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MathGroup Archive 2000

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Re: AW: Re: Dirichlet function plot

  • To: mathgroup at smc.vnet.net
  • Subject: [mg23609] Re: AW: [mg23573] Re: [mg23552] Dirichlet function plot
  • From: David Ong <do226 at is2.nyu.edu>
  • Date: Wed, 24 May 2000 02:16:09 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Thanks. It's quite beautiful.

David

On Mon, 22 May 2000, Wolf Hartmut wrote:

> See below
> 
> 	-----Ursprüngliche Nachricht-----
> 	Von:	David Ong [SMTP:do226 at is2.nyu.edu]
> 	Gesendet am:	Samstag, 20. Mai 2000 23:45
> 	An:	mathgroup at smc.vnet.net
> 	Betreff:	[mg23573] Re: [mg23552] Dirichlet function plot
> 
> 
> 	Sorry, I misstated the function.
> 	f(x)=1/q when x is an element of p/q in lowest terms and f(x)=0
> 	otherwise. This is a strange looking function because it is
> continous at
> 	every irrational and discontinuous at every rational.
> 
> 	On Sat, 20 May 2000, Andrzej Kozlowski wrote:
> 
> 	> The problem with your question is that the concept of an
> "irrational number"
> 	> does not really make sense in relation to a present day computer.
> Neither
> 	> Mathematica not any other computer program can distinguish between
> rationals
> 	> and irrationals and no sensible concept of an "irrational" number
> can be
> 	> implemented. Of course you could invent a new  Mathematica
> function,
> 	> IrrationalQ, an tell Mathematica it should return True for some
> well known
> 	> irrationals, e.g. Pi, E, Sqrt[2], this would not get you very far.
> It is
> 	> well known that there can be no algorithm which would decide
> whether any
> 	> given (constructible) real number is rational or not.  One can
> easily
> 	> generate arbitrary long sequences consisting entirely of
> irrationals, e.g.,
> 	> anything of the form p^(1/n) where p is a prime and n a  positive
> integer,
> 	> or  any real number of the form (1-x^n)^(1/n), where x is any
> rational s.t.
> 	> 0<x<1, and n a positive integer>2, but no  computer can check
> this.
> 	> 
> 	> However, in spite of all the above,  it is very easy to  plot your
> function.
> 	> You simply take the union of the graph of 1/x  and the real axis
> (you must
> 	> exclude 0 since your function has no value there). This is as good
> an
> 	> approximation as one can ever hope for!
> 	> 
> 	> 
> 	> -- 
> 	> Andrzej Kozlowski
> 	> Toyama International University
> 	> JAPAN
> 	> 
> 	> http://platon.c.u-tokyo.ac.jp/andrzej/
> 	> http://sigma.tuins.ac.jp/
> 	> 
> 	> on 5/20/00 4:10 PM, David Ong at do226 at is2.nyu.edu wrote:
> 	> 
> 	> > Hi, 
> 	> > 
> 	> > Would anyone know of an easy way to plot some approximation of
> this
> 	> > variant of the Dirichlet function?
> 	> > f(x)=1/x if x is an element of the rationals and 0 if x is not
> an element
> 	> > of the rationals.
> 	> > 
> 	> > 
> 	> > 
> 	> 
> 	> 
> 	
> As Andrzej already has pointed out, this is nothing to be "seen".  Yet there
> still may be some aspect to be visualized; so it might be interesting to
> "look at" this part of the Dirichlet function:
> 
> 	tt[n_] := Table[{x, 1/Denominator[Rationalize[x]]}, {x, 0, 1, 1/n}];
> 
> 	t = Join @@ tt /@ Prime /@ Range[100] // Union;
> 
> 	ListPlot[t, PlotJoined -> True, PlotRange -> {{0, 1}, {0, 1}}]
> 
> It makes up quite a pleasant graph.
> 
> Kind regards,   Hartmut
> 



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