You guys are probably sick of people who know little math posting here, but there's something that's been bugging me. I've bought The Feynman Lectures on Physics and have been reading through it slowly, and I'm up to the part where he talks about probability amplitudes of the electrons/photons. Now, I know that eix = cos x + i sin x. And when talking about adding up these probability amplitudes, the book often says things like "you have to add the real part of eix1 and the real part of eix2, then add the imaginary part of eix1 and the imaginary part of eix2, then square each and add them together to get the final probability." And the whole time, I think to myself, geez, why not just say add the horizontal components and the vertical components. Instead of saying, "the real part of eix" why not just say "cos x?" I get that Euler's Formula is pretty and that Feynman likes it (that much is on the very top of the wiki[/PLAIN] [Broken] article) but it seems to me that all this talk of imaginary numbers doesn't mean much. All they need is a 2-dimensional space to add up vectors, and they just so happened to choose the Argand Plane. I'm not upset about it, but is there a real reason to use i, and not just some arbitrary Cartesian 2 dimensional space? Is it actually necessary that the unit of one of the components be the square root of negative one? I'm just curious if this will pay off. Thanks.