Download e-book for kindle: Abu Kamil Algèbre et analyse diophantienne. Édition, by Roshdi Rashed

By Roshdi Rashed

ISBN-10: 3110295679

ISBN-13: 9783110295672

The mathematical works of Ab? K?mil (floruit circa 880) have been produced generations after the works of Al-Khwarizm?, the founding father of algebra. They unfolded fields of analysis that proved fertile up until eventually the 17th century, and have been quickly to develop into either a reference and a version. Their impression was once decisive at the improvement of algebra in Arabic at least in Latin and Hebrew. there'll be present in the current booklet the 1st conscientiously severe variation of Ab? K?mil s works, in addition to the 1st ever translation right into a sleek language. textual content and translation are preceded via an exhaustive remark, instantaneously mathematical and historical."

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Additional resources for Abu Kamil Algèbre et analyse diophantienne. Édition, traduction et commentaire

Sample text

On notera qu’Abû Kâmil n’hésite pas à considérer la somme d’une aire et d’un segment ou bien à multiplier une aire par une autre. A notre connaissance, ce chapitre, à la différence de tous les autres, ne sera ni repris, ni développé, par les algébristes arithméticiens tels qu’alKaraji et ses successeurs. L’algèbre géométrique prendra son essor à partir de l’étude des problèmes solides. En revanche, on rencontre dans les traités de hisàb un chapitre sur les polygones réguliers et la détermination de leurs aires.

2 i 4 ^2 1 4 I V ZJ ou encore, après multiplication par z^ '■ Comme les deux équations sont homogènes, la solution générale est 10 000 = 100 Z* + Z*. z L’unique solution positive est Remarque : Dans ce problème, l’énoncé parle de trois mal que l’on aurait pu traduire par « fortune » pour ne pas confondre avec le mal carré algébrique qui apparaît au cours de la résolution du problème, une fois spécifiée l’inconnue choisie par Abü Kâmil (l’inconnue z ; mal désigne alors z} tandis que chacune des trois inconnues du problème dans son énoncé initial est elle-même un mal).

Si on met au carré chaque membre, on a yVô + V26. - 40 - On considère l’équation 3x^ = 100 + 4x^ - 40x. Mais cette mise au carré introduit des solutions superflues. Pour les reconnaître ultérieurement, retenons que l’on doit aussi avoir 10 - 2;r > 0. x46 •x 4 s + IOa:^ + 20 = jc\ Changement de variable y = x^. ^|^ x^ +100 = 40x x =2o± ^^m . Abü Kâmil ne donne que la racine 20 - V300 qui est bien la solution du problème (en effet l’autre racine de l’équation quadratique ne vérifie pas la condition 10 - 2 a: > 0).

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Abu Kamil Algèbre et analyse diophantienne. Édition, traduction et commentaire by Roshdi Rashed


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