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 >