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square root of 2
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(Definition)
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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 $$1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \ldots}}}},$$ 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 $$\sqrt{2} = \prod_{i=0}^\infty\frac{(4i+2)^2}{(4i+1)(4i+3)},$$ another is $$\sqrt{2} = \sum_{i=0}^\infty (-1)^{i+1} \frac{(2i-3)!!}{(2i)!!}.$$
- 1
- Flannery, David. The square root of 2 : a dialogue concerning a number and a sequence. New York: Copernicus, 2006.
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"square root of 2" is owned by MathNerd.
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See Also: surd
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Pythagoras' constant |
This object's parent.
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Cross-references: infinite product, number, simple continued fraction, OEIS, sequence, decimal expansion, irrational number
There are 7 references to this entry.
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 1862 times total.
Classification:
| AMS MSC: | 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas) |
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Pending Errata and Addenda
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