By V. S. Varadarajan

ISBN-10: 082180989X

ISBN-13: 9780821809891

This article bargains a different account of Indian paintings in diophantine equations through the sixth via twelfth centuries and Italian paintings on recommendations of cubic and biquadratic equations from the eleventh via sixteenth centuries. the quantity strains the historic improvement of algebra and the idea of equations from precedent days to the start of contemporary algebra, outlining a few sleek issues equivalent to the basic theorem of algebra, Clifford algebras, and quarternions. it's aimed at undergraduates who've no heritage in calculus.

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**Example text**

Fo r thos e reader s wh o ar e a littl e mor e verse d i n th e elements of prime factorizations o f numbers, I have indicated i n the exercise s belo w how th e proof s o f the statement s mad e abov e ca n b e developed . NOTES AN D EXERCISE S Several ne w concept s hav e bee n mentione d i n th e text , an d i t i s a goo d ide a t o wor k throug h a fe w simpl e exercise s t o becom e familia r wit h them . W e begi n wit h th e techniqu e o f expressin g any numbe r i n severa l scales . ALGEBRA I N ANCIEN T AN D MODER N TIME S 15 1.

Whil e the algorith m i s straightforward, on e can often reduc e the numbe r of step s i f on e take s som e shor t cuts . Thes e shor t cuts , base d o n th e bhavana, g o back t o Brahmagupt a althoug h h e di d no t posses s th e algorithm . The y sho w tha t if someho w on e ca n com e u p wit h a solutio n i n positiv e integer s t o (P m ) wher e m = - 1 , ± 2 , ± 4 , the n on e ca n ge t a solutio n i n positiv e integer s fo r (Pi) , an d hence a n infinit y o f solution s b y th e bhavana.

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. Finally , a s I have already mentioned, ther e is n o indicatio n tha t th e Indian s ha d a proo f tha t thi s algorith m wil l alway s lea d to a solution . A s w e shal l see presently , the y worke d ou t ver y difficul t numerica l cases like N = 61 ,6 7 which suggest tha t the y ha d grea t convictio n tha t thi s proces s will alway s lea d t o a solution .

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

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