By V. S. Varadarajan
This article bargains a different account of Indian paintings in diophantine equations throughout 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 ancient improvement of algebra and the speculation of equations from precedent days to the start of contemporary algebra, outlining a few sleek topics resembling the basic theorem of algebra, Clifford algebras, and quarternions. it really is aimed at undergraduates who've no heritage in calculus.
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Extra resources for Algebra in Ancient and Modern Times
For mor e detail s se e th e accoun t give n i n th e translatio n b y H . T . Colebrook e o f th e work s o f Brahmagupta an d Bhaskar a : Algebra, with arithmetic and mensuration, from the Sanscrit of Brahmagupta and Bhascara, London, 1 81 7. 1090/mawrld/012/04 4 IRRATIONAL NUMBER S : CONSTRUCTIO N AND APPROXIMATIO N The Greek s alread y kne w tha t no t al l segment s tha t aris e i n geometrica l con structions hav e length s whic h ar e rationa l fractions . Thu s y/2 i s no t rational , al though i t i s the lengt h o f th e diagona l o f a squar e o f sid e 1 .
Prov e tha t i f 771 and n hav e n o common factor , the n n and 77 1 canno t hav e a common factor . Deduc e tha t i f n > 1 and 77 1 and Tl hav e n o commo n factor, 7 7 cannot divid e 77 1 . 16 V. S . VARADARAJA N 10. Sho w tha t i f th e positiv e intege r k i s no t th e squar e o f anothe r positiv e integer , i t canno t be th e squar e o f a rationa l numbe r either , s o tha t Vk i s a n irrationa l number . (Hin t : If k = 77 1 jn wher e 771 , 77 are positiv e integer s withou t a commo n factor , deduc e fro m kn = 77 1 tha t 7 7 divides 77 1 an d us e th e previou s exercis e t o sho w tha t 7 7 must b e 1 .
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 . W e ca n als o d o thi s i n th e manne r o f Bhaskara , wit h th e Xi chose n differentl y a s explained . NOTES A N D EXERCISE S 1. Trea t th e cas e N = 6 1 b y th e cakravala without usin g shor t cuts , proceedin g a s i n th e example above , an d sho w that on e gets the solutio n X = 1 76631 9049 , Y in either method .