Take the 2-minute tour ×  The golden ratio
$\phi=\frac{1+\sqrt5}2$ is sometimes said to be one of the most
difficult numbers to approximate with rational numbers, because
its continued fraction development $$\phi = 1 + \cfrac{1}{1 +
\cfrac{1}{1 + \cfrac{1}{1 + \cdots}}}$$has all entries equal to
$1$, so that its convergents are as far from $\phi$ as possible.
This can be quantified and made precise (I guess: I'd love to
know the precisely how!) It shares this property with all the
numbers $\frac{a+b\phi}{c+d\phi}$ with $\left(\begin{smallmatrix
}a&b\\\c&d\end{smallmatrix}\right)\in\mathrm{SL}(2,\mathbb Z)$
(and no others?)  Are there other numbers which are
characterized by such extremal properties? Is there a second
worst aproximable number?  share|improve this question
en.wikipedia.org/wiki/… –  Qiaochu Yuan Apr 26 '10 at 2:32
Qiaochu, that's precisely the link I gave! :) –  Mariano Suárez-
Alvarez Apr 26 '10 at 2:42 Oh. Well, in that case, doesn't the
theorem stated count as a quantification of the statement