> Q is indeed dense in R, but firstly it’s very clear that there isn’t an equal number of them because rational numbers are a subset of the real numbers and there exists at least one irrational number (I pick “e”) that is in R but not in Q. So R must be at least bigger than Q.

This isn't a correct explanation, because I can use the same explanation to show that there are more integers than that there are even integers.

"it’s very clear that there isn’t an equal number of them because even numbers (let's call it E) are a subset of the natural numbers (let's call that N) and there exists at least one odd number (I pick 1) that is in N but not in E. So N must be at least bigger than E."