I don't much care, I don't want to define my own math, I just want to know why we mix the real and imaginary parts of a product in one specific case.
Z1*Z2=r1r2(cos a + isin a)(cosb + isin b)
= r1r2 (cosa*cosb)-(sina*sinb) + i(sina*cosb + cosa*sinb)
= r1r2 (cos(a + b) + isin(a + b))
You could indeed say that "entity of your choice = 1/0". But you need a
compelling reason why such a quantity is useful, and various proofs on how
it is derived and used. All math depends on certain assumptions and those
assumptions are different in different circumstances - aka Euclidian
geometry and topology.
Actually, you can say 1+2 = 12 ... and if you are putting together a
string in a programming language, that makes sense and is useful, it is a
valid operation. Many object oriented programming languages let you
redefine basic symbols such as "+ - * /". But string addition wouldn't be
very helpful for doing audio math calculations.
So if you want to define your own i, go ahead, but don't you think your
calculations are going to turn out better if you follow the normal math
definitions?
~David
> What I'm getting at is that expressing rotation as complex numbers is no
> different than using cartesian coordinates. Why, when you multiply two
> points, would one of the multiples turn negative?
> I see no reason you couldn't say i = 1/0. Then 4*0*i=4. That makes as much
> sense.
>