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AW: Re: Dirichlet function plot
*To*: mathgroup at smc.vnet.net
*Subject*: [mg23607] AW: [mg23573] Re: [mg23552] Dirichlet function plot
*From*: Wolf Hartmut <hwolf at debis.com>
*Date*: Wed, 24 May 2000 02:16:07 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
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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