We're taught that |z|=|exp(i*x)|=1, for real x. But this doesn't work so well for rational approximations, especially when we rationalize the real & imaginary parts separately. So I'm seeking rational approximations to exp(i*x), x real, such that |exp(i*x)| is identically zero. Let t=rationalize(tan(x/2)) be a rational approximation to tan(x/2). Then z = ((t^2-1) + i*2*t)/(t^2+1) is rational, and |z| is identically 1. Question: Is this the best (or perhaps only) way to do this? Are there closer rational approximations that don't correspond to one of these forms? By 'closer', I mean closer for a given denominator size.