       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, 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 // Union;

ListPlot[t, PlotJoined -> True, PlotRange -> {{0, 1}, {0, 1}}]

It makes up quite a pleasant graph.

Kind regards,   Hartmut

```

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