatrix}$ and then you get $i+j\ne 0$ but $(i+j)^2 = 0$, which isn't very much like numbers any more. –  MJD Apr 25 '13 at 20:18

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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 $B((x_1,x_2),(y_1,y_2))=x_1y_1-x_2y_2$.

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