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
I've seen pictures of, e.g., all the roots of monic polynomials up through degree n (around 16) having all coefficients with absolute value <= 1. These also show complex and mysterious patterns with regions of avoidance. Is this what you have in mind, Adam, re the algebraics -- or something else? --Dan On 2013-08-16, at 5:17 AM, Adam P. Goucher wrote:
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Is this what you had in mind? http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.77.7281&rep=rep1&ty... Victor On Fri, Aug 16, 2013 at 11:07 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I've seen pictures of, e.g., all the roots of monic polynomials up through degree n (around 16) having all coefficients with absolute value <= 1. These also show complex and mysterious patterns with regions of avoidance.
Is this what you have in mind, Adam, re the algebraics -- or something else?
--Dan
On 2013-08-16, at 5:17 AM, Adam P. Goucher wrote:
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
-
Adam P. Goucher -
Dan Asimov -
Victor Miller