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MathGroup Archive 2000

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Re: Dirichlet function plot

  • To: mathgroup at smc.vnet.net
  • Subject: [mg23576] Re: [mg23552] Dirichlet function plot
  • From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
  • Date: Sat, 20 May 2000 17:44:32 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Sorry, the definition should of course have been:

f[x_] := If[IntegerQ[Numerator[Rationalize[N[x]]]],
 1/Denominator[Rationalize[x]], 0]

on 00.5.21 0:07 AM, Andrzej Kozlowski at andrzej at tuins.ac.jp wrote:

> This is of course a somewhat different matter. One can define a reasonable
> "simulation" of this function and in principle plot its graph but I think it
> will be  hard to see anything interesting. The definition is simple enough:
> 
> f[x_] := If[IntegerQ[Numerator[Rationalize[N[x]]]],
> Denominator[Rationalize[x]], 0]
> 
> f is a good approximation to what you want:
> 
> In[3]:=
> f[Pi]
> 
> Out[3]=
> 0
> 
> In[4]:=
> f[1.4]
> 
> Out[4]=
> 5
> 
> In[5]:=
> f[Sqrt[2]]
> 
> Out[5]=
> 0
> 
> Unfortunately it is hard to get a meaningful graph. You could try using
> ListPlot and do something like:
> 
> In[6]:=
> l1 = Table[{x, f[x]}, {x, 3.0, Pi + 0.1, 0.001}];
> 
> In[7]:=
> l2 = Table[{x, f[x]}, {x, Pi - 0.14, Pi + 0.1, 0.001}];
> 
> In[8]:=
> ListPlot[Union[l1, l2]]
> 
> But the result does not seem t me to be very instructive. In principle I think
> what I wrote in my first reply still holds and I do not think computers are
> suitable tools for investigating this sort of phenomena.
> 
> on 00.5.20 11:02 PM, David Ong at do226 at is2.nyu.edu wrote:
> 
>> 
>> 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.
>>>> 
>>>> 
>>>> 
>>> 
>>> 
>> 

-- 
Andrzej Kozlowski
Toyama International University
JAPAN

http://platon.c.u-tokyo.ac.jp/andrzej/
http://sigma.tuins.ac.jp/



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