[AmishDude]

No. Transcendental is a subset of irrational. I don't know how many transcendental numbers there are, whether it's the cardinality of the integers or the reals.

e and pi are the only numbers I know to be transcendental, but sqrt(n) for any n not a perfect square is irrational but not transcendental.

[supercat]

No. Transcendental is a subset of irrational. I don't know how many transcendental numbers there are, whether it's the cardinality of the integers or the reals.

The algebraic numbers have the same cardinality as integers. Consequently, since real numbers have a higher cardinality than integers and all real numbers are either algebraic or trancendental, trancendental numbers have a higher cardinality than algebraic ones. This despite the fact that between any two trancendental numbers there is at least one algebraic number.

[Nogbad]

e and pi are the only numbers I know to be transcendental

Any number of the form Alpha^Beta is transcendental,
if Alpha and Beta are non-rational algebraic numbers.
This theorem was proven in 1926 by Gelfond.