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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- Aug 12th 2009, 12:23 PMRapid_WProve 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? - Aug 12th 2009, 12:38 PMalunw
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.

- Aug 12th 2009, 12:39 PMred_dog
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. - Aug 12th 2009, 12:46 PMRapid_W