Keith Thompson <ks...@mib.org> wrote: > The radian is defined as a ratio of lengths. That ratio > is the same regardless of the size of the circle. The > choice of 1/(2*pi) of the circumference isn't arbitrary > at all; there are sound mathematical reasons for it.
Yes, but what is pi then? > Mathematicians could have chosen to set the full > circumference to 1, for example, but then a lot of > computations would contain additional multiplications > and/or divisions by 2*pi. The formula: circumference = 2 x pi x radius is taught in primary schools, yet it's actually a very difficult formula to prove! Most attempts by students collapse because they assume the formula in advance (pi is the ratio of circle semi-circumference to its radius.) Perhaps the most elegant approach is to define arctan x as the integral from 0 to x of dz/(1 + z^2). The trig functions and relationships, and their application to normal geometry, can all be defined and derived from that alone without any reference to pi. Pi is simply the constant 4 times arctan 1. -- Peter -- http://mail.python.org/mailman/listinfo/python-list
