Results 1 to 3 of 3

Math Help - explicit sequence

  1. #1
    Member
    Joined
    Jun 2008
    Posts
    170

    explicit sequence

    Let  \bold{Z} \times (\bold{Z}- \{0 \}) = \{(a,b): a,b \in \bold{Z}, b \neq 0 \} . This is a countable set. Now let us suppose that we have a function  f: \bold{Z} \times (\bold{Z}- \{0 \}) \to \bold{Q} defined by  f(a,b) := a/b . Then  f(\bold{Z} \times (\bold{Z}- \{0 \})) = \bold{Q} which is at most countable. Since  \bold{Q} in infinite, then it must be countable. Then we can arrange the rational numbers in a sequence:  \bold{Q} = \{a_{0}, a_{1}, a_{2}, a_{3}, \ldots \} .

    How would you come up with a sequence, such that every element of the sequence is different from every other element, and the elements of the sequence exhaust  \bold{Q} ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Jun 2008
    Posts
    170
    Is there an explicit formula?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by particlejohn View Post
    Is there an explicit formula?
    Yes. It is not nice looking, one such formula can take advantage of unique factorization of the numbers into primes.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help with Explicit Sequence Formula
    Posted in the Algebra Forum
    Replies: 3
    Last Post: May 24th 2010, 07:33 PM
  2. Implicit to Explicit
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: February 3rd 2010, 07:01 PM
  3. explicit formula
    Posted in the Algebra Forum
    Replies: 2
    Last Post: November 19th 2008, 12:23 PM
  4. find the explicit formula for sequence
    Posted in the Algebra Forum
    Replies: 1
    Last Post: September 26th 2008, 04:24 PM
  5. Explicit form
    Posted in the Differential Equations Forum
    Replies: 6
    Last Post: July 1st 2008, 12:07 PM

Search Tags


/mathhelpforum @mathhelpforum