query: Showing irrationality of a sum/field
positive_passages.docid: 20315
positive_passages.text: $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational
negative_passages.docid: 125876
negative_passages.text: Sum and Product of Infinite Radicals
negative_passages.docid: 158368
negative_passages.text: Irrationals: A Group?
negative_passages.docid: 125267
negative_passages.text: Prove that $2^{1/2}$ is irrational
negative_passages.docid: 141424
negative_passages.text: $\log_7 n$ is either an integer or an irrational number
negative_passages.docid: 62884
negative_passages.text: Prove the series $ \sum_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number
negative_passages.docid: 63960
negative_passages.text: proving "e" irrational using convergent series
negative_passages.docid: 129433
negative_passages.text: $\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$ is Irrational
negative_passages.docid: 65895
negative_passages.text: Irrationality proofs not by contradiction
negative_passages.docid: 148310
negative_passages.text: Showing rational numbers are algebraic
negative_passages.docid: 127830
negative_passages.text: proof that $\sqrt(2)$ is irrational / why is the fraction irreducible
negative_passages.docid: 132115
negative_passages.text: Verify my proof of the irrationality of $\sqrt{6}$.
negative_passages.docid: 22762
negative_passages.text: How can adding an infinite number of rationals yield an irrational number?
negative_passages.docid: 140949
negative_passages.text: Proving that for each prime number $p$, the number $\sqrt{p}$ is irrational
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: 118374
negative_passages.text: Is there a proof that $\pi$ is an irrational number?
negative_passages.docid: 150088
negative_passages.text: Prove that $\sqrt {n − 1} +\sqrt {n + 1}$ is irrational for every integer $n ≥ 1$