I may be asking a stupid question, but what is the co-relation between the complex plane and the real plane? I know Euler's equation ei\pi+1=0 relates them, but graphically, how are they related?
For example, the numbers 1, 2, 3, 4, 5 on the real line can denote a box, two boxes, three boxes, etc. The equation [tex]y=a sin(\omega t+\phi)[/tex] can be used to represent the simple harmonic motion of a particle. Similarly, with respect to the world around us (real world), what do complex numbers represent?
For example, there are countless places where I've used the equation [tex]i(t)=Asin(\omega t+\phi) +iBcos(\omega t+\phi)[/tex], but what does the complex current denote? In any such equation which is used to define some aspect of the world around us, what do complex quantities denote? What do they mean?