query: prove a number is irrational
positive_passages.docid: 1877
positive_passages.text: Can an irrational number raised to an irrational power be rational?
negative_passages.docid: 124641
negative_passages.text: Prove that $5^{1/3}+7^{1/2}$ is irrational
negative_passages.docid: 67605
negative_passages.text: Different proof that $\sqrt{2}$ is irrational
negative_passages.docid: 118374
negative_passages.text: Is there a proof that $\pi$ is an irrational number?
negative_passages.docid: 30411
negative_passages.text: The closure of the irrational numbers on $\mathbb{R}$
negative_passages.docid: 55077
negative_passages.text: prove that $2\sqrt5 +\sqrt{11}$ is irrational
negative_passages.docid: 30523
negative_passages.text: A binary irrational with bits defined by primes
negative_passages.docid: 10882
negative_passages.text: irrationality of $\sqrt{2}^{\sqrt{2}}$.
negative_passages.docid: 48035
negative_passages.text: Prove that square root of 2 is irrational using the principle of Mathematical Induction
negative_passages.docid: 53945
negative_passages.text: The square root of a prime number is irrational
negative_passages.docid: 101028
negative_passages.text: Irrational numbers and Borel Sets
negative_passages.docid: 158730
negative_passages.text: Is $\ln(\ln(n))$ irrational for any integer $n>1$?
negative_passages.docid: 19676
negative_passages.text: Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational?
negative_passages.docid: 69264
negative_passages.text: Is there an irrational number $a$ such that $a^a$ is rational?
negative_passages.docid: 139755
negative_passages.text: Prove that $\sqrt{2}+\sqrt{3}+\sqrt{5}$ is irrational. Generalise this.
negative_passages.docid: 127830
negative_passages.text: proof that $\sqrt(2)$ is irrational / why is the fraction irreducible
negative_passages.docid: 143733
negative_passages.text: Is this proof that $\sqrt 2$ is irrational correct?