Results 1 to 6 of 6

Math Help - derichlet function question..

  1. #1
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    derichlet function question..

    derihlet function return 1 if the input is rational.
    and 0 if the input is irrational.

    prove that Dirichlet function is periodic

    and prove that every rational number is its period
    ??
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by transgalactic View Post
    derihlet function return 1 if the input is rational.
    and 0 if the input is irrational.

    prove that Dirichlet function is periodic

    and prove that every rational number is its period
    ??
    I think all that needs is the following observation:

    For a fixed rational r,
    1) Every rational can be written as sum of another rational and r, uniquely.
    2) Every irrational can be written as sum of another irrational and r, uniquely.

    f(x+r) = f(x) for all x, where f is the Dirichlet function.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    f(x+r) = f(x)


    how this expression proves the its periodic?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by transgalactic View Post
    f(x+r) = f(x)


    how this expression proves the its periodic?
    That is the definition of periodicity I know... What is yours?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    thanks you are correct

    how to prove the second part that
    each rational number is its period?

    i think i can use f(x+r) = f(x)and say that
    for every rational number "r"

    i get same number

    is it ok?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    i think i can use f(x+r) = f(x)and say that
    for every rational number "r"
    Exactly!

    For every r, f(x+r) = f(x) and thus r is the period of the function.

    Thus every rational is the period of the function.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 12
    Last Post: October 2nd 2011, 06:07 AM
  2. function inside function proofe question
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 13th 2010, 11:32 AM
  3. Replies: 1
    Last Post: May 30th 2009, 02:59 PM
  4. Replies: 1
    Last Post: August 29th 2008, 10:17 AM
  5. Replies: 2
    Last Post: March 30th 2008, 02:28 PM

Search Tags


/mathhelpforum @mathhelpforum