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: Smallest positive irrational number

  1. Apr 30, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that there is no smallest positive irrational number

    2. Relevant equations

    3. The attempt at a solution

    I have no idea how to do this, please help walk me through it.
  2. jcsd
  3. Apr 30, 2010 #2


    Staff: Mentor

    Do you know how to do a proof by contradiction? If so, assume that there is a smallest positive irrational number, and then produce another one that's even smaller.
  4. Apr 30, 2010 #3
    I thought the way to do it was by contradiction. But I'm confused as to how to produce a generalized irrational number, and then like you say, get one smaller than that.
  5. May 1, 2010 #4
    Label your arbitrary irrational number a. Starting with a, can you think of a way to get a number smaller than a? There may be many ways to do this. Once you have a candidate idea in mind, try to prove that the number you get is always irrational.
  6. May 1, 2010 #5
    You could try using the density of rationals in R
  7. May 1, 2010 #6


    User Avatar
    Science Advisor

    If r is a positive irrational number, then r/2 is a smaller positive irrational number.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook