query: Show that there is a rational number and an irrational number between any two real numbers.??
positive_passages.docid: 142000
positive_passages.text: Prove that there is an irrational number and a rational number between any two distinct real numbers
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: 28347
negative_passages.text: Prove that $\sqrt 2 + \sqrt 3$ is irrational
negative_passages.docid: 19676
negative_passages.text: Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational?
negative_passages.docid: 95702
negative_passages.text: Remarkable/unexpected rational numbers
negative_passages.docid: 98666
negative_passages.text: Continuity defined for rational and irrational numbers
negative_passages.docid: 158066
negative_passages.text: Prove $2^{1/3}$ is irrational.
negative_passages.docid: 152512
negative_passages.text: What is the ratio of rational to irrational real numbers?
negative_passages.docid: 32
negative_passages.text: Is $n^{th}$ root of $2$ an irrational number?
negative_passages.docid: 96204
negative_passages.text: Irrational number $\sqrt2$ proof
negative_passages.docid: 114043
negative_passages.text: Prove the existence of a real number satisfying a property
negative_passages.docid: 148310
negative_passages.text: Showing rational numbers are algebraic
negative_passages.docid: 96630
negative_passages.text: Let a and b be real numbers such that 0  \sqrt{ab} > \frac{2ab}{a + b}$
negative_passages.docid: 4946
negative_passages.text: Non-existence of irrational numbers?
negative_passages.docid: 3790
negative_passages.text: Rational + irrational = always irrational?
negative_passages.docid: 118374
negative_passages.text: Is there a proof that $\pi$ is an irrational number?
negative_passages.docid: 127830
negative_passages.text: proof that $\sqrt(2)$ is irrational / why is the fraction irreducible