You could represent the infinity of rationals by displaying Ford circles on the real line. However, I'm tempted to agree with Henry Baker: if we include the algebraic numbers, it would be far more pleasing to zoom in on the complex plane. There are beautiful patterns (including fractals) of algebraic numbers, for instance: http://math.ucr.edu/home/baez/roots/ I'm not sure whether it's easy to create a real-time inverter to find algebraic numbers within an interval. It's very easy to find rational numbers (degree-1) and quadratic irrationals (degree-2) using continued fractions and the Minkowski ? function, but I don't know whether we can explore algebraic numbers of higher degree. We could make a database of algebraic numbers, but of course that would be limited. Sincerely, Adam P. Goucher http://cp4space.wordpress.com/ ----- Original Message ----- From: "Charles Greathouse" <charles.greathouse@case.edu> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Thursday, August 30, 2012 6:43 PM Subject: Re: [math-fun] zooming in on the reals
Expanding the space between each pair of rationals to include infinitely many more rationals would be fairly straightforward but boring; but how would you represent the infinite number of irrationals between each of the infinitely many rationals you can't even see yet?
Presumably you show the interesting numbers. This might be from a database like Plouffe's inverter (many numbers with little known about them), from the OEIS (few numbers with relatively large amounts known), from RIES (numbers with low Kolmogorov complexity), etc.
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Thu, Aug 30, 2012 at 1:34 PM, Dave Dyer <ddyer@real-me.net> wrote:
At 09:42 AM 8/30/2012, James Propp wrote:
That would certainly be easy to code.
We don't know the identities of very many interesting numbers. I suppose it would be easy and moderately interesting to zoom in on the zone containing pi e e^pi pi^e and so on.
Expanding the space between each pair of rationals to include infinitely many more rationals would be fairly straightforward but boring; but how would you represent the infinite number of irrationals between each of the infinitely many rationals you can't even see yet?
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