Re: [math-fun] rationalize exp(%i*x), for floating point x.

Dan Asimov wrote:

That's a beautiful and intriguing map of the Moon that Fred posted. Those areas of avoidance almost look like parts of a Petri dish where antibiotics keep a mold from growing. It looks as if it would take a lot of mathematical explaining to understands why it looks that way.

You get the same phenomenon in the algebraics, and indeed in the rationals (c.f. Ford circles). I think that the proof is essentially the same as showing that numbers with fast-growing convergents (e.g. Liouville's number) are transcendental. See, for example, this Wikipedia image: http://upload.wikimedia.org/wikipedia/en/d/d1/Algebraicszoom.png Sincerely, Adam P. Goucher http://cp4space.wordpress.com