query: Type of irrational number in $R$
positive_passages.docid: 147534
positive_passages.text: The set of irrational numbers is not a $F_{\sigma}$ set.
negative_passages.docid: 142000
negative_passages.text: Prove that there is an irrational number and a rational number between any two distinct real numbers
negative_passages.docid: 56464
negative_passages.text: Proving the irrationality of $e^n$
negative_passages.docid: 100319
negative_passages.text: Functional equation with strange property about irrational numbers
negative_passages.docid: 1877
negative_passages.text: Can an irrational number raised to an irrational power be rational?
negative_passages.docid: 138977
negative_passages.text: Infinitely many rational numbers between two different real numbers
negative_passages.docid: 34825
negative_passages.text: Intuitive explanation for how could there be "more" irrational numbers than rational?
negative_passages.docid: 141424
negative_passages.text: $\log_7 n$ is either an integer or an irrational number
negative_passages.docid: 40009
negative_passages.text: Show that $e^{\sqrt 2}$ is irrational
negative_passages.docid: 67605
negative_passages.text: Different proof that $\sqrt{2}$ is irrational
negative_passages.docid: 96204
negative_passages.text: Irrational number $\sqrt2$ proof
negative_passages.docid: 31744
negative_passages.text: Proving that there is an irrational number between any two unequal rational numbers.
negative_passages.docid: 4381
negative_passages.text: Proof that the set of irrational numbers is dense in reals
negative_passages.docid: 19676
negative_passages.text: Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational?
negative_passages.docid: 4946
negative_passages.text: Non-existence of irrational numbers?
negative_passages.docid: 30622
negative_passages.text: Proving $\sqrt 3$ is irrational.
negative_passages.docid: 92226
negative_passages.text: Prove $\sqrt6$ is irrational