[Aurelius]

Maybe I should stop, but will take up cardinality. As AmishDude said the rational numbers have the same cardinality as the integers; it can be shown that the algebraic numbers also have that cardinality. So the transcendental numbers have to "make up the difference" and have the same cardinality as the real numbers. It is sort of amusing that we know so few of them.

[AmishDude]

Just trying to prove a number is transcendental is ridiculously hard.