query: Baire's Theorem and Irrationals
positive_passages.docid: 11710
positive_passages.text: How to show that $\mathbb{Q}$ is not $G_\delta$
negative_passages.docid: 158368
negative_passages.text: Irrationals: A Group?
negative_passages.docid: 65895
negative_passages.text: Irrationality proofs not by contradiction
negative_passages.docid: 31744
negative_passages.text: Proving that there is an irrational number between any two unequal rational numbers.
negative_passages.docid: 5529
negative_passages.text: Proving that $x$ is irrational if $x-\lfloor x \rfloor + \frac1x - \left\lfloor \frac1x \right\rfloor = 1$
negative_passages.docid: 57233
negative_passages.text: Proof of continuity of Thomae Function at irrationals.
negative_passages.docid: 111898
negative_passages.text: Is $\{ (e^{2\pi i a n},e^{2\pi i b n }) : n \in \mathbb{Z} \}$ dense in the torus, where $a,b$ irrationals such that $a/b$ irrational?
negative_passages.docid: 30105
negative_passages.text: Multiples of an irrational number forming a dense subset
negative_passages.docid: 158066
negative_passages.text: Prove $2^{1/3}$ is irrational.
negative_passages.docid: 2926
negative_passages.text: The contradiction method used to prove that the square root of a prime is irrational
negative_passages.docid: 101162
negative_passages.text: Square root of $2$ is irrational
negative_passages.docid: 84558
negative_passages.text: Prove that the square root of a positive integer is either an integer or irrational
negative_passages.docid: 125812
negative_passages.text: Is sin(x) necessarily irrational where x is rational?
negative_passages.docid: 50555
negative_passages.text: Approximation of real number to the power of irrational
negative_passages.docid: 6275
negative_passages.text: How can you prove that the square root of two is irrational?
negative_passages.docid: 30622
negative_passages.text: Proving $\sqrt 3$ is irrational.
negative_passages.docid: 32
negative_passages.text: Is $n^{th}$ root of $2$ an irrational number?