Results 1 to 2 of 2

Thread: Square root Irrational

  1. #1
    Senior Member I-Think's Avatar
    Joined
    Apr 2009
    Posts
    288

    Square root Irrational

    Quick proof veracity test for this question

    Let $\displaystyle d\in{N}$ that is not the square of another natural number. Prove that $\displaystyle \sqrt{d}$ is not a rational number

    My proof
    Assume $\displaystyle \sqrt{d}\in{Q}$, hence $\displaystyle \sqrt{d}=\frac{a}{b}$, where $\displaystyle a,b\in{N}$ and $\displaystyle gcd(a,b)=1$

    So $\displaystyle d=\frac{a^2}{b^2}$, so $\displaystyle db^2=a^2$. Note $\displaystyle d|a^2$, so d|a
    Hence$\displaystyle a=dn$
    and $\displaystyle a^2=d^2n^2$

    Therefore
    $\displaystyle d=\frac{d^{2}n^{2}}{b^2}$, divide by b

    $\displaystyle 1=\frac{dn^2}{b^2}$

    So $\displaystyle b^2=n^{2}d$ and $\displaystyle d=\frac{b^2}{n^2}$
    So $\displaystyle \sqrt{d}=\frac{b}{n}$
    Thus $\displaystyle \sqrt{d}=\frac{a}{b}=\frac{b}{n}$
    Hence
    $\displaystyle \frac{a}{b}=\frac{b}{n}\rightarrow{b^2=an}\rightar row{\frac{b^2}{a}=n}$
    Hence $\displaystyle a|b$
    But this gives a contradiction as $\displaystyle gcd(a,b)=1$
    Hence our initial assumption is wrong and $\displaystyle \sqrt{d}$ is irrational
    QED
    Is this proof 100% correct?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Pim
    Pim is offline
    Member
    Joined
    Dec 2008
    From
    The Netherlands
    Posts
    91
    To me it seems correct. Except for one thing: you say "divide by b" , but you're actually dividing by d. All the steps are correct though.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 11
    Last Post: Oct 25th 2009, 06:45 PM
  2. Replies: 2
    Last Post: May 26th 2009, 09:09 AM
  3. Replies: 12
    Last Post: Nov 22nd 2008, 12:41 PM
  4. Proof of square root being irrational
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Sep 14th 2007, 01:07 PM
  5. How to prove square root 2 is irrational?
    Posted in the Math Topics Forum
    Replies: 8
    Last Post: Jun 24th 2007, 07:40 AM

Search Tags


/mathhelpforum @mathhelpforum