L'extraction de la racine \(n\)-ième et l'invention des fractions décimales (XI\(^e\) -- XII\(^e\) siècles)
type: Item
instance of: scholarly article
title: L'extraction de la racine \(n\)-ième et l'invention des fractions décimales (XI\(^e\) -- XII\(^e\) siècles)
zbMATH Open document ID: 0389.01003
author: Roshdi Rashed
published in: Archive for History of Exact Sciences
publication date: 1978-01-01T00:00:00Z
review text: The Sumerians discovered the (sexagesimal) positional system and developed it perfectly. The decimal positional system is much inferior to the sexagesimal system and it is more a concession to the arithmetically lower level of the greater and greater becoming classes which had to compute that the decimal system arose, regrettably. Astronomers never forgot the excellent qualities of the basis 60 and continued to use the sexagesimal system till modern times. We should not forget that the ``invention'' of the decimal system is ``a step backwards''. Simon Stevin was a propagandist of the decimal system, never called himself the inventor of this system. Indeed many people as Rudolff, Apianus, Bonfils have been indicated as using the decimal system before Stevin. Al-Kâshî explains both the sexagesimal and the decimal system in his ``Key of Arithmetics'' and the present author emphasizes that not al-Kâshî (\(\dagger\) 1437) or Bonfils (1335) following A. Saidan but al-Uqlîdisi in the Xth century is the first who is known to write on the decimal system. The author considers the works of the school of al-Karaji and in particular of as-Samaw'al (1172) for the application of the decimal system in computing \(n\)-th roots. Also al-Tûsi's work on more general equations is elucidated. The author discusses in detail the extraction of a fifth root by as-Samaw'al and sees -- as is en vogue during some decades -- ``Ruffini-Horner''-methods, and a ``Horner-triangle''. Horner's paper (Phil. Trans. 1819) is mirabile dictu, concerned with the remark that one should never use more than two lines of the algorithm named nowadays after him, that one should not compute one more digit at each step, that one should not use more digits than necessary, i.e. for an equation of degree\(n\) never more than \(1/n\)-th of the digits wished in the result. Al-Kâshî and as-Samaw'al are therefore doing just what Horner combats: computing too much and too many numbers without importance. Indeed when as-Sam…
Mathematics Subject Classification ID: 01A30
zbMATH DE Number: 3604853
zbMATH Keywords: root extraction
MaRDI profile type: Publication
Recommended article: The Invention of the Decimal Fractions and the Application of the Exponential Calculus by Immanuel Bonfils of Tarascon (c. 1350)
description: scientific article; zbMATH DE number 3604853