how there can be more irrational numbers than rational numbers if there’s an infinite amount of both?

Question

In terms of group theory, it’s been proven that the set of rational numbers is countable, specifically, infinitably countable. Now I understand that the set of irrational numbers is not countable because there’s no way you can line them up such that you can list every single one without skipping, but apparently this means that you could say that there are more irrational numbers than there are rational. How is this possible then, if there’s is an infinite amount of each?

In: Mathematics

Answer 1

If I have bunch of kids and bunch of chairs, how do I know there’s the same number of both? Well, you could count them, but a simple way is to just make one kid sit on one chair. If you have no chairs and no kids left without a pair, there were equal number of both.

With infinite sets, it works much the same. If you can pair up elements of one set with another, so that none are left out, then surely we can say there are equally many elements in both sets.

And it turns out, you cannot pair up irrational numbers with rational numbers.

Now, as to “why”, you gotta realize that in some ways the set of all rational numbers is restricted. It’s not just formless blob of infinity, the numbers are of specific form(ratios of two integers) and this means the elements could in a way run out, despite being infinite.

Like, the usual example of countably infinite set is natural numbers. With natural numbers, you have each number be finite. There are infinitely many integers, each larger than the one before it, but they are all finite. That’s kinda weird. One of the usual attempts at “counterproof” of different sizes of infinity is trying to prove integers have same size as reals between 0 and 1. You pair each integer with real that’s “0.” added in front of it, so 345th integer turns into 0.345 for example. But you never get a number like 1/3 = 0.333… because 333… isn’t an integer.

To prove reals are more numerous, the typical way is to notice that if you had paired all integers with some reals, you can find real number that isn’t the same as the first one, not the same as the second one, not the same as the third one, etc. So very concretely, you run out of natural numbers. And that’s kinda because natural numbers aren’t just formless infinity, they’re a well-defined collection of things, and this structure should be looked at beyond just observing “there are infinitely many of them”