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Thread: Prove sqrt(50) irrational

  1. #1
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    Prove sqrt(50) irrational

    Just a little queations really

    $\displaystyle \sqrt{50} = 5\times\sqrt{2}$

    Now i can prove that $\displaystyle \sqrt{2}$ is irrational, can i just say that $\displaystyle 5\times\sqrt{2}$ is irrational with no extra proof?
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  2. #2
    Member alunw's Avatar
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    I'd be convinced, but it is perhaps even more obvious if you do things the other way around. If $\displaystyle \surd 50$ were rational then you would have $\displaystyle \surd 2$ = $\displaystyle \frac{\surd 50}{5}$ so that then $\displaystyle \surd 2$ would be rational as well.
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  3. #3
    MHF Contributor red_dog's Avatar
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    Suppose that $\displaystyle 5\sqrt{2}$ is rational.

    Then $\displaystyle 5\sqrt{2}=q, \ q\in\mathbb{Q}\Rightarrow \sqrt{2}=\frac{q}{5}\in\mathbb{Q}$, contradiction. Therefore, the supposition is false and $\displaystyle 5\sqrt{2}$ is irational.
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  4. #4
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    Quote Originally Posted by red_dog View Post
    Suppose that $\displaystyle 5\sqrt{2}$ is rational.

    Then $\displaystyle 5\sqrt{2}=q, \ q\in\mathbb{Q}\Rightarrow \sqrt{2}=\frac{q}{5}\in\mathbb{Q}$, contradiction. Therefore, the supposition is false and $\displaystyle 5\sqrt{2}$ is irational.
    i don't really understand this, can you add a few more words explaining please

    edit-ignore this question, i just reread it a few times and it makes complete sense now thanks!
    Last edited by Rapid_W; Aug 12th 2009 at 01:01 PM.
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