By V. S. Varadarajan
This article bargains a different account of Indian paintings in diophantine equations in the course of the sixth via twelfth centuries and Italian paintings on ideas of cubic and biquadratic equations from the eleventh via sixteenth centuries. the amount lines the historic improvement of algebra and the idea of equations from precedent days to the start of contemporary algebra, outlining a few glossy issues corresponding to the basic theorem of algebra, Clifford algebras, and quarternions. it's aimed at undergraduates who've no history in calculus.
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27 ALGEBRA I N ANCIENT AN D MODERN TIME S 1. Find po , #o? ™o s o tha t PQ
Bu t w e shal l no t tak e thi s shor t cu t bu t procee d strictl y a s require d b y the cakravala. The n 0 < x\ < 3 and 3 + X\ i s to b e divisibl e b y 4 giving x\ = 1 . S o Pi=4,g i= l,rai= 3 giving u s (4,1 ;3) . The n 1 < x 2 < 3 an d 4 + x 2 i s t o b e divisibl e b y 3 , s o tha t x2 = 2 . S o P2 = 7 , q 2 = 2 , ra 2 = - 3 giving us (7 , 2; —3). Proceedin g i n thi s manne r w e get i n successio n (11,3; 4), 1( 8,5 ; - 1), 1 1 ( 9,33 (256,71; 3), (393, 1 09 ; 4), 1( 37,38 ; - 4 ), (649, 1 80 ; -3) ; 1) We hav e alread y see n tha t (649,180) is a solutio n i n thi s case .
X^ n,... replac e y/X, &X, ... J/X, .... There wa s anothe r obstacl e face d b y th e mathematician s eve n upt o th e 1 6 th century, namel y the unwillingness t o use negative numbers, eve n though th e Hindu s had successfull y introduce d the m centurie s earlier . Thus , Al~Khwarizm i , with th e convention o f writin g onl y positiv e quantities , ha d t o distinguis h betwee n differen t types o f quadrati c equation s suc h a s X2 = BX + C , X 2 + BX = C , X 2 4 - C = BX Moreover thes e woul d hav e t o b e referre d t o i n descriptiv e for m a s square equal s thin g an d numbe r square an d thin g equa l numbe r square an d numbe r equa l thin g ALGEBRA I N ANCIENT AN D MODER N TIME S 45 Al-Khwarizmi gav e rule s fo r solvin g thes e equations , presente d example s t o illustrate hi s rules , an d essentiall y als o givin g demonstration s o f th e rule s throug h these example s a s wel l a s geometrically .