[eru]

> There is an infinite quantity of both rational and irrational numbers, so isn't it therefore impossible for there to be more of one than of the other?

Mathematicians can even meaningfully compare infinities.

See eg https://www.cantorsparadise.com/this-may-seem-more-irrationa... or https://math.stackexchange.com/questions/474415/intuitive-ex...

You can also look at eg a uniform random variable on the interval between 0 to 1. The probability of hitting a rational number is 0%. The probability of hitting an irrational number is 100%.

> Or is the reasoning that, because there is an infinite quantity of irrational numbers between any two given rational numbers, there are therefore many more irrational numbers than rational numbers?

No, that's not enough. There are also an infinitely many rational numbers between any two given irrational numbers.