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Wrapping my head around the mathematical definition of infinity and just curious here: Are there more real numbers than irrational numbers? It would intuitively seem so, but they are both just uncountable infinite sets, right? Some I guess there would be the same number of irrational numbers as real numbers? But I can't imagine a "bijection" between them. (A term I just learned about five minutes ago...)


marked as duplicate by José Carlos Santos, Dietrich Burde number-theory Nov 12 '18 at 19:03

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  • $\begingroup$ The set of all irrational numbers is a subset of all real numbers. So the set of real numbers has all irrational numbers plus some other numbers (i.e. 1, 0.25, etc). But if you are talking about cardinality, both sets have the same. $\endgroup$ – Vasya Nov 12 '18 at 18:58
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    $\begingroup$ I'm not sure this is a duplicate. It seems to be asking for an example of a bijection between $\mathbb R \setminus\mathbb Q$ and $\mathbb R$, not why $|\mathbb R\setminus\mathbb Q| > |\mathbb Q|$ $\endgroup$ – eyeballfrog Nov 12 '18 at 19:09
  • $\begingroup$ Here is an answer that gives a bijection between $\mathbb R \setminus \mathbb Q$ and $\mathbb R$. $\endgroup$ – Daniel Mroz Nov 12 '18 at 19:13
  • $\begingroup$ This isn't a duplicate question. I'm not comparing rational numbers to irrational numbers, I'm comparing irrational numbers to all of the real numbers. The set of real numbers contain the irrational numbers, but both are uncountable infinite sets. So my question is, does that mean that those infinities are equal? $\endgroup$ – John Nov 12 '18 at 20:46