A sum of irrational numbers ending rational

Question

Let $x$ be a positive irrational number

I know that there exists $y$ such that:

$$\begin{cases} y>0 \\ x+y\in \mathbb Q.\end{cases}$$

How would you construct explicitly such $y$ ?

For instance if $y=\sqrt 2$ ?

Answer 1

Hint: Let $n\in \mathbb N$ such that $n > x$. Take $y=n-x$.

Make sure to justify that such an $n$ exists and that $y$ is irrational.