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August 25th, 2015, 06:43 AM   #1
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The Least Rational Number there is?

I've just read the following sentence.
Quote:
 [The Golden Ratio (or Golden mean) is] the most irrational number there is, that is to say, the one the least well approximated by a fraction.
Would anyone like to make a guess as to what this claim means and how it might be proved?

My best guess is that the claim means that in order to approximate the Golden Ration, $\alpha$ by the rational $p \over q$ such that $0 \lt |\alpha - {p \over q} \lt \epsilon$ we require $q$ to be larger than for any other irrational. How we'd prove it I'm far from sure, although I did find this reference on approximation theory which I haven't time to read in its entirety now.

The focus of my interest is that I'm looking for fun, but unusual maths to try to stimulate interest in high school students.

 August 25th, 2015, 01:26 PM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,913 Thanks: 1113 Math Focus: Elementary mathematics and beyond ... who would, of course, know that $\displaystyle \cos36^\circ=\dfrac{\varphi}{2}$. Thanks from topsquark
 August 25th, 2015, 02:18 PM #3 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms No time for an explanation, but look here: https://en.wikipedia.org/wiki/Hurwit...(number_theory)
December 19th, 2015, 01:24 AM   #4
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Quote:
 Originally Posted by CRGreathouse No time for an explanation, but look here: https://en.wikipedia.org/wiki/Hurwit...number_theory))
now it is clickable

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