$\mathbb{Q}$ dense in $\mathbb{R}$ does also $\mathbb{Q^2}$ dense in $\mathbb{R^2}$?

Question

It is well known that density of Rational numbers in Real numbers a well known result which means that there exist infinity many rational numbers between two arbitrary real numbers , Now I ask simply is $\mathbb{Q^2}$ also dense in $\mathbb{R^2}$? ?

Answer 1

Yes, this is true, and follows almost immediately from the density of $\mathbb{Q}$ in $\mathbb{R}$. For instance, given any open rectangle $(a,b) \times (c,d)$ in $\mathbb{R}^2$, by the density of $\mathbb{Q}$ in $\mathbb{R}$ you can find rationals $p \in (a,b)$ and $q \in (c,d)$. Now $(p,q)$ is a point of $\mathbb{Q}^2$ that is in the given rectangle.