TITLE: The sum of two irrational square roots
QUESTION [5 upvotes]: This is very similar to this question, but I was wondering if there was a simpler proof.  In particular, a proof that would prove that $\sqrt{x}+\sqrt{y}$ is an irrational number if both $\sqrt{x}$ and $\sqrt{y}$ are irrational.

REPLY [10 votes]: If $x$ and $y$ are integers then so is $x - y$ and further if $\sqrt x + \sqrt y$ is rational then $\sqrt x - \sqrt y = \frac{x - y}{\sqrt{x} + \sqrt{y}}$ is also rational. Then by sums and differences so are both $\sqrt{x}$ and $\sqrt{y}$. Therefore if we assume that at least one of $\sqrt{x}$ and $\sqrt{y}$ are irrational than so is their sum.