By V. S. Varadarajan

ISBN-10: 082180989X

ISBN-13: 9780821809891

This article deals a unique account of Indian paintings in diophantine equations throughout the sixth via twelfth centuries and Italian paintings on strategies 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 smooth topics comparable to the basic theorem of algebra, Clifford algebras, and quarternions. it really is aimed toward undergraduates who've no history in calculus.

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Alread y b y rul e I V w e ca n ge t th e solution. 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 .

1 84) . 2. Prov e th e tw o identitie s 2 \Ja±\/b=\j{a+ y/a - b)/2 ± ^J{a - y/a 2 - b)/2 3. Th e basi c approximatio n i n computing squar e root s i s y a 2 -\- b ~ a -\r b/2a. Prov e tha t b9 a ± — > az ± b 2a Use this repeatedl y t o get ^ ~ 1+ 3 + 1 2-40 8 (Hint: Th e firs t applicatio n give s 3/2 > y/2. The n 2 = ( 3 / 2 ) 2 - (1 /4 ) an d s o a secon d application give s 3/2 — 1 /1 2 > v 2 . A third applicatio n give s the require d result. ) A P P R O X I M A T I O N S T OT T Throughout th e histor y o f mathematics, goin g back t o th e mos t ancien t times , the numbe r 7r , which denote s th e rati o o f the circumferenc e o f a circl e to it s diame ter, ha s exerte d a tremendou s fascinatio n an d generate d surprisingl y profoun d an d fertile questions .

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 .

### Algebra in Ancient and Modern Times by V. S. Varadarajan

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