query: Why is $e$ irrational?
positive_passages.docid: 148035
positive_passages.text: Is it possible for a sequence of rational numbers to have a non rational limit?
negative_passages.docid: 34825
negative_passages.text: Intuitive explanation for how could there be "more" irrational numbers than rational?
negative_passages.docid: 19076
negative_passages.text: If a.b.c,d are rationals and x is irrational number, then prove that $\frac{ax+b}{cx+d}$ is usually a irrational number. When do exceptions occur?
negative_passages.docid: 77129
negative_passages.text: Density of irrationals
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: 48035
negative_passages.text: Prove that square root of 2 is irrational using the principle of Mathematical Induction
negative_passages.docid: 126199
negative_passages.text: Proof of $\pi$ not being a quadratic irrational number.
negative_passages.docid: 2786
negative_passages.text: The staircase paradox, or why $\pi\ne4$
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: 37642
negative_passages.text: Why is $\pi$ irrational if it is represented as $C/D$?
negative_passages.docid: 141424
negative_passages.text: $\log_7 n$ is either an integer or an irrational number
negative_passages.docid: 125267
negative_passages.text: Prove that $2^{1/2}$ is irrational
negative_passages.docid: 7801
negative_passages.text: Can you raise a number to an irrational exponent?
negative_passages.docid: 131634
negative_passages.text: Prove the irrationality of $0.235711131719...$
negative_passages.docid: 35277
negative_passages.text: Why is $\mathbb{E}[X] = 1 + \sum^\infty_{k=1}\mathbb{P}(X > k)$ true?
negative_passages.docid: 63960
negative_passages.text: proving "e" irrational using convergent series
negative_passages.docid: 64885
negative_passages.text: Irrational and rational sequence proof