query: Can sum of rationals be irrational?
positive_passages.docid: 22762
positive_passages.text: How can adding an infinite number of rationals yield an irrational number?
negative_passages.docid: 56464
negative_passages.text: Proving the irrationality of $e^n$
negative_passages.docid: 68456
negative_passages.text: Prove that there exists irrational numbers $x$ and $y$ such that $x + y$ is rational, without using subtraction
negative_passages.docid: 141273
negative_passages.text: When is $\sin(x)$ rational?
negative_passages.docid: 143054
negative_passages.text: Can you raise $\pi$ to a real power to make it rational?
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: 7801
negative_passages.text: Can you raise a number to an irrational exponent?
negative_passages.docid: 95787
negative_passages.text: Prove or find a counterexample: For all real numbers x and y it holds that x + y is irrational if, and only if, both x and y are irrational.
negative_passages.docid: 138162
negative_passages.text: Do Irrationals form a group?
negative_passages.docid: 152512
negative_passages.text: What is the ratio of rational to irrational real numbers?
negative_passages.docid: 42444
negative_passages.text: Irrationality of $\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$
negative_passages.docid: 3790
negative_passages.text: Rational + irrational = always irrational?
negative_passages.docid: 24706
negative_passages.text: Is there a rational number between any two irrationals?
negative_passages.docid: 10882
negative_passages.text: irrationality of $\sqrt{2}^{\sqrt{2}}$.
negative_passages.docid: 70315
negative_passages.text: Process to show that $\sqrt 2+\sqrt[3] 3$ is irrational
negative_passages.docid: 21237
negative_passages.text: $\sqrt{6}-\sqrt{2}-\sqrt{3}$ is irrational
negative_passages.docid: 158368
negative_passages.text: Irrationals: A Group?