Can square roots of imperfect square such as $ \sqrt{2}$ , $ \sqrt{3}$.....$ \sqrt{n}$ be written as sum of other real numbers or other imperfect square roots which are not linear combinations with multiple of $ \sqrt{n}$ as one of its term, where n is the imperfect square whose root need to be represented. I believe it can't be, but is there any theorem which states that ? To put it even simply. Does there exist $ a,b \in R $ such that
$$ a+b= \sqrt{n}$$ where $ n$ is an imperfect square and a,b are not linear combinations using multiples of $\sqrt{n}$ one of their terms.
If "real numbers" is your only restriction, then this is simple (and almost feels like cheating, if that makes any sense). Let $a = \pi$ and $b = \sqrt n - \pi$. Then $$ a + b = \pi + \sqrt n - \pi = \sqrt n $$