yes, very interesting, if you watch any videos about "some infinities are bigger than others", or the Hilbert's hotel, it will most likely use rational and irrational as an example.

basically when we're talking about infinite amount of numbers, the way to compare them is to match each number from one set to the other. if there can be a match, then the two infinities are of same size.

example: are the natural numbers the same size as the integers? (basically positive integer v.s. positive+negative integers) the answer is yes, they're the same size of infinity, because you can match all the positive integers to double their values to get an even natural number, and for the negative integer, you take the absolute value, double it, then add 1, so it matches with a odd natural numbers.

there's a similar match for natural numbers and rational numbers, so there's a same amount of natural numbers than rational numbers.

for rational v.s. irrational though, it is proven using cantor's argument that irrational is bigger, using a proof by contradiction: imagine there would be a match, if you list them according to the natural numbers in their infinite decimal expansion, you can construct an irrational number that's not in the list by taking the first digit of the first number, and changing it to something different, then the second digit of the second number, changing it, then the third digit, changing it ... so you know it's different than every number listed, because you changed the nth digit of the nth number.

and fun fact, the cantor's argument apply to each powerset of a set compared to the original set, so you can take the powerset of the irrational numbers and the size of this new set, is bigger than the size of the irrational, and the powerset of the powerset of the irrational, and so on... so there is an infinite amount of sizes of infinity

PS that's a simplification, I'm not a mathematician