query: Why does $\frac{p+2q}{p+q}$ approximate $\sqrt 2$?
positive_passages.docid: 148163
positive_passages.text: Show that if m/n is a good approximation of $\sqrt{2}$ then $(m+2n)/(m+n)$ is better
negative_passages.docid: 99146
negative_passages.text: Find $\sqrt a + \sqrt b + \sqrt c$ only in terms of $p$ .
negative_passages.docid: 150981
negative_passages.text: Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit?
negative_passages.docid: 62758
negative_passages.text: How find this value $\frac{a^3}{(a-p)(a-q)}+\frac{b^3}{(b-p)(b-q)}+\frac{c^3}{(c-p)(c-q)}$
negative_passages.docid: 147307
negative_passages.text: How to evaluate $\sum_{\gcd (p,q)=1} \frac{1}{p^2q^2}$?
negative_passages.docid: 113334
negative_passages.text: If $x+\frac{1}{x}=\frac{1+\sqrt{5}}{2}$ then $x^{2000}+\frac{1}{x^{2000}}= $?
negative_passages.docid: 39504
negative_passages.text: Why does $ \frac{2x}{2+x}$ provide a particularly tight lower bound for $\ln(1+x)$ for small positive values of $x$?
negative_passages.docid: 139674
negative_passages.text: Prove $\ln\frac{p}{q}\leq \frac{p-q}{\sqrt{pq}}$ for $0<q\leq p$
negative_passages.docid: 121655
negative_passages.text: What is the importance of the Diophantine inequality $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$?
negative_passages.docid: 91101
negative_passages.text: Proving $\sqrt2$ is irrational
negative_passages.docid: 126484
negative_passages.text: Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$?
negative_passages.docid: 53958
negative_passages.text: General misconception about $\sqrt x$
negative_passages.docid: 34430
negative_passages.text: Why does $(\frac{p-1}{2}!)^2 = (-1)^{\frac{p+1}{2}}$ mod $p$?
negative_passages.docid: 54539
negative_passages.text: Quadratic substitution question: applying substitution $p=x+\frac1x$ to $2x^4+x^3-6x^2+x+2=0$
negative_passages.docid: 136403
negative_passages.text: Prove that $\sqrt{a}\leq\frac{1+a}{2}$
negative_passages.docid: 50930
negative_passages.text: Proving the continued fraction representation of $\sqrt{2}$
negative_passages.docid: 148876
negative_passages.text: What is the meaning of $\mathbf{Q}(\sqrt{2},\sqrt{3})$