The intersection of the set of rational numbers and the set of irrational numbers is empty.

Here's why:

* Rational numbers can be expressed as a fraction of two integers (a/b, where b is not zero).

* Irrational numbers cannot be expressed as a fraction of two integers. They have decimal representations that go on forever without repeating.

Since these definitions are mutually exclusive, there is no number that can be both rational and irrational.