What is a point in $\mathbb{R}^2\setminus \mathbb{Q}^2$ that only has one irrational coordinate?

Question

I am confused why wouldn't every single point in $\mathbb{R}^2\setminus \mathbb{Q}^2$ be irrational.

It seems that whenever we have a pair of rational value or pair of integer, it is removed. Then the stuff between two removed point has all be irrational. But why can there exist points where only one coordinate is irrational?

Answer 1

The point $\left\langle\sqrt2,2\right\rangle$ is in $\Bbb R^2$, since $\sqrt2$ and $2$ are both real numbers, and it is not in $\Bbb Q^2$, since $\sqrt2$ is not a rational number, so it is in $\Bbb R^2\setminus\Bbb Q^2$. The points that are removed from $\Bbb R^2$ are those with both coordinates rational, so you’ve kept every point that has at least one irrational coordinate. That includes the points with one rational and one irrational coordinate.