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August 25th, 2015, 05:43 AM  #1  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,659 Thanks: 2635 Math Focus: Mainly analysis and algebra  The Least Rational Number there is?
I've just read the following sentence. Quote:
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, 12:26 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,935 Thanks: 1129 Math Focus: Elementary mathematics and beyond 
... who would, of course, know that $\displaystyle \cos36^\circ=\dfrac{\varphi}{2}$.

August 25th, 2015, 01:18 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, 12:24 AM  #4  
Senior Member Joined: Dec 2015 From: holland Posts: 162 Thanks: 37 Math Focus: tetration  Quote:
 

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