sum of irrational numbers - are there nontrivial examples? I know that the sum of irrational numbers does not have to be irrational. For example $\sqrt2+\left(-\sqrt2\right)$ is equal to $0$. But what I am wondering is there any example where the sum of two irrational numbers isn't obviously rational like an integer and yet after, say 50 digits after the decimal point it turns out to be rational. Are there any such examples with something non-trivial going on behind the scenes?

If $a+b=q$, where $a,b\notin\mathbb{Q}$ and $q\in \mathbb{Q}$, then $a=q-b$, so just choose
a rational number whith sufficiently long period of decimals and you will get what you want.
On the other hand, this is still quite trivial, since here we just sum up $b$ and $q-b$ (in your question, $q=0$).
