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Homework Help: Prove there is an irrational between any two rationals

  1. Dec 17, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove that there exists an irrational between any two rationals.



    2. Relevant equations



    3. The attempt at a solution

    How would one do this? So far I've proven there is an irrational between any rational and irrational, any irrational and rational, that there's a rational between any two reals, or a irrational between any two reals in the attempt; I realize that it can be taken as a case of the last two; but I haven't been able to start with just p,q such that p>q, and p and q rational, and arrive at this result. It's pissing me off.

    Thanks in advance.
     
  2. jcsd
  3. Dec 17, 2008 #2
    Let p ,q be rationals with p>q

    Let e be a positive real. Then for e < |p-q|, q+e is in the open interval (p, q). Due to the density of the irrationals in R (as you pointed out, between any two reals is a rational), there is a positive irrational, call it a, that is less than e. Now take q+a. What can you say about that value?
     
  4. Dec 17, 2008 #3

    lurflurf

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    Homework Helper

    Say the two rationals are p and q.
    define
    x=[(p+q)/2]+[(p-q)/2]r/n
    where r is any irrational and n is any integer such that
    |r|<n

    show
    (1/2)(p+q-|p-q|)<x<(1/2)(p+q+|p-q|)
    or
    |2x-p-q|<|p-q|
    and
    x is irrational
     
    Last edited: Dec 17, 2008
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