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

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.

*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 e^{ix}= 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 e^{ix1}and the real part of e^{ix2}, then add the imaginary part of e^{ix1}and the imaginary part of e^{ix2}, 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 e^{ix}" 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.

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