2π is an irrational number, and 1 is a rational number. When an irrational number is added to a rational number, the result is that the irrational number 2π (Pi)+1 is an irrational number.
The root number 2 is irrational and 1 is rational. When an irrational number is added to a rational number, the result is irrational, so √2+ 1 is irrational.