query: Proof any periodic number is rational.
positive_passages.docid: 46561
positive_passages.text: How can I prove that all rational numbers are either terminating decimal or repeating decimal numerals?
negative_passages.docid: 32848
negative_passages.text: Prove that the product of a non-zero rational and irrational number 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: 104548
negative_passages.text: Geometric proof for irrationality of $\pi$
negative_passages.docid: 150088
negative_passages.text: Prove that $\sqrt {n − 1} +\sqrt {n + 1}$ is irrational for every integer $n ≥ 1$
negative_passages.docid: 153652
negative_passages.text: Countable ordinals are embeddable in the rationals $\Bbb Q$ -- proofs and their use of AC
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: 99656
negative_passages.text: Prove that the set of all algebraic numbers is countable
negative_passages.docid: 65895
negative_passages.text: Irrationality proofs not by contradiction
negative_passages.docid: 96204
negative_passages.text: Irrational number $\sqrt2$ proof
negative_passages.docid: 145397
negative_passages.text: How to prove that for every $i$, $a_i = i$
negative_passages.docid: 118374
negative_passages.text: Is there a proof that $\pi$ is an irrational number?
negative_passages.docid: 43268
negative_passages.text: How to prove that the set of rational numbers are countable?
negative_passages.docid: 120906
negative_passages.text: Is there a proof that $\pi \times e$ is irrational?
negative_passages.docid: 108394
negative_passages.text: Proof of that every interval is connected
negative_passages.docid: 150037
negative_passages.text: Finding the simplest rational in a closed interval
negative_passages.docid: 91101
negative_passages.text: Proving $\sqrt2$ is irrational