prove i^2 =-1 and hence that (u+iv)(x+iy) = (ux-yv) + i(xv +uy)
hint prove that if i^2 =A+iB, then A=-1 and B=0.
I'm not sure I entirely understand the question, since it's asking you to prove a definition, but I'll give it a shot.
i is defined as the square root of -1. i^2 = (-1)^(1/2)^(2) = -1 ^(2/2) = -1 ^1 = -1.
So, given (u+iv)(x+iy), we can use the FOIL method to get ux + xvi + uyi + yvi^2 = ux + i(xv + uy) + yv * -1 = (ux -yv) + (xv+uy)