Continued Fraction - History of Continued Fractions

History of Continued Fractions

  • 300 BC Euclid's Elements contains an algorithm for the greatest common divisor which generates a continued fraction as a by-product
  • 499 The Aryabhatiya contains the solution of indeterminate equations using continued fractions
  • 1579 Rafael Bombelli, L'Algebra Opera – method for the extraction of square roots which is related to continued fractions
  • 1613 Pietro Cataldi, Trattato del modo brevissimo di trovar la radice quadra delli numeri – first notation for continued fractions
Cataldi represented a continued fraction as & & & with the dots indicating where the following fractions went.
  • 1695 John Wallis, Opera Mathematica – introduction of the term "continued fraction"
  • 1737 Leonhard Euler, De fractionibus continuis dissertatio – Provided the first then-comprehensive account of the properties of continued fractions, and included the first proof that the number e is irrational.
  • 1748 Euler, Introductio in analysin infinitorum. Vol. I, Chapter 18 – proved the equivalence of a certain form of continued fraction and a generalized infinite series, proved that every rational number can be written as a finite continued fraction, and proved that the continued fraction of an irrational number is infinite.
  • 1761 Johann Lambert – gave the first proof of the irrationality of π using a continued fraction for tan(x).
  • 1768 Joseph Louis Lagrange – provided the general solution to Pell's equation using continued fractions similar to Bombelli's
  • 1770 Lagrange – proved that quadratic irrationals have a periodic continued fraction expansion
  • 1813 Carl Friedrich Gauss, Werke, Vol. 3, pp. 134–138 – derived a very general complex-valued continued fraction via a clever identity involving the hypergeometric function
  • 1892 Henri Padé defined Padé approximant
  • 1972 Bill Gosper – First exact algorithms for continued fraction arithmetic.

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