The square root of -4 is the same thing as the square root of -1 times the square root of 4. The square root of 4 is 2, but the square root is an imaginary number, i. Since it is an imaginary number, it is indeterminate whether it is a rational or irrational (given current number theory) so the square root of negative 4 is neither rational nor irrational. It is imaginary.
