A complex number x C is called transcendental if x is not al

A complex number x C is called transcendental if x is not algebraic. Prove that the set T of transcendental numbers is uncountable.

(Remark: It is very hard to prove that a given complex number x is transcendental. Essentially the only known examples are e and . We do not even know if is transcendental. However, this problem shows that there are “more” transcendental numbers than algebraic numbers.)

Solution

We say that x C is transcendental if it is not algebraic.

Theorem

The set of all transcendental numbers in R is uncountable.

Proof. I Let T be the set of all transcendental numbers in R.

I Let A be the set of all algebraic numbers in R.

I If T is countable,

then R = A T is countable.

But R is not countable by Cantor’s theorem.