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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.
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