As adjectives the difference between irrational and radical
is that
irrational is not rational; unfounded or nonsensical while
radical is favoring fundamental change, or change at the root cause of a matter.
As nouns the difference between irrational and radical
is that
irrational is a real number that can not be expressed as the quotient of two integers, an irrational number while
radical is a member of the most progressive wing of the liberal party; someone favouring social reform (but generally stopping short of socialism).