How do we prove the existence of uncountably many transcendental numbers? I know how to prove the countability of sets using equivalence relations to other sets, but I'm not sure how to go about proving the uncountability of the transcendental numbers (i.e., numbers that are not algebraic).

If a number $t$ is algebraic, it is the root of some polynomial with integer coefficients. There are only countably many such polynomials (each having a finite number of roots), so there are only countably many such $t$.  Since there are uncountably many real (or complex) numbers, and only countably many of them are algebraic, uncountably many of them must be transcendental.
