Results 1 to 3 of 3

Math Help - Proving irrationality

  1. #1
    Newbie
    Joined
    Feb 2007
    Posts
    7

    Proving irrationality

    Hi,

    Can anyone help with this one:
    Establish the following facts:
    √p is irrational for any prime p.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by lyla View Post
    Hi,

    Can anyone help with this one:
    Establish the following facts:
    √p is irrational for any prime p.
    Euclid's proof.
    --------------
    Assume,
    \sqrt{p}=a a positive integer.
    By definition it is equivalent to say,
    p=a^2.
    Now the prime decomposition of a^2 has an even amount of prime factors because of the square. While p does not. By uniquness this is impossible.

    Pythagorus' Proof.
    -------------------
    Assume that \sqrt{p}=\frac{n}{m} where n,m is a reduced fraction, meaning no common factors.
    Then,
    p=\frac{n^2}{m^2}
    m^2p=n^2.
    Note, the right hand side is divisible by p because the left hand side. Meaning p divides n^2. But then n itself is divisible by p by properties of prime numbers. That is n=pk.
    Subsitute,
    m^2p=k^2p^2
    m^2=k^2p
    But then the left hand side is divisble by p, that is o divides m^2. But then p divides m. Hence n and n have common factors, contrary to assumption.
    (This is also a similar approach to Fermat's principle of infinite descent.)

    Eisenstein Proof.
    -----------------
    The polynomial x^2-p\in \mathbb{Z}[x] fits the conditions of Eisenstein irreducibility criterion for p. Thus, there are no solution in \mathbb{Z} and hence none in \mathbb{Q}.

    Rational Roots Proof.
    ----------------
    The polynomial x^2-p can only have zeros for \pm 1,\pm p. None of which work. Thus, it is irrational.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Feb 2007
    Posts
    7
    I have to thank you very much for the help.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: October 5th 2010, 12:39 PM
  2. Replies: 8
    Last Post: July 21st 2010, 06:57 AM
  3. Irrationality
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: February 16th 2010, 07:06 PM
  4. Irrationality and Division
    Posted in the Algebra Forum
    Replies: 3
    Last Post: December 18th 2008, 05:42 AM
  5. [SOLVED] Proving the irrationality of √3.
    Posted in the Algebra Forum
    Replies: 2
    Last Post: September 28th 2008, 03:34 PM

Search Tags


/mathhelpforum @mathhelpforum