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 options of cubic and biquadratic equations from the eleventh via sixteenth centuries. the quantity lines the ancient improvement of algebra and the speculation of equations from precedent days to the start of contemporary algebra, outlining a few sleek issues equivalent to the elemental theorem of algebra, Clifford algebras, and quarternions. it really is aimed toward undergraduates who've no historical past in calculus.

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U, v; ±1) = » (2u 2 T 1 , 2uv; 1) II. (p,q;±2)=>(p 2 Tl,pq;l) III. (p , q; 4),p eve n == > ( | p 2 - 1 , \pq : l) (p,q;4),p od d = • (ip(p 2 - 3) , icz(p 2 - 1 ) ; l) IV. (p,q; - 4 ) , p eve n =* • ( | ( p 2 + 2) , \pq; l ) ( p , g ; - 4 ) , p o d d =*{P,Q; 1 ) where ^ = ^ ( p 2 + 2){(p 2 + l ) ( p 2 + 3 ) - 2 } , Q = \pq(p2 + l)(p 2 + 3) For I w e note tha t b y formin g th e compositio n o f (u , v) wit h itsel f w e ge t ([/ , V; 1) where U = u 2+Nv29 V = 2uv 22 V. S . ) with itsel f w e get (P , Q; 4) wher e P = p 2 + Nq 2 = 2p 2 - (p 2 - Nq 2) = 2(p 2 = F 1 ) , Q = 2pq Now w e us e th e principl e tha t i f a2 - Nb 2 = m 2 then so that , takin g m — 2, a = P , 6 = Q w e ge t (P2TI,OT;1) For II I th e sam e principl e give s (p/2 , qr/2; 1 ) an d so , b y I , w e have , takin g u = p/2,v = q/2, If p i s even w e stop her e a s the number s abov e ar e positiv e integers .

S . VARADARAJA N 18 What woul d hav e bee n Fermat' s astonishment , i f som e missionary , jus t bac k fro m India, ha d tol d hi m tha t hi s proble m ha s bee n successfull y tackle d ther e b y nativ e mathematicians almos t si x centurie s earlier ! The first definitiv e treatmen t o f Pell' s equatio n wa s give n b y th e grea t Frenc h mathematician Lagrang e (1 736-1 81 3) . Nevertheless on e shoul d no t minimiz e th e India n achievemen t becaus e o f th e absence of any indication that the y were concerned abou t provin g that thei r metho d always le d t o a solution o r tha t i t gav e al l the solutions .

VARADARAJA N 18 What woul d hav e bee n Fermat' s astonishment , i f som e missionary , jus t bac k fro m India, ha d tol d hi m tha t hi s proble m ha s bee n successfull y tackle d ther e b y nativ e mathematicians almos t si x centurie s earlier ! The first definitiv e treatmen t o f Pell' s equatio n wa s give n b y th e grea t Frenc h mathematician Lagrang e (1 736-1 81 3) . Nevertheless on e shoul d no t minimiz e th e India n achievemen t becaus e o f th e absence of any indication that the y were concerned abou t provin g that thei r metho d always le d t o a solution o r tha t i t gav e al l the solutions .

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

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