1. ## 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?

2. 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.

3. 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.

4. Originally Posted by red_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.
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!