Re: Dirichlet function plot

*To*: mathgroup at smc.vnet.net*Subject*: [mg23576] Re: [mg23552] Dirichlet function plot*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>*Date*: Sat, 20 May 2000 17:44:32 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Sorry, the definition should of course have been: f[x_] := If[IntegerQ[Numerator[Rationalize[N[x]]]], 1/Denominator[Rationalize[x]], 0] on 00.5.21 0:07 AM, Andrzej Kozlowski at andrzej at tuins.ac.jp wrote: > 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/

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