Why are frequencies represented as complex numbers?

Question

In a FFT, the resulting frequencies represent both magnitude and phase. Since each frequency element in the output array of an FFT essentially just describes the SIN wave at each frequency interval, shouldn't it just be magnitude that we need? What is the significance of the phase represented in the imaginary part of the complex number?

To clarify my question, to be able to put a meaning to the phase of a wave, I need a reference point or reference wave.

When an FFT reports the phase for each sin wave in the resulting frequency domain output, what is the reference wave with respect to which it is reporting the phase?

Answer 1

Because the phase of different components affects the total signal. The two functions in the plot are both summed from sine waves with periods of pi and 2pi, but the phase of the p=2pi sine waves are different. As you can see, the outputs are not the same.

Answer 2

Well in layman's words: magnitude tells you how much of that frequency is there, and phase tells you where it is.