Results 1 to 3 of 3

Math Help - Proof....rational numbers

  1. #1
    Member
    Joined
    Jan 2011
    Posts
    87

    Proof....rational numbers

    Hey all,
    how can i prove that if a and b are non-zero rational numbers, then a + b\sqrt{2} \not\in \mathbb{Q}

    ...using the fact that \sqrt{2} \not\in \mathbb{Q}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,404
    Thanks
    1293
    Does this need proof? It should be obvious that a rational times an irrational is irrational, so \displaystyle b\sqrt{2} is irrational.

    The sum of a rational and an irrational is also irrational, so \displaystyle a + b\sqrt{2} is irrational.


    If it requires proof, then you probably need to use the fact that the set of rationals is closed under addition and multiplication...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Oiler View Post
    Hey all,
    how can i prove that if a and b are non-zero rational numbers, then a + b\sqrt{2} \not\in \mathbb{Q}

    ...using the fact that \sqrt{2} \not\in \mathbb{Q}
    Let $$a \ne 0 and $$b \ne 0 be rational and suppose a+b\sqrt{2} is rational. Hence there exists a rational $$c such that:

    c=a+b\sqrt{2}

    But that rationals are closed under addition so:

    d=c+a=b\sqrt{2}

    is rational. Also the rationals are closed under division (by non-zero elements anyway) so:

    e=d/b=\sqrt{2}

    is rational, which is a contradiction so our hypothesis that a+b\sqrt{2} is rational fails, hence a+b\sqrt{2} is irrational.

    CB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof: If p and q are rational numbers then p + q...
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: March 16th 2011, 09:17 PM
  2. Proof: All rational numbers are algebraic numbers
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: September 5th 2010, 10:26 AM
  3. Replies: 0
    Last Post: April 17th 2009, 11:36 AM
  4. rational numbers proof
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: March 6th 2009, 12:33 AM
  5. rational numbers proof
    Posted in the Algebra Forum
    Replies: 1
    Last Post: September 9th 2008, 10:13 AM

Search Tags


/mathhelpforum @mathhelpforum