MaplePrimes - comments on Post, Dirichlet function

MaplePrimes - comments on Post, Dirichlet function http://www.mapleprimes.com/posts/38691-Dirichlet-Function en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Tue, 16 Jun 2026 08:09:17 GMT Tue, 16 Jun 2026 08:09:17 GMT The latest comments added to the Post, Dirichlet function http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - comments on Post, Dirichlet function http://www.mapleprimes.com/posts/38691-Dirichlet-Function how would you tell? http://www.mapleprimes.com/posts/38691-Dirichlet-Function?ref=Feed:MaplePrimes:Dirichlet function:Comments#comment69764 How would you be able to distinguish the difference between a plot of that function and the plot of the solid lines y=0 and y=1? <pre> plot([0,1],x=-5..5,color=red,thickness=5); </pre> Remember, the rationals are a dense subset of the reals. Between any two reals there exists some rational. And, given a nonzero distance and some real number, there is a rational to be found within that distance from it. So, how could you plot the rationals without it "looking" like a solid line? acer The latest comments added to the Post, Dirichlet function 69764 Sat, 13 Sep 2008 22:33:14 Z acer acer Dirichlet function -- modified version http://www.mapleprimes.com/posts/38691-Dirichlet-Function?ref=Feed:MaplePrimes:Dirichlet function:Comments#comment69763 Here is a modifed version, said to be graphable: f(x) =  {  0,     for x irrational            {  1/b,  for x = a/b a reduced fraction Alla             The latest comments added to the Post, Dirichlet function 69763 Sat, 13 Sep 2008 23:18:55 Z alla alla It does not make much sense. Anyway ... http://www.mapleprimes.com/posts/38691-Dirichlet-Function?ref=Feed:MaplePrimes:Dirichlet function:Comments#comment69761 <pre>   restart;   Digits:=14;   d:=proc(x)     local q,a,b;     q:=convert(evalf(x),rational);     b:=1/denom(q);   end proc;     plot('d'(x),x=0.1..Pi, thickness=3, numpoints=100); </pre> The latest comments added to the Post, Dirichlet function 69761 Sat, 13 Sep 2008 23:35:50 Z Axel Vogt Axel Vogt the hard part http://www.mapleprimes.com/posts/38691-Dirichlet-Function?ref=Feed:MaplePrimes:Dirichlet function:Comments#comment82850 The hard part for graphing this is not dealing with the rationals, it is dealing with the irrationals. One difficulty is in making f plot as a function (1-to-1, not 1-to-many). Each irrational is only representable in fixed precision floating-point as a value which happens to also correspond (perfectly) to some rational. And that rational should here be plotted also as a rational (using the rule in f for rationals). But if the same point x gets plotted twice, by both schemes, then the plot doesn't really represent a function (because it's 1-to-2). Also, how does one make the line along y=0 appear to be not merely solid and yet still be faithful to the spirit of the given definition? Omitting the line along y=0 altogether doesn't seem to encapsulate that important part of it all (to me). acer The latest comments added to the Post, Dirichlet function 82850 Sun, 14 Sep 2008 07:14:38 Z acer acer numpoints http://www.mapleprimes.com/posts/38691-Dirichlet-Function?ref=Feed:MaplePrimes:Dirichlet function:Comments#comment69734 When you say numpoints=100, doesn't that cause evaluation at 100 equally-spaced points (which, in this case, are all rational)??? --- G A Edgar The latest comments added to the Post, Dirichlet function 69734 Thu, 18 Sep 2008 19:44:03 Z edgar edgar My version http://www.mapleprimes.com/posts/38691-Dirichlet-Function?ref=Feed:MaplePrimes:Dirichlet function:Comments#comment69731 Here is a reasonable approximation to the graph. <pre>> plots[pointplot]([seq(seq([a/b,1/b], a=select((t->igcd(t,b)=1), [$1..b])),b=1..200)], symbolsize=4,colour=red); </pre> <img src="/view.aspx?sf=69731/plot1.gif" alt=""> The latest comments added to the Post, Dirichlet function 69731 Thu, 18 Sep 2008 21:44:11 Z Robert Israel Robert Israel