...
No, the square of a negative number is a positive number.
Sure, I understand that is correct in terms of pure math. What I am trying to describe is that real world can and does differ from a pure math model.
By using sinewaves and squares in my example it has brought to your mind pure math solutions (since both are pure math constructs), which I don't think has made the point. I'm not saying the math is wrong, I'm saying the real world can and does work differently to the math.
Forget sines and squares and consider a physical object, say a beam with a centre coord and a right and left side. You can arbitrarily assign + and - coords to either side of the beam as you wish.
Now there are 3 holes on either side of the beam their positions referenced with relative coords to the beam centre point, so that the position of 3rd hole xc is determined by the positions of the first 2 holes;
xa = 2"
xb = 3"
xc = (xa*xb) = (2*3) = 6"
and on the other side of the beam;
xa = -2"
xb = -3"
xc = (xa*xb) = (-2*-3) = -6"
So where in pure math we assume that (-a*-b) = +c in the real world the sign can be arbitrarily assigned to either side of something real, and the sign has a much lower importance than the requirement that an *identical operation* be carried out on both sides of something with mirror symmetry.
So I guess my premise is that in pure math we need (-a*-b) = +c but in the real world the case is very commonly that the same operation must be carried out on both sides of the zero, and the sign is much less important and can be arbitrarily assigned to either side of the mirror. Either before or after the operation or inverted as needed without affecting the operation.
And so within a real world "mirror symmetry" model I have no problem with the square root of -9 being -3.