Since the set of algebraic numbers is countable, it follows that the set of transcendental numbers is uncountable. In other words, almost all of the real numbers are transcendental. Nonetheless, it is extremely difficult to find a transcendental number, or to prove that a number is transcendental.
It was not until 1873 that e was proved to be transcendental (by Hermite), and not until 1882 that pi was proved to be transcendental (by Lindemann)