(a) A real number x is said to be algebraic (over the rationals) if it satisfies some poly...

(a) A real number x is said to be algebraic (over the rationals) if it satisfies some polynomial equation of positive degree

with rational coefficients ai. Assuming that each polynomial equation has only finitely many roots, show that the set of algebraic numbers is countable.

(b) A real number is said to be transcendental if it is not algebraic. Assuming the reals are uncountable, show that the transcendental numbers are uncountable. (It is a somewhat surprising fact that only two transcendental numbers are familiar to us: e and n. Even proving these two numbers transcendental is highly nontrivial.)