system
Given the following post, respond as if you're commenting on Reddit:
human
Do imaginary components of equations ever play a physical role?  Are there deep reasons to ignore them, or is it all empirical?

I understand from Euler's equation how imaginary components fall out from periodic systems, but I have always been bothered by the fact that real world solutions only take the real component.  Are there cases where the imaginary component plays a role in reality?

Any insight would be greatly appreciated, thank you.

**Eureka Edit:**

These responses have been excellent, and I thank you all.  I particularly appreciate the effort made by duetosymmetry, RobotRollCall, and others in clarifying my misconceptions about the of complex numbers in general.

Should anyone be monitoring this post further, I would like to present what I learned here and see if I'm on the right track.

Imaginary numbers are a series of numbers orthogonal to the real number line.  A complex number, therefore defines a number plane.  This additional dimension always (almost?) contains meaningful information about a system, but it is not always relevant for the questions you want to ask.  Also, some systems can be fully described using real numbers, so why complicate things...

When dealing with periodic systems, the "imaginary" component represents information that is not explicitly stated in a trig function, but is there nonetheless, such as phase. 

Again I thank you all, and if I didn't reply specifically to your comment please don't take it personally.
gpt
As platypuskeeper mentioned, it comes up when looking at the impedance of an AC circuit. For instance, a resistor and an inductor in a circuit might have an impedance of 50+30I Ohms. The reason the imaginary part goes away in quantum mechanics is because pretty much any measureable quantity involves multiplying the wavefunction by its conjugate.