Are there more transcendental numbers or irrational numbers that are not transcendental?

Solution 1:

The non-transcendental numbers (otherwise known as the algebraic numbers – Wikipedia link) comprise a countably infinite set, whereas the transcendental numbers are uncountably infinite.

(Why are there only countably many algebraic numbers? Because we can group them according to what polynomial in $\mathbb{Q}[x]$ they are a root of, and any such polynomial has finitely many roots, and there are only countably many such polynomials.)

The point is: in colloquial terms, there are more transcendental numbers than algebraic numbers.

Therefore, there are certainly more transcendental numbers than there are algebraic numbers that also are not rational.

Solution 2:

The set of algebraic numbers $\mathbb A$ is countable, so $\mathbb A\cap (\mathbb R\setminus \mathbb Q)$ is also countable. On the other hand, the set of transcendental numbers $\mathbb R\setminus\mathbb A$ must be uncountable, so $$|\mathbb R\setminus\mathbb A|>|\mathbb A\cap(\mathbb R\setminus\mathbb Q)|.$$