roots, so the set of all possible roots of all polynomials with integer coefficients is a countable union of finite sets, hence at most countable. It is obvious that the set is not finite, so the set of all algebraic numbers are countable.

Are algebra numbers countable?

All integers and rational numbers are algebraic, as are all roots of integers. ... The set of complex numbers is uncountable, but the set of algebraic numbers is countable and has measure zero in the Lebesgue measure as a subset of the complex numbers. In that sense, almost all complex numbers are transcendental.

How do you prove that a set of algebraic numbers is countable?

We can prove the theorem by a cardinality argument, counting the number of such polynomials and roots. By Set of Polynomials over Infinite Set has Same Cardinality, the set Q[x] of polynomials over Q is countable. Next, note that A can be written as the union of the set of roots of each polynomial.