Edit made on August 13, 2011 by ColinWright at 17:02:31
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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 EQN:e , EQN:\pi , sin(1) , log(2) and EQN:e^\pi
Whether EQN:2^e , EQN:2^\pi or EQN:\pi^e are transcendental or not has still to be decided.