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
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.
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