Guy Wallet - Integer calculus on the Harthong-Reeb Line

arima:1911 - Revue Africaine de Recherche en Informatique et Mathématiques Appliquées, August 8, 2008, Volume 9, 2007 Conference in Honor of Claude Lobry, 2008 -
Integer calculus on the Harthong-Reeb LineArticle

Authors: Guy Wallet 1

In this work, we give a presentation of the so-called Harthong-Reeb line. Only based on integer numbers, this numerical system has the striking property to be roughly equivalent to the continuous real line. Its definition requires the use of a natural number w which is infinitely large in the meaning of nonstandard analysis. Following the idea of G. Reeb, we show how to implement in this framework the Euler scheme. Then we get an exact representation in the Harthong-Reeb line of many real functions like the exponential. Since this representation is given with the help of an explicit algorithm, it is natural to wonder about the global constructivity of this numerical system. In the conclusion, we discuss this last point and we outline some new directions for getting analogous systems which would be more constructive

Volume: Volume 9, 2007 Conference in Honor of Claude Lobry, 2008
Published on: August 8, 2008
Submitted on: January 22, 2008
Keywords: nonstandard analysis, arithmetization, Euler scheme, discrete line, constructive mathematics.,analyse nonstandard,arithmétisation,schéma d’Euler,droite discrète,mathématiques constructives,[INFO] Computer Science [cs],[MATH] Mathematics [math]

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