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MathGroup Archive 2000

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Re: Dirichlet function plot

  • To: mathgroup at smc.vnet.net
  • Subject: [mg23573] Re: [mg23552] Dirichlet function plot
  • From: David Ong <do226 at is2.nyu.edu>
  • Date: Sat, 20 May 2000 17:44:30 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

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.
> > 
> > 
> > 
> 
> 



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