# square root of 2

The *square root of 2 ^{}* is an irrational number, the first to have been proved irrational. Its decimal expansion begins 1.41421356237309504880168872420969807856… (sequence http://www.research.att.com/ njas/sequences/A002193A002194 in Sloane’s OEIS) Its simple continued fraction

^{}is

$$1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\mathrm{\dots}}}}},$$ |

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

$$\sqrt{2}=\prod _{i=0}^{\mathrm{\infty}}\frac{{(4i+2)}^{2}}{(4i+1)(4i+3)},$$ |

another is

$$\sqrt{2}=\sum _{i=0}^{\mathrm{\infty}}{(-1)}^{i+1}\frac{(2i-3)!!}{(2i)!!}.$$ |

## References

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

Title | square root of 2 |
---|---|

Canonical name | SquareRootOf2 |

Date of creation | 2013-03-22 17:29:12 |

Last modified on | 2013-03-22 17:29:12 |

Owner | MathNerd (17818) |

Last modified by | MathNerd (17818) |

Numerical id | 10 |

Author | MathNerd (17818) |

Entry type | Definition |

Classification | msc 11A25 |

Synonym | Pythagoras’ constant |

Related topic | Surd |