Irrationals: A Group? I understand that the set of irrational numbers with multiplication does not form a group (clearly, $\sqrt{2}\sqrt{2}=2$, so the set is not closed). But is there a proof or a counter-example that the irrationals with addition form (or do not form) a group?
Thank you!
Edit: In particular, I am wondering if the set is closed with respect to addition.

$0$ is not irrational. The set of irrationals is not even closed under addition: 
$$(1-\sqrt{2})+\sqrt{2}=1$$
