query: How can define irrational numbers in general
positive_passages.docid: 151325
positive_passages.text: (How to/Can I) show irrational numbers?
negative_passages.docid: 67031
negative_passages.text: use contradiction to prove that the square root of $p$ is irrational
negative_passages.docid: 131634
negative_passages.text: Prove the irrationality of $0.235711131719...$
negative_passages.docid: 53945
negative_passages.text: The square root of a prime number is irrational
negative_passages.docid: 11069
negative_passages.text: Deciding whether $2^{\sqrt2}$ is irrational/transcendental
negative_passages.docid: 30523
negative_passages.text: A binary irrational with bits defined by primes
negative_passages.docid: 44600
negative_passages.text: How can I prove: if $p $ is prime and $n>1$, then $ p^{\frac1n} $ is irrational?
negative_passages.docid: 86988
negative_passages.text: Cardinality of the Irrationals
negative_passages.docid: 1383
negative_passages.text: How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?
negative_passages.docid: 101162
negative_passages.text: Square root of $2$ is irrational
negative_passages.docid: 100319
negative_passages.text: Functional equation with strange property about irrational numbers
negative_passages.docid: 68976
negative_passages.text: Given real numbers: define integers?
negative_passages.docid: 152172
negative_passages.text: Does there exist a positive irrational number $\alpha $, such that for any positive integer $n$ the number $\lfloor n\alpha \rfloor$ is not a prime?
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: 30105
negative_passages.text: Multiples of an irrational number forming a dense subset
negative_passages.docid: 10124
negative_passages.text: how can i show an irrational number to 100 decimal places in python?
negative_passages.docid: 10882
negative_passages.text: irrationality of $\sqrt{2}^{\sqrt{2}}$.