On the number line, are there more rational numbers or irrational numbers? I was told that there are equally many rational and irrational numbers. Is this correct? How could we prove that?
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2No, the person who told you that is wrong. There are more irrational numbers than rational numbers. Indeed, there are [uncountably](http://en.wikipedia.org/wiki/Uncountable_set) many irrational numbers but only [countably](http://en.wikipedia.org/wiki/Countable_set) many rational numbers. – user1729 May 09 '14 at 09:17

As far as rational numbers and irrational numbers alternating is concerned, there is the fact that between any two rational numbers you can find an irrational number, and vice versa. – J W May 09 '14 at 09:26
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Hint The cardinality of the rational numbers, $\Bbb Q)$, is countable, but the cardinality of the real numbers, $\Bbb R$, is uncountable. How many irrational numbers $\Bbb R \setminus \Bbb Q$ must there be?
naslundx
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@Yukulélé You can make the same argument for [0,1]. And [0,1[ just has one less element. – naslundx Oct 04 '18 at 13:28