In
mathematics, a
transcendental number is a
real or
complex number that is not
algebraic—that is, it is not a
root of a non-zero
polynomial equation with
rational coefficients. The best-known transcendental numbers are
p and
e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed,
almost all real and complex numbers are transcendental, since the algebraic numbers are
countable while the sets of real and complex numbers are both
uncountable. All real transcendental numbers are
irrational, since all rational numbers are algebraic. The
converse is not true: not all irrational numbers are transcendental; e.g., the
square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation .