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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.
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August 25th, 2015, 01:26 PM   #2
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... who would, of course, know that $\displaystyle \cos36^\circ=\dfrac{\varphi}{2}$.
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August 25th, 2015, 02:18 PM   #3
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No time for an explanation, but look here:
https://en.wikipedia.org/wiki/Hurwit...(number_theory)
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December 19th, 2015, 01:24 AM   #4
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Originally Posted by CRGreathouse View Post
No time for an explanation, but look here:
https://en.wikipedia.org/wiki/Hurwit...number_theory))
now it is clickable
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