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

By Roshdi Rashed

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 spread out fields of study that proved fertile up until eventually the 17th century, and have been quickly to turn into either a reference and a version. Their impression was once decisive at the improvement of algebra in Arabic not less than in Latin and Hebrew. there'll be present in the current book the 1st conscientiously serious variation of Ab? K?mil s works, in addition to the 1st ever translation right into a smooth language. textual content and translation are preceded by way of an exhaustive observation, without delay mathematical and historical."

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3 3x + 4y/x^ - 3 x = 2 0 . On multiplie tout par - et on divise par ^Jx^-3x =-^(20-3x). 3x = - . 2 On met tout au carré : AlorsX 2 : 70 Chapitre II Autre méthode (due à un certain Abu Yûsuf) ; On pose )>-£■. Alors Le calcul algébrique et ses applications /1 Abu Kâmil ne détaille pas la résolution de cette équation x^=î 6. |y-3Vÿ = )’ Notons que l’équation initiale admet aussi les racines x = 0 et x = 3 qui annulent le radical. VV = I - 23 - On considère l’équation 3x + 4^/x^ - 3x = x^ +4. - 21 - On considère l’équation __ 2 /2x ' =x -7x '-2 3-3 j~ 3 On pose yr - 3x.

T 4 d’après la première équation, et la seconde donne : V ? T 4 ? = 4. 10 Voir plus loin p. 486. )-"(2^/2Ü + 2 V 2 Ô ^ ) = 400 - 80y + 4y^ (4 + V 8 Ü y =400 - 80y. On multiplie tout par + — —1 qui est l’inverse de 4 + V M . V8-8 4-8 2'SJ Chapitre II 104 Le calcul algébrique et ses applications / + (VÏ25 - 5 ))-= VÎÎ25 - 25 105 Autre méthode : À partir de (**), on peut aussi procéder comme suit : 5 - x = (x--V5 + xj . On développe, on isole le radical et on met chaque membre au carré ; - 6 7 - O n considère le système d’équations (2-j5x^~+ + V= 10 [u + 2Vm = V- 2-Vv j =(x ^+ 2x) ^ On développe et on obtient ; 16x^ = x'^ Abû Kâmil pose M= 5 - A:etv = 5 + ^ .

Abû Kâmil, plutôt que de développer le carré (10 - xŸ, introduit un procédé d’addition membre à membre que l’on représentera désormais sous la forme : - 28 - On considère l’équation ^^^((10-jc)-jc) = 24. Abu Kâmil effectue l’addition membre à membre ; ^^"■*-(10- a:) = 9 X 10-A 1 0 -JC JC= 10-JC lO -jc -(10-2x) = 24 X 10-X 10 = 19-jc. 10-A 2x = 20 - 2x 10 = 4 4 - 2 a . Finalement 10-JC , 9 1 ------ = 1 + --------- JC. JC 10 10 Abu Kâmil renvoie, pour la suite, au problème 24, équation (*). JC= 4.

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