1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Homework Helper

    Say the two rationals are p and q.
    where r is any irrational and n is any integer such that

    x is irrational
    Last edited: Dec 17, 2008
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook