March 18th, 2018, 05:47 PM  #21 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,305 Thanks: 2443 Math Focus: Mainly analysis and algebra  
March 18th, 2018, 06:14 PM  #22 
Senior Member Joined: Jun 2014 From: USA Posts: 363 Thanks: 26  So you’re asserting if we moved the paper a distance of $r’r$, we would then need to ‘nudge’ to the right? Why not the left? There is no first rational as you stated though, and we don’t need one (which is evident when $r’r \in \mathbb{Q}$, as it makes no difference in this case either).

March 18th, 2018, 07:35 PM  #23  
Senior Member Joined: Sep 2016 From: USA Posts: 377 Thanks: 205 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
The proof of this is not hard. Let $a$ be an arbitrary irrational number. Then $a \in \mathbb{Q}[a]$ and the set of real numbers which are algebraic over $\mathbb{Q}[a]$ is countable (since this is a field extension of degree 2). Since the irrational reals are uncountable, it follows that if $b$ is an irrational chosen arbitrary, then it is algebraic over this field with probability zero or in other words, $a,b$ are algebraically independent with probability 1.  
March 18th, 2018, 07:50 PM  #24 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,305 Thanks: 2443 Math Focus: Mainly analysis and algebra  
March 18th, 2018, 08:01 PM  #25  
Senior Member Joined: Jun 2014 From: USA Posts: 363 Thanks: 26  Quote:
 
March 18th, 2018, 08:23 PM  #26  
Senior Member Joined: Sep 2016 From: USA Posts: 377 Thanks: 205 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
If you are particularly interested in this one result, I would guess that the result I mentioned before (classification of homeomorphisms of the circle) can be adapted to prove exactly the question you are asking about. A quick google search turned up the following: http://math.bu.edu/people/cew/preprints/introkam.pdf The first 11 pages contain a complete proof of the classification result for diffeomorphisms of the circle and also seems to contain essentially all of the important technical tools required to address your problem.  
March 19th, 2018, 06:01 AM  #27  
Senior Member Joined: Jun 2014 From: USA Posts: 363 Thanks: 26  Quote:
Ask yourself, what fills the space between the unaligned rationals when $r$ and $r'$ are aligned? That is more or less the starting point for the problem. My longer exposition to a8archie then follows, as he went right to that point (which is what anyone should do...).  
March 19th, 2018, 06:48 AM  #28  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,305 Thanks: 2443 Math Focus: Mainly analysis and algebra  Quote:
The question you seem to be trying to ask is "what is there between consecutive rationals?", forgetting that such beasties don't exist. Last edited by v8archie; March 19th, 2018 at 06:51 AM.  
March 19th, 2018, 08:09 AM  #29  
Senior Member Joined: Jun 2014 From: USA Posts: 363 Thanks: 26  Quote:
On the flip side, you (we) question what is between $r$ and $r’$ when the rationals are aligned in the game. That distance would be the same distance that separates the philosophically unaligned rationals when $r$ and $r’$ are aligned, right?  
March 19th, 2018, 09:44 AM  #30 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,305 Thanks: 2443 Math Focus: Mainly analysis and algebra 
Not necessarily. I see it as just a distance, not any sort of minimum possible distance. To me it has no philosophical meaning because no such "minimum" exists. All these questions boil down to asking about the bottom turtle. 

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