pratt@CS.STANFORD.EDU (Vaughan Pratt) (11/17/90)
I'd appreciate a pointer to any reference containing the induction hypothesis of the following argument, which surely deserves wider dissemination than it has apparently received. It is not mentioned in any of several histories of pi that I've seen, including most recently Remmert's in Ebbinghaus et al, Numbers, Springer-Verlag 1990, nor in any definition of angular measure or exp(z) and ln(z) I'm aware of. Besides simplifying the exposition of Archimedes' hexagon subdivision process, it also makes Vieta's product formula for 2/pi more transparent (consider the ratios obtained by scaling at each step to hold z on the unit circle, derivable without trigonometry (exercise)). Theorem. "complex z := i; do forever z := (z + |z|)/2" converges to 2/pi. Proof. Each step halves both y = Im(z) and theta = arg(z), making y/theta constant. Initially y/theta is 2/pi, in the limit it is z. QED More generally, if z is initialized to x0+iy0 for y0>0, then the process converges to y0/theta0. Given that conversion of angular to linear measure with just ruler and compass must be an infinitary process, the package consisting of this process and its proof would appear to be close to the simplest possible *plausible* definition of angle. Expressing the complex arithmetic as real, or less symbolically and more visually (recently returned to fashion by graphics workstations) as operations with straightedge and compass, should make it also close to the most elementary possible. Vaughan Pratt CSD, Stanford University