Fourier transform and Ludolph van Ceulen

The author would like to dedicate this paper to the memory of Ludolph van Ceulen (1540-1610), the famous Dutch mathematician, who spent almost his whole life to calculate the first 35 decimals of p. Now, 400 years later, the known number of digits of p exceeds 200 billion and increasing. Highly efficient and very fast converging methods have been developed in the last decades. Without the number crunching power of digital computers it would never have been possible to calculate the billions of digits of p. Amazingly, the key element in calculating the billions of digits is an efficient multiplication method. And here we can find an unexpected application from signal processing and control theory: the Fast Fourier transform (FFT). The paper presents the most efficient algorithms and describes how the Fourier transform makes fast multiplication possible.