NewRPL: Complex Numbers in Cartesian Form r[x,y] , ...

[Claudio L.]

(03-17-2017 04:08 PM)The Shadow Wrote:  I've always been a little dubious of using parentheses for complex numbers. For one thing, ordered pairs have plenty of other uses, which don't necessarily multiply in the complex manner. (Or even multiply at all.)

For that reason and others, I'd support using C(re, im) even on the stack. Or just always using the a+b*i form.

EDIT: Oh, and while we're on the topic, how about a system variable or flag to change what i^2 evaluates to? +1 for split-complex, 0 for dual.

And as long as I'm howling for the moon, how about adding complex j and k, so we can do quaternions? These could also perhaps be modified like i above, giving a wide range of capabilities. (Heck, you could store everything about i, j, and k in a single 3x3 matrix representing the multiplication table.)

Not gonna ask for octonions, though. They make my head hurt.

Actually, quaternions only have one challenge: make sure the CAS supports non-commutative multiplication.
You can define rules (which you can apply whenever you need to simplify the expressions), for ij=k, k^2=-1, etc., that would be covered.
But non-commutative multiplication is a challenge not only for quaternions, every time a variable in an expression contains a matrix or vector, how does the machine know that it's supposed to use non-commutative multiplication?
For example the expression 'A*B+C' can be manipulated in many ways by a CAS, but the CAS has no way of knowing that A and B are matrices, so how do we tell the CAS:
* If the variable is real (REALASSUME)
* If the variable is complex
* If the variable is a matrix
It seems this property should be embedded somehow in the expression, to help solvers and the CAS make the right decisions.

More and more to think...