Why does FFT produce complex numbers instead of real numbers?

Question

All the FFT implementations we have come across result in complex values (with real and imaginary parts), even if the input to the algorithm was a discrete set of real numbe

Answer 1

The FFT provides you with amplitude and phase. The amplitude is encoded as the magnitude of the complex number (sqrt(x^2+y^2)) while the phase is encoded as the angle (atan2(y,x)). To have a strictly real result from the FFT, the incoming signal must have even symmetry (i.e. x[n]=conj(x[N-n])).

If all you care about is intensity, the magnitude of the complex number is sufficient for analysis.