Let the root number 2 be a rational number, then the root number 2 can be expressed in the form of p/q, and p\q is a approximate fraction (that is, p and q are coprime). The square of P 2 = 2Q 2, obviously p is an even number. Let p=2r be substituted into q 2 = 2r 2, so q is even, so p and q have a common divisor of 2, which contradicts that p/q is a cross fraction.

So the root number 2 is irrational.