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
> 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.
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