query: The Irrationality of 2
positive_passages.docid: 6275
positive_passages.text: How can you prove that the square root of two is irrational?
negative_passages.docid: 95787
negative_passages.text: Prove or find a counterexample: For all real numbers x and y it holds that x + y is irrational if, and only if, both x and y are irrational.
negative_passages.docid: 37642
negative_passages.text: Why is $\pi$ irrational if it is represented as $C/D$?
negative_passages.docid: 39021
negative_passages.text: Proving that $\sqrt{2}+\sqrt{3}$ is irrational
negative_passages.docid: 7801
negative_passages.text: Can you raise a number to an irrational exponent?
negative_passages.docid: 32
negative_passages.text: Is $n^{th}$ root of $2$ an irrational number?
negative_passages.docid: 11069
negative_passages.text: Deciding whether $2^{\sqrt2}$ is irrational/transcendental
negative_passages.docid: 43985
negative_passages.text: Why must $a$ and $b$ both be coprime when proving that the square root of two is irrational?
negative_passages.docid: 53945
negative_passages.text: The square root of a prime number is irrational
negative_passages.docid: 1877
negative_passages.text: Can an irrational number raised to an irrational power be rational?
negative_passages.docid: 95702
negative_passages.text: Remarkable/unexpected rational numbers
negative_passages.docid: 34710
negative_passages.text: Cardinality of Irrational Numbers
negative_passages.docid: 77129
negative_passages.text: Density of irrationals
negative_passages.docid: 67605
negative_passages.text: Different proof that $\sqrt{2}$ is irrational
negative_passages.docid: 21237
negative_passages.text: $\sqrt{6}-\sqrt{2}-\sqrt{3}$ is irrational
negative_passages.docid: 19676
negative_passages.text: Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational?
negative_passages.docid: 34825
negative_passages.text: Intuitive explanation for how could there be "more" irrational numbers than rational?