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MathGroup Archive 2000

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Re: Dirichlet function plot

  • To: mathgroup at smc.vnet.net
  • Subject: [mg23571] Re: [mg23552] Dirichlet function plot
  • From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
  • Date: Sat, 20 May 2000 17:44:28 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

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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