so we know that $$i^2 = -1$$ and from Hamilton who said that $$i^2=j^2=k^2=-1$$ we get $$ ij = -ji = k$$ $$ki = -ik = j$$ $$jk = -kj = i$$
but what if I am in the complex plane of i and j and the z axis is the set of reals. What does ij equal? My guess, from the intuition that multiplication by i results in a 90 degree rotation, is that $$ij = ji$$ But that is so unsatisfying. It looks like a simple commutation. Somehow I feel like I want to get a single symbol out of that operation for it to mean something.
EDIT: I am sorry for the confusing presentation. Let me start anew. Consider the number system consisting of real numbers, i, and j. Define a 3D cartesian(?) coordinate system where the 'z' axis is the set of reals and the 'x' axis is the set of i numbers and the 'y' axis is the set of j numbers. Every point in this space can be represented as
$$A + B i + C j$$
What is the result of
$$(3 + 2 i) * 4 j$$
I get
$$12 j + 8 ij$$
So what does the quantity $ij$ reduce to so I can graph it?
Is it $$-j$$ I am looking for the results of the following multiplications table: $$ij=?$$ $$ji=?$$