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Physics math without $\sqrt{-1}$

When I produce a complex final solution to a problem that began without complex coefficients at all, I have so far (with my limited expertise) tended to discard it as unphysical, unless I had to tinker some more, in which case the $i$s may square and real-ity be restored. Is this acceptable?

I understand that the Wessel plane is useful in modelling planetary motion, and many other things, but the fact (?) that you could do it all (albeit more awkwardly) without them seems to be indicative that they are a little contrived.

In short, is there a counter-example?; a physical theory (not a particular, neat, mathematical model of a physical theory) in classical mechanics that could not be formulated without complex numbers?

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Possible duplicate: physics.stackexchange.com/q/11396/2451 –  Qmechanic Nov 23 '12 at 22:51
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marked as duplicate by Qmechanic, Manishearth, Emilio Pisanty Dec 10 '12 at 10:17

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2 Answers

The answer is "It depends how you use complex numbers"!

If you are modelling motion on the plane, you could use $\mathbb{C}$ instead of $\mathbb{R}^{2}$ perfectly fine. In this case $1$ and $i$ are "unit vectors".

If you are working with, e.g., waves in a plane...complex numbers are quite natural.

This is markedly different from quantum theory, though.

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Also complex numbers show up in problems that result in harmonic functions (sine, cosine, etc) depending on how they are formulated. Mass-spring systems are a classic example. –  tpg2114 Nov 23 '12 at 22:40
    
yes but those examples can be formulated without complex numbers, which is what I think constitutes the main point of the question –  Jorge Nov 23 '12 at 22:42
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@Burzum: But classical mechanics can be formulated without calculus...does this mean calculus is contrived? –  Alex Nelson Nov 23 '12 at 23:05
    
@Alex good question I obviously need to think more about it. –  Jorge Nov 23 '12 at 23:57
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@Burzum The question poses "Are complex numbers unphysical" and the answer is a definitive no. Granted at the end it does ask if solutions are not possible without complex numbers, but I recall doing very complicated mass-spring systems that would not be possible to solve without Laplace transforms and imaginary numbers. Also, control theory requires complex numbers for things like stability analysis, which still falls under classical mechanics (sort of, it's analysis of classical mechanical systems). –  tpg2114 Nov 24 '12 at 1:48
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No. For example, complex numbers are used in an essential way in classical electromagnetism to study networks of conductors, capacitors, and inductors.

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