query: how $\pi$ is irrational if it is a ratio
positive_passages.docid: 37642
positive_passages.text: Why is $\pi$ irrational if it is represented as $C/D$?
negative_passages.docid: 36546
negative_passages.text: For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$
negative_passages.docid: 126199
negative_passages.text: Proof of $\pi$ not being a quadratic irrational number.
negative_passages.docid: 81150
negative_passages.text: Positive integer multiples of an irrational mod 1 are dense
negative_passages.docid: 77129
negative_passages.text: Density of irrationals
negative_passages.docid: 20315
negative_passages.text: $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational
negative_passages.docid: 140949
negative_passages.text: Proving that for each prime number $p$, the number $\sqrt{p}$ is irrational
negative_passages.docid: 48975
negative_passages.text: Transcendental numbers and irrationality measure
negative_passages.docid: 59391
negative_passages.text: $p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.
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: 22408
negative_passages.text: Jutifying: irrational + rational = irrational
negative_passages.docid: 48035
negative_passages.text: Prove that square root of 2 is irrational using the principle of Mathematical Induction
negative_passages.docid: 131634
negative_passages.text: Prove the irrationality of $0.235711131719...$
negative_passages.docid: 41191
negative_passages.text: How to prove this $\pi$ formula?
negative_passages.docid: 34710
negative_passages.text: Cardinality of Irrational Numbers
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: 125267
negative_passages.text: Prove that $2^{1/2}$ is irrational