Re: Dirichlet function plot

*To*: mathgroup at smc.vnet.net*Subject*: [mg23575] Re: [mg23552] Dirichlet function plot*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>*Date*: Sat, 20 May 2000 17:44:31 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

This is of course a somewhat different matter. One can define a reasonable "simulation" of this function and in principle plot its graph but I think it will be hard to see anything interesting. The definition is simple enough: f[x_] := If[IntegerQ[Numerator[Rationalize[N[x]]]], Denominator[Rationalize[x]], 0] f is a good approximation to what you want: In[3]:= f[Pi] Out[3]= 0 In[4]:= f[1.4] Out[4]= 5 In[5]:= f[Sqrt[2]] Out[5]= 0 Unfortunately it is hard to get a meaningful graph. You could try using ListPlot and do something like: In[6]:= l1 = Table[{x, f[x]}, {x, 3.0, Pi + 0.1, 0.001}]; In[7]:= l2 = Table[{x, f[x]}, {x, Pi - 0.14, Pi + 0.1, 0.001}]; In[8]:= ListPlot[Union[l1, l2]] But the result does not seem t me to be very instructive. In principle I think what I wrote in my first reply still holds and I do not think computers are suitable tools for investigating this sort of phenomena. on 00.5.20 11:02 PM, David Ong at do226 at is2.nyu.edu wrote: > > 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. >>> >>> >>> >> >> > -- Andrzej Kozlowski Toyama International University JAPAN http://platon.c.u-tokyo.ac.jp/andrzej/ http://sigma.tuins.ac.jp/