Prove that $x+y$ is irrational if x and y are irrational and positive.

Question

I haven't seen really a straightforward proof towards this question. All of them regarding this topic focus on the fact that $x+y$ can be rational even if x and y are irrational because you could set y as the negative of x, but there isn't really anything about if both are positive and have no minus signs "inside" the variable (say 10-sqrt2 is not valid).

Answer 1

Not true.

$x = \sqrt{2}, y = 2-\sqrt{2}$.

Answer 2

You may take $\sqrt 2$ and $10-\sqrt 2$, they are both positive and irrational and their sum is $10$

Answer 3

Counterexample: let $x \in \mathbb{R}^+ \setminus \mathbb{Q}$ be an arbitrary positive irrational, and define $y=\lceil x \rceil - x$. Then $y$ is itself a strictly positive irrational (why?) and $x+y=\lceil x \rceil \in \mathbb{Z^+}$.