Prove sqrt(50) irrational

**Rapid_W** · August 12th 2009, 12:23 PM

Just a little queations really

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

Now i can prove that $\sqrt{2}$ is irrational, can i just say that $5\times\sqrt{2}$ is irrational with no extra proof?

**alunw** · August 12th 2009, 12:38 PM

I'd be convinced, but it is perhaps even more obvious if you do things the other way around. If $\surd 50$ were rational then you would have $\surd 2$ = $\frac{\surd 50}{5}$ so that then $\surd 2$ would be rational as well.

**red_dog** · August 12th 2009, 12:39 PM

Suppose that $5\sqrt{2}$ is rational.

Then $5\sqrt{2}=q, \ q\in\mathbb{Q}\Rightarrow \sqrt{2}=\frac{q}{5}\in\mathbb{Q}$ , contradiction. Therefore, the supposition is false and $5\sqrt{2}$ is irational.

**Rapid_W** · August 12th 2009, 12:46 PM

Originally Posted by red_dog

Suppose that $5\sqrt{2}$ is rational.

Then $5\sqrt{2}=q, \ q\in\mathbb{Q}\Rightarrow \sqrt{2}=\frac{q}{5}\in\mathbb{Q}$ , contradiction. Therefore, the supposition is false and $5\sqrt{2}$ 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!

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