square root of 2SquareRootOf22007-08-18 15:02:472007-09-06 16:30:12Definitionsquare root% this is the default PlanetMath preamble. as your knowledge
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The \emph{square root of 2} is an irrational number, the first to have been proved irrational. Its decimal expansion begins 1.41421356237309504880168872420969807856... (sequence \PMlinkexternal{A002194}{http://www.research.att.com/~njas/sequences/A002193} in Sloane's OEIS) Its simple continued fraction is $$1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \ldots}}}},$$ periodically repeating the 2. Some call this number \emph{Pythagoras' constant}.
There are several different ways to express $\sqrt{2}$ as an infinite product. One way is $$\sqrt{2} = \prod_{i=0}^\infty\frac{(4i+2)^2}{(4i+1)(4i+3)},$$ another is $$\sqrt{2} = \sum_{i=0}^\infty (-1)^{i+1} \frac{(2i-3)!!}{(2i)!!}.$$
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\bibitem{flannery} Flannery, David. {\it The square root of 2 : a dialogue concerning a number and a sequence}. New York: Copernicus, 2006.
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