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.
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?