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

When mathematicians talk about ‘an infinite amount’ they are using formal definitions of size.

Two sets A and B are the same size if you can pair each object in A to an object in B, and there are no objects left over in either set. For example the sets {cat, sheep, dog} is the same size as {3,2,8} because I can pair the object in them like (cat,8), (sheep,3), (dog,2). But {cat, sheep, dog} is NOT the same size as {9,2} because there will always be objects left in the first set no matter how I make the pairs.

> 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

Listing is important, because if you number the list (1,2,3,4…) you are essentially creating a pair of counting numbers and whatever you are listing. If you can create such a list (even if it never ends) you are showing that the set of counting numbers and the set of things you are listing are both the same size.

For example, consider the all the ‘words’ made up of the letter ‘a’ repeating 1 to infinite number of times. You can easily create a numbered list for these words that will allow all such words to be paired with a counting number: 1.a, 2. aa, 3.aaa, 4.aaaa, and so on.

You can do the [same with rational numbers](https://www.homeschoolmath.net/teaching/rational-numbers-countable.php). It is actually pretty exciting that you can create a numbered list of all rational numbers in a way that you WON’T miss any. Since you can create this numbered list, the size of the set of rational numbers and the set of counting numbers is the same, i.e. both are countably infinite.

Irrational numbers however can NOT be listed. This means they cannot be the same size as counting numbers. Since there will always be irrational numbers left over no matter how you list them, there are more of them than the counting numbers, and the rational numbers, i.e. they are uncountable.

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”

The answers here are good, so I won’t rehash them. But I will offer another perspective to try to answer your main question:

> 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?

If you were to pick a number between 0 and 1, with all possible numbers having the same chance of being picked, the probability that you would get a rational number is effectively zero. The probability that you would get an irrational number is effectively 100%.

If you picked a million numbers between 0 and 1, again with all possible numbers having the same chance of being picked, the probability that you get even a single rational number is effectively zero. You could pick a billion, or a trillion, or a googol numbers this way and the probability that you would even get a single rational number is effectively zero.

By “effectively zero”, I mean that there is no positive number that is smaller than the probability that you will get a rational. I know that’s not satisfying – it would be better if I could just say “the probability is zero”. But we’re dealing with infinity here so things aren’t always quite so tidy.

The reality is that the set of irrationals is not “sort of” bigger than the set of rationals. The reality is that the set of irrationals is so much bigger than the set of rationals that if you put them next to each other, the rationals all but disappear.

> if there’s is an infinite amount of each

It’s a weird concept to get your head around, having different cardinalities that are all infinite, but one is “bigger” than the other.

But that’s the core idea. You can pair up all rational numbers to a single irrational number, and you’ll still have irrational numbers left over that don’t have a partner. You can’t arrange your list in such a way that you have matched up all irrational numbers and still have rational numbers left over. Therefore there are “more” irrational numbers.