I'm trying to get straight some basic complex number fundamentals. First we have z=x+iy. Ok, my question is what does the z in the equation represent? Does it represent a "point" on the complex plane? Does it represent a vector from the origin to where the x and iy values add? Does it represent the "length" of that resultant vector? I'm guessing it doesn't represent the length of the vector, z, because I read that the length of that vector is the squared modulus of z^2=x^2+(iy)^2. I'm guessing that if we didn't take the modulus, we'd end up with an equation which gave us a length of z=√x^2-y^2. Why are we allowed just to take the modulus here and not worry about it? Just because it gives us a Pythagorean "real" looking length? It seems incorrect. It seems as though the length should be as the equation states, z=√x^2-y^2. What am I missing here? But my main question is what does the z in the z=x+iy represent? And could you give me the equivalent on how that would look on just the ordinary "real" x-y plane. I know this sounds elementary, but I just haven't been able to find a straightforward explanation anywhere.