query: Proving that $\pi$ and $e$ are rational numbers
positive_passages.docid: 22762
positive_passages.text: How can adding an infinite number of rationals yield an irrational number?
negative_passages.docid: 59391
negative_passages.text: $p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.
negative_passages.docid: 10008
negative_passages.text: Proof that the nth root of a rational s/t is irrational unless s and t are perfect nth powers
negative_passages.docid: 129821
negative_passages.text: Prove that the Gaussian rationals is the field of fractions of the Gaussian integers
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: 39943
negative_passages.text: Proving $\mathrm e <3$
negative_passages.docid: 37642
negative_passages.text: Why is $\pi$ irrational if it is represented as $C/D$?
negative_passages.docid: 28347
negative_passages.text: Prove that $\sqrt 2 + \sqrt 3$ is irrational
negative_passages.docid: 34017
negative_passages.text: Show that $n^{\pi\left(2n\right)-\pi\left(n\right)}2$
negative_passages.docid: 118374
negative_passages.text: Is there a proof that $\pi$ is an irrational number?
negative_passages.docid: 99311
negative_passages.text: Prove $2^{1/3} + 2^{2/3}$ is irrational
negative_passages.docid: 21003
negative_passages.text: Comparing $\pi^e$ and $e^\pi$ without calculating them
negative_passages.docid: 44400
negative_passages.text: $e^{2\pi i x} = (e^{2\pi i})^x$: What happens if x is rational?
negative_passages.docid: 96328
negative_passages.text: Proving $e^{i\pi} = -1$ without proving $e^{ix} = \cos x + i\sin x$
negative_passages.docid: 44600
negative_passages.text: How can I prove: if $p $ is prime and $n>1$, then $ p^{\frac1n} $ is irrational?
negative_passages.docid: 63804
negative_passages.text: Proof that every repeating decimal is rational
negative_passages.docid: 41191
negative_passages.text: How to prove this $\pi$ formula?