Are there more transcendental numbers or irrational numbers that are not transcendental? This is not a question of counting (obviously), but more of a question of bigger vs. smaller infinities. I really don't know where to even start with this one whatsoever. Any help? Or is it unsolvable?

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.