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