The set of all algebraic numbers is countable.

The story behind the answer

True. An algebraic number is any complex number that solves a nonzero polynomial with integer coefficients. There are only countably many such polynomials (for each degree, the list of integer coefficient tuples is countable; a countable union of countable sets is countable). Each polynomial has finitely many roots, so the set of all algebraic numbers—being a countable union of finite sets—is countable.

Context: This contrasts sharply with the real numbers, which are uncountable, so “most” real numbers are transcendental (not algebraic), like π and e. Liouville first proved the existence of transcendental numbers in the 19th century. Memory tip: “Count equations, count roots”—since you can list all integer-coefficient polynomials, you can, in principle, list all their roots.