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[parent] square root of 2 (Definition)

The square root of 2 is an irrational number, the first to have been proved irrational. Its decimal expansion begins 1.41421356237309504880168872420969807856... (sequence A002194 in Sloane's OEIS) Its simple continued fraction is

$\displaystyle 1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \ldots}}}},$
periodically repeating the 2. Some call this number Pythagoras' constant.

There are several different ways to express $ \sqrt{2}$ as an infinite product. One way is

$\displaystyle \sqrt{2} = \prod_{i=0}^\infty\frac{(4i+2)^2}{(4i+1)(4i+3)},$
another is
$\displaystyle \sqrt{2} = \sum_{i=0}^\infty (-1)^{i+1} \frac{(2i-3)!!}{(2i)!!}.$

Bibliography

1
Flannery, David. The square root of 2 : a dialogue concerning a number and a sequence. New York: Copernicus, 2006.



"square root of 2" is owned by MathNerd.
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See Also: surd

Other names:  Pythagoras' constant

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Attachments:
proof that $\sqrt{2}$ is irrational (Proof) by Wkbj79
alternative proof that $\sqrt{2}$ is irrational (Proof) by Wkbj79
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Cross-references: product, infinite, number, simple continued fraction, OEIS, sequence, decimal expansion, irrational number
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This is version 7 of square root of 2, born on 2007-08-18, modified 2007-09-06.
Object id is 9874, canonical name is SquareRootOf2.
Accessed 976 times total.

Classification:
AMS MSC: 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
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