Rational numbers are all numbers expressible as #p/q# for some integers #p# and #q# with #q != 0#.

#pi# is not expressible as #p/q# for some integers #p#, #q# with #q != 0#, though there are some good approximations of that form. So it is not rational and is irrational.

The Chinese discovered that #355/113# was a good approximation for #pi# about 15 centuries ago.

#355/113 ~= 3.1415929#

See https://en.wikipedia.org/wiki/Mil%C3%BC

#pi# is not only irrational, it is what is called a transcendental number: It is not a root of any polynomial equation with integer coefficients.

Though almost all real numbers are transcendental numbers, it is not easy to determine that any given number is transcendental. For example, it has been proved that #pi# and #e# are transcendental numbers, but it is not known whether #pi+e# is transcendental.