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!

Irrationality Proof

  1. Aug 18, 2009 #1

    jgens

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data

    Prove that [itex]\log_{10}(2)[/itex] is irrational.

    2. Relevant equations

    N/A

    3. The attempt at a solution

    Suppose not, then [itex]\log_{10}(2) = p/q[/itex] where p and q are integers. This implies that [itex]2 = 10^{p/q}[/itex] or similarly, [itex]2^q = 10^p[/itex]. However, this is a contradiction since each number's prime factorization is unique - [itex]2^q[/itex] contains only 2's as prime factors while [itex]10^p[/itex] contains both 2's and 5's. Therefore, our assumption that [itex]\log_{10}(2)[/itex] was rational must have been incorrect. This completes the proof.

    I'm really bad at these irrationality proofs so I was wondering if someone could comment on the validity of my method. Thanks!
     
  2. jcsd
  3. Aug 18, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    That looks like a perfectly valid proof to me!
     
  4. Aug 18, 2009 #3

    jgens

    User Avatar
    Gold Member

    Swell! Thank you very much!
     
  5. Aug 19, 2009 #4
    this is really clever!
    i would have had no idea what to have done.
     
  6. Aug 19, 2009 #5

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    This comment is probably somewhat pedantic, but I think it's worth saying anyways.

    [itex]2^q = 10^p[/itex] is not quite a contradiction -- it can be satisfied when p=q=0. Of course, it's easy to derive a contradiction from that possibility.
     
  7. Aug 19, 2009 #6

    jgens

    User Avatar
    Gold Member

    Perhaps it's a bit pedantic but I definately should have considered that case. Thanks for your input Hurkyl!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Irrationality Proof
  1. Irrationality Proofs (Replies: 9)

Loading...