Binary expansion of $frac{1}{pi}tan^{-1}left(frac{5}{12}right)$












8












$begingroup$



Prove that the binary expansion of $dfrac{1}{pi}tan^{-1}left(dfrac{5}{12}right)$ has strings of $0$s or $1$s of arbitrary length.




I didn't see how we can calculate the binary expansion of $tan^{-1}(x)$ or $pi$. Is there some other way of solving this question?






number-theory trigonometry decimal-expansion







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    I do not know if this is really relevant, but since $arctan(z)=text{Im}log(1+iz)$ and $5+12i=(3+2i)^2$, the question is more or less equivalent to showing that $(3+2i)^{2^k}$ is often close to a real number (i.e.has a small imaginary part, compared to the real part).
    $endgroup$
    – Jack D'Aurizio
    Nov 6 '16 at 15:30






  • 3




    $begingroup$
    where did you find this question ?
    $endgroup$
    – mercio
    Nov 6 '16 at 17:59






  • 1




    $begingroup$
    @mercio I was solving a problem and needed to prove this as a step in the solution.
    $endgroup$
    – user19405892
    Nov 6 '16 at 22:39






  • 1




    $begingroup$
    this could very well be the sort of thing that is "obviously true" (because a number taken randomly in a unit interval will have this property with probability $1$) but completely hopeless to prove.
    $endgroup$
    – mercio
    Nov 6 '16 at 22:46






  • 1




    $begingroup$
    I knew this was a hidden duplicate : math.stackexchange.com/questions/1615708/…
    $endgroup$
    – mercio
    Dec 2 '16 at 14:10
















8












$begingroup$



Prove that the binary expansion of $dfrac{1}{pi}tan^{-1}left(dfrac{5}{12}right)$ has strings of $0$s or $1$s of arbitrary length.




I didn't see how we can calculate the binary expansion of $tan^{-1}(x)$ or $pi$. Is there some other way of solving this question?






number-theory trigonometry decimal-expansion







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    I do not know if this is really relevant, but since $arctan(z)=text{Im}log(1+iz)$ and $5+12i=(3+2i)^2$, the question is more or less equivalent to showing that $(3+2i)^{2^k}$ is often close to a real number (i.e.has a small imaginary part, compared to the real part).
    $endgroup$
    – Jack D'Aurizio
    Nov 6 '16 at 15:30






  • 3




    $begingroup$
    where did you find this question ?
    $endgroup$
    – mercio
    Nov 6 '16 at 17:59






  • 1




    $begingroup$
    @mercio I was solving a problem and needed to prove this as a step in the solution.
    $endgroup$
    – user19405892
    Nov 6 '16 at 22:39






  • 1




    $begingroup$
    this could very well be the sort of thing that is "obviously true" (because a number taken randomly in a unit interval will have this property with probability $1$) but completely hopeless to prove.
    $endgroup$
    – mercio
    Nov 6 '16 at 22:46






  • 1




    $begingroup$
    I knew this was a hidden duplicate : math.stackexchange.com/questions/1615708/…
    $endgroup$
    – mercio
    Dec 2 '16 at 14:10














8












8








8


4



$begingroup$



Prove that the binary expansion of $dfrac{1}{pi}tan^{-1}left(dfrac{5}{12}right)$ has strings of $0$s or $1$s of arbitrary length.




I didn't see how we can calculate the binary expansion of $tan^{-1}(x)$ or $pi$. Is there some other way of solving this question?






number-theory trigonometry decimal-expansion







share|cite|improve this question











$endgroup$





Prove that the binary expansion of $dfrac{1}{pi}tan^{-1}left(dfrac{5}{12}right)$ has strings of $0$s or $1$s of arbitrary length.




I didn't see how we can calculate the binary expansion of $tan^{-1}(x)$ or $pi$. Is there some other way of solving this question?






number-theory trigonometry decimal-expansion




number-theory trigonometry decimal-expansion






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 28 '18 at 16:47









Martin Sleziak

44.9k10121274




44.9k10121274










asked Nov 6 '16 at 14:05









user19405892user19405892

7,63131056




7,63131056








  • 1




    $begingroup$
    I do not know if this is really relevant, but since $arctan(z)=text{Im}log(1+iz)$ and $5+12i=(3+2i)^2$, the question is more or less equivalent to showing that $(3+2i)^{2^k}$ is often close to a real number (i.e.has a small imaginary part, compared to the real part).
    $endgroup$
    – Jack D'Aurizio
    Nov 6 '16 at 15:30






  • 3




    $begingroup$
    where did you find this question ?
    $endgroup$
    – mercio
    Nov 6 '16 at 17:59






  • 1




    $begingroup$
    @mercio I was solving a problem and needed to prove this as a step in the solution.
    $endgroup$
    – user19405892
    Nov 6 '16 at 22:39






  • 1




    $begingroup$
    this could very well be the sort of thing that is "obviously true" (because a number taken randomly in a unit interval will have this property with probability $1$) but completely hopeless to prove.
    $endgroup$
    – mercio
    Nov 6 '16 at 22:46






  • 1




    $begingroup$
    I knew this was a hidden duplicate : math.stackexchange.com/questions/1615708/…
    $endgroup$
    – mercio
    Dec 2 '16 at 14:10














  • 1




    $begingroup$
    I do not know if this is really relevant, but since $arctan(z)=text{Im}log(1+iz)$ and $5+12i=(3+2i)^2$, the question is more or less equivalent to showing that $(3+2i)^{2^k}$ is often close to a real number (i.e.has a small imaginary part, compared to the real part).
    $endgroup$
    – Jack D'Aurizio
    Nov 6 '16 at 15:30






  • 3




    $begingroup$
    where did you find this question ?
    $endgroup$
    – mercio
    Nov 6 '16 at 17:59






  • 1




    $begingroup$
    @mercio I was solving a problem and needed to prove this as a step in the solution.
    $endgroup$
    – user19405892
    Nov 6 '16 at 22:39






  • 1




    $begingroup$
    this could very well be the sort of thing that is "obviously true" (because a number taken randomly in a unit interval will have this property with probability $1$) but completely hopeless to prove.
    $endgroup$
    – mercio
    Nov 6 '16 at 22:46






  • 1




    $begingroup$
    I knew this was a hidden duplicate : math.stackexchange.com/questions/1615708/…
    $endgroup$
    – mercio
    Dec 2 '16 at 14:10








1




1




$begingroup$
I do not know if this is really relevant, but since $arctan(z)=text{Im}log(1+iz)$ and $5+12i=(3+2i)^2$, the question is more or less equivalent to showing that $(3+2i)^{2^k}$ is often close to a real number (i.e.has a small imaginary part, compared to the real part).
$endgroup$
– Jack D'Aurizio
Nov 6 '16 at 15:30




$begingroup$
I do not know if this is really relevant, but since $arctan(z)=text{Im}log(1+iz)$ and $5+12i=(3+2i)^2$, the question is more or less equivalent to showing that $(3+2i)^{2^k}$ is often close to a real number (i.e.has a small imaginary part, compared to the real part).
$endgroup$
– Jack D'Aurizio
Nov 6 '16 at 15:30




3




3




$begingroup$
where did you find this question ?
$endgroup$
– mercio
Nov 6 '16 at 17:59




$begingroup$
where did you find this question ?
$endgroup$
– mercio
Nov 6 '16 at 17:59




1




1




$begingroup$
@mercio I was solving a problem and needed to prove this as a step in the solution.
$endgroup$
– user19405892
Nov 6 '16 at 22:39




$begingroup$
@mercio I was solving a problem and needed to prove this as a step in the solution.
$endgroup$
– user19405892
Nov 6 '16 at 22:39




1




1




$begingroup$
this could very well be the sort of thing that is "obviously true" (because a number taken randomly in a unit interval will have this property with probability $1$) but completely hopeless to prove.
$endgroup$
– mercio
Nov 6 '16 at 22:46




$begingroup$
this could very well be the sort of thing that is "obviously true" (because a number taken randomly in a unit interval will have this property with probability $1$) but completely hopeless to prove.
$endgroup$
– mercio
Nov 6 '16 at 22:46




1




1




$begingroup$
I knew this was a hidden duplicate : math.stackexchange.com/questions/1615708/…
$endgroup$
– mercio
Dec 2 '16 at 14:10




$begingroup$
I knew this was a hidden duplicate : math.stackexchange.com/questions/1615708/…
$endgroup$
– mercio
Dec 2 '16 at 14:10










1 Answer
1






active

oldest

votes


















0












$begingroup$

NOT A SOLUTION:



Something that may be of use



begin{equation}
arctanleft(frac{x + y}{1 - xy}right) = arctan(x) + arctan(y)
end{equation}



Here,



begin{equation}
frac{x + y}{1 - xy} = frac{5}{12}
end{equation}



Which has the integer solutions $x,y = -5$



and so,



begin{equation}
frac{1}{pi}left[arctanleft(frac{5}{12}right)right] = frac{1}{pi}left[arctan(-5) + arctan(-5)right] = -frac{2}{pi}arctan(5)
end{equation}



As before, unsure if this will be of help.






share|cite|improve this answer









$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2001949%2fbinary-expansion-of-frac1-pi-tan-1-left-frac512-right%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    NOT A SOLUTION:



    Something that may be of use



    begin{equation}
    arctanleft(frac{x + y}{1 - xy}right) = arctan(x) + arctan(y)
    end{equation}



    Here,



    begin{equation}
    frac{x + y}{1 - xy} = frac{5}{12}
    end{equation}



    Which has the integer solutions $x,y = -5$



    and so,



    begin{equation}
    frac{1}{pi}left[arctanleft(frac{5}{12}right)right] = frac{1}{pi}left[arctan(-5) + arctan(-5)right] = -frac{2}{pi}arctan(5)
    end{equation}



    As before, unsure if this will be of help.






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      NOT A SOLUTION:



      Something that may be of use



      begin{equation}
      arctanleft(frac{x + y}{1 - xy}right) = arctan(x) + arctan(y)
      end{equation}



      Here,



      begin{equation}
      frac{x + y}{1 - xy} = frac{5}{12}
      end{equation}



      Which has the integer solutions $x,y = -5$



      and so,



      begin{equation}
      frac{1}{pi}left[arctanleft(frac{5}{12}right)right] = frac{1}{pi}left[arctan(-5) + arctan(-5)right] = -frac{2}{pi}arctan(5)
      end{equation}



      As before, unsure if this will be of help.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        NOT A SOLUTION:



        Something that may be of use



        begin{equation}
        arctanleft(frac{x + y}{1 - xy}right) = arctan(x) + arctan(y)
        end{equation}



        Here,



        begin{equation}
        frac{x + y}{1 - xy} = frac{5}{12}
        end{equation}



        Which has the integer solutions $x,y = -5$



        and so,



        begin{equation}
        frac{1}{pi}left[arctanleft(frac{5}{12}right)right] = frac{1}{pi}left[arctan(-5) + arctan(-5)right] = -frac{2}{pi}arctan(5)
        end{equation}



        As before, unsure if this will be of help.






        share|cite|improve this answer









        $endgroup$



        NOT A SOLUTION:



        Something that may be of use



        begin{equation}
        arctanleft(frac{x + y}{1 - xy}right) = arctan(x) + arctan(y)
        end{equation}



        Here,



        begin{equation}
        frac{x + y}{1 - xy} = frac{5}{12}
        end{equation}



        Which has the integer solutions $x,y = -5$



        and so,



        begin{equation}
        frac{1}{pi}left[arctanleft(frac{5}{12}right)right] = frac{1}{pi}left[arctan(-5) + arctan(-5)right] = -frac{2}{pi}arctan(5)
        end{equation}



        As before, unsure if this will be of help.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 25 '18 at 4:09









        DavidGDavidG

        1




        1






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2001949%2fbinary-expansion-of-frac1-pi-tan-1-left-frac512-right%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Probability when a professor distributes a quiz and homework assignment to a class of n students.

            Image
            0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
            Read more

            Aardman Animations

            Image
            Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
            Read more

            Are they similar matrix

            Image
            0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
            Read more