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MathGroup Archive 2000

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Re: Dirichlet function plot

  • To: mathgroup at smc.vnet.net
  • Subject: [mg23575] Re: [mg23552] Dirichlet function plot
  • From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
  • Date: Sat, 20 May 2000 17:44:31 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

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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