The fast Fourier transform is investigated. It is proved that the number of real (as opposed to complex) multiplications necessary to implement the algorithm M for complex input sequence of length N = 2 is 2N(M -7/2) + 12. Methods which do not avoid the unnecessary multiplications predict 2N(M-1) or 2NM. It is shown experimentally that for at least one implementation of the algorithm, it is faster to take advantage of the multiplication savings mentioned above. Some theorems regarding computational savings when transforming real data are presented. A system of subroutines for calculating finite discrete Fourier transforms by the fast Fourier transform method is given. The results of applying this system to two specific problems is presented.