2. A paper punch can be placed at any point on a plane. When it works, it can punch out points that are at an irrational distance from it. What is the minimum number of paper punches needed to punch out all points on the plane?
2. Obviously, two paper punch machines are not enough. To prove that three paper punch machines are sufficient, it is enough to prove that among the distances from any point on the plane to these three points, at least one is an irrational number. Therefore, it is sufficient to construct a sum involving three distances, such that if the entire sum is an irrational number, then one of the distances is an irrational number. For this purpose, place the three paper punch machines at $A(0,0), B(d, 0)$, $C(2 d, 0)$. where $d=\sqrt[5]{2}$.

Let $P$ be any point on the plane, and the distances from $P$ to $A, B, C$ be $u, b, c$ respectively. Since $b$ is the median length of $\triangle P A C$, we have $a^{2}+c^{2}-2 b^{2}=2 \sqrt{2}$. Thus, $a, b, c$ cannot all be rational numbers, meaning that point $P$ will be punched by at least one paper punch machine.