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 many irrational numbers but only countably many rational numbers. – user1729 May 09 '14 at 09:17
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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