A number which is not an algebraic number is called transcendental.

Transcendental numbers are a strict subset of the irrational numbers.

Although "almost all" real numbers are transcendental it is difficult to prove that a given number that occurs naturally is transcendental. The usual weapon of choice is the Gelfond-Schneider Theorem.

The size of the set of transcendental numbers is the size of the continuum c. (See Uncountable sets).

Some numbers shown to be transcendental are $e$ , $\pi$ , sin(1) , log(2) and $e^\pi$

Whether $2^e$ , $2^\pi$ or $\pi^e$ are transcendental or not has still to be decided.