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'square root of 2'
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| Title of object: |
square root of 2 |
| Canonical Name: |
SquareRootOf2 |
| Type: |
Definition |
| Created on: |
2007-08-18 15:02:47 |
| Modified on: |
2007-08-23 18:12:29 |
| Classification: |
msc:11A25 |
| Synonyms: |
square root of 2=Pythagoras' constant
square root of 2=Pythagoras's constant |
Revision comment (for changes between this and next version):
| Changes for correction #13036 ('several minor issues'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
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Content:
The \emph{square root of 2} is an irrational number, the first to have been proved irrational. Its decimal expansion begins 1.4142... Its simple continued fraction is 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ... Some call it \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 $$\sqrt{2} =
\prod_{i=0}^\infty\left(1+\frac{1}{4i+1}\right)\left(1-\frac{1}{4i+3}\right).$$
\begin{thebibliography}{1}
\bibitem{flannery} Flannery, David. {\it The square root of 2 : a dialogue concerning a number and a sequence} New York: Copernicus, 2006
\end{thebibliography}
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