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Is there a meaningful way to define ij, where i is the imaginary unit and j is the split-complex unit?

Original question by Will Bolden on math.stackexchange

At the moment I’m treating ij=ji as its own quantity and reducing it where possible, as in (ij)^2=−1 and e^(ij)=cos(j)+isin(j)=cosh(i)+jsinh(i).

From the first identity seems like you can say ij=√−1=i, and this even agrees with cos(j)+isin(j)=cos(1)+isin(1) and cosh(i)+jsinh(i)=cosh(i)+1sinh(i) unless I made a mistake, but it still seems like a strange definition.

So, Is there a way to represent ij as a linear combination of 1, i, and j, and does the definition I came up with here break anything I haven’t noticed?

Answer by Anixx

No. What you are talking about is tessarines, a 4-dimensional algebra that combines complex and split-complex numbers.

In that system ij is a separate unit vector, similar to i in that (ij)^2=−1. But not equal to i. It is irreducible.

The algebra of tessarines is commutative and associative.