### Abstract

Language | English |
---|---|

Pages | 105-129 |

Number of pages | 25 |

Journal | Journal of Formalized Reasoning |

Volume | 7 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2014 |

### Fingerprint

### Keywords

- Archimedes
- arctangent
- Coq
- Gregory's formula
- pi

### Cite this

*Journal of Formalized Reasoning*,

*7*(1), 105-129. https://doi.org/10.6092/issn.1972-5787/4343

}

*Journal of Formalized Reasoning*, vol. 7, no. 1, pp. 105-129. https://doi.org/10.6092/issn.1972-5787/4343

**Views of pi : definition and computation.** / Bertot, Yves; Allais, Guillaume.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Views of pi

T2 - Journal of Formalized Reasoning

AU - Bertot, Yves

AU - Allais, Guillaume

PY - 2014

Y1 - 2014

N2 - We study several formal proofs and algorithms related to the number pi in the context of Coq's standard library. In particular, we clarify the relation between roots of the cosine function and the limit of the alternated series whose terms are the inverse of odd natural numbers (known as Leibnitz' formula).We give a formal description of the arctangent function and its expansion as a power series. We then study other possible descriptions of pi, first as the surface of the unit disk, second as the limit of perimeters of regular polygons with an increasing number of sides.In a third section, we concentrate on techniques to effectively compute approximations of pi in the proof assistant by relying on rational numbers and decimal representations.

AB - We study several formal proofs and algorithms related to the number pi in the context of Coq's standard library. In particular, we clarify the relation between roots of the cosine function and the limit of the alternated series whose terms are the inverse of odd natural numbers (known as Leibnitz' formula).We give a formal description of the arctangent function and its expansion as a power series. We then study other possible descriptions of pi, first as the surface of the unit disk, second as the limit of perimeters of regular polygons with an increasing number of sides.In a third section, we concentrate on techniques to effectively compute approximations of pi in the proof assistant by relying on rational numbers and decimal representations.

KW - Archimedes

KW - arctangent

KW - Coq

KW - Gregory's formula

KW - pi

UR - https://jfr.unibo.it/index

U2 - 10.6092/issn.1972-5787/4343

DO - 10.6092/issn.1972-5787/4343

M3 - Article

VL - 7

SP - 105

EP - 129

JO - Journal of Formalized Reasoning

JF - Journal of Formalized Reasoning

SN - 1972-5787

IS - 1

ER -