Views of pi: definition and computation

Yves Bertot, Guillaume Allais

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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.
Original languageEnglish
Pages (from-to)105-129
Number of pages25
JournalJournal of Formalized Reasoning
Issue number1
Publication statusPublished - 2014


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

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