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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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