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.
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