What does imaginary Component of a bin's phase in the Fourier Transform mean? I have been recently working with the FFT and realized that each frequency within a bin corresponds to a phase. This concept I have no problem with because if you wanted to construct the summation of sinusoidal waves back into the original signal, you would need the phases of those sinusoidal waves. 

My Questions:


Why must there be an imaginary component to each of the bins corresponding phases? It makes no sense to me why a phase would need an imaginary component. 
In reference to DSP with the FFT, how does this imagery component come into play when characterizing a signal?
It simple trigonometry.  A vector (FFT result bin) can have a magnitude and a phase.  Or you can specify the same vector identically with X,Y coordinates.  The complex representation simply uses X for the real part and Y for the imaginary component.  Without an imaginary component, there’s no place to put the Y, and your vector would be under specified.

The phase ends up being atan2(I,Q) == atan2(Y,X) == atan2(im(z),re(z))

As for why use a complex representation rather than 2 strictly real quantities?  The complex math is more elegant, and requires less writing on the chalkboard to demonstrate similar things.