Can we construct three irrational numbers $a,b,c$ such that $a+b+c \in \mathbb Q$?

Question

This is rather easily shown to be possible if no constraint is put on $a,b,c$.

However, is it also possible under the following constraint: $a, b$ and $c$ can not be rational multiples of each other.

If that constraint is too tight to allow such a construction, would it be possible if we loosen it a bit so that $a,b,c$ can be rational- but not natural multiples of each other?

Answer 1

There are such triples. An example would be $\sqrt{2},\sqrt{3},-\sqrt{2}-\sqrt{3}$. It's fairly easy to show that neither is a rational multiple of another.

Answer 2

the general solution to this is addition of any of two irrational numbers is equal to the negative of third irrational number. example as given by Wojowu so general representation of numbers $\sqrt{a},\sqrt{b},-[\sqrt{a}+\sqrt{b}]$ so there sum is $0$ where the numbers arent rational or complex numbers and it is all.