By V. S. Varadarajan

This article bargains a unique account of Indian paintings in diophantine equations in the course of the sixth via twelfth centuries and Italian paintings on suggestions of cubic and biquadratic equations from the eleventh via sixteenth centuries. the amount strains the ancient improvement of algebra and the speculation of equations from precedent days to the start of recent algebra, outlining a few smooth subject matters resembling the elemental theorem of algebra, Clifford algebras, and quarternions. it's aimed toward undergraduates who've no heritage in calculus.

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**Example text**

I have followed Wei l [W ] in describing the variant a s well as the argumen t fo r the existenc e of Xi+\\ fo r Bhaskar a an d hi s successor s th e argumen t fo r th e choic e o f x i+\ wa s a little mor e involved . Th e Indian s als o use d th e shor t cut s mentione d earlier , base d on Brahmagupta' s rules , t o reduc e th e computatio n (th e technolog y o f writing wa s not advance d i n Indi a an d s o ther e wa s a premiu m o n shor t calculation s a s wel l as 28 V. S. ) ; howeve r thes e shor t cut s ar e no t strictl y necessar y becaus e on e can show that th e cakravala , applie d i n a straightforward manner , alway s leads to a solution, an d i n fact, t o all the solutions.

I have followed Wei l [W ] in describing the variant a s well as the argumen t fo r the existenc e of Xi+\\ fo r Bhaskar a an d hi s successor s th e argumen t fo r th e choic e o f x i+\ wa s a little mor e involved . Th e Indian s als o use d th e shor t cut s mentione d earlier , base d on Brahmagupta' s rules , t o reduc e th e computatio n (th e technolog y o f writing wa s not advance d i n Indi a an d s o ther e wa s a premiu m o n shor t calculation s a s wel l as 28 V. S. ) ; howeve r thes e shor t cut s ar e no t strictl y necessar y becaus e on e can show that th e cakravala , applie d i n a straightforward manner , alway s leads to a solution, an d i n fact, t o all the solutions.

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 . 1090/mawrld/012/03 3 ARYABHATA-BRAHMAGUPTA-BHASKARA Aryabhata (476-c . 550) , Brahmagupt a (c .