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Is split-complex j=i+2ϵ?

Original post on math.stackexchange by Anixx

In matrix representation imaginary unit i=

(0 −1
 1 0)

dual numbers unit ϵ=

(0 0
 1 0)

split-complex unit j=

(0 1
 1 0)

Given this definition, does not it follow that

j=i+2ϵ

and as such, one of these systems can be fully expressed through others?

Answer by Milo Brandt

If, ignoring the means by which you reached this conclusion (which was well-addressed in epimorphic's answer), we supposed that

j=i+2ε

then it follows that (assuming commutativity)

j^2 = (i+2ε)^2 = i^2+4εi+4ε^2

which, replacing each by the definition of their square:

1 = −1+4εi

which only works if we define

εi=½

This is a pretty long shot from any “reasonable” definition, since our intuition about i and ε should certainly not lead us to this point. Moreover, making the definition εi=½ breaks very important properties of multiplication - for instance, it makes the operation not associative since

(ε^2)i ≠ ε(εi)
0i ≠ ε½
0 ≠ ½ε

which poses a rather major difficulty for algebra. Moreover, the equation j=i+2ε is not even particularly special; setting j=i+ε (i.e. by making ε correspond to twice the matrix representation you suggest - which is an equally valid matrix representation of the dual numbers) yields that we want εi=2 — but this doesn't solve the lack of associativity. In fact, if we want associativity, we conclude that εi must not be a linear combination of 1, i, and ε with real coefficients (since εi can't be invertible given that ε^2=0 and εi can’t be a multiple of ε as that would cause i⋅i⋅ε to break associativity) — which implies that (i+aε)2=1=j^2 must have no solution, so we can’t reasonably express j in such a system.

The fundamental issue with this is that the expression i+2ε doesn’t even make sense without additional structure. Though we can happily add the terms together in a formal sum (i.e. where we write every number as a+bi+cε without allowing any simplification) and this sometimes yields meaningful results, this leaves us with the issue of multiplication. Eventually the term εi will come up, and, unless we wish to break important properties of multiplication, we have to consider it as an entirely new thing — and we can prove that, in any extension of R which still obeys certain algebraic properties, but contains a new element ε squaring to 0 and i squaring to −1, there would be only two solutions to x^2=1, and they are 1 and −1 — there is no extra root j, so j cannot meaningfully play into that system.