Get Algebra in Ancient and Modern Times PDF

By V. S. Varadarajan

ISBN-10: 082180989X

ISBN-13: 9780821809891

This article bargains 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 strains the old improvement of algebra and the speculation of equations from precedent days to the start of recent algebra, outlining a few sleek subject matters equivalent to the elemental theorem of algebra, Clifford algebras, and quarternions. it's aimed at undergraduates who've no heritage in calculus.

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Suppos e tha t (a , 6, c) i s a Pythagorea n triple t whic h i s primitive . Suppos e first tha t bot h a and b are even. The n c 2 = a 2 + b 2 is even, whic h make s c even. S o 2 divide s al l thre e o f them , violatin g ou r assumptio n o f primitivity . O n th e othe r hand, suppos e tha t a and b are bot h odd . The n a 2 an d b 2 ar e bot h odd , s o that c 2 is even, showin g that c must b e even. Bu t a 2 an d b 2 must bot h leav e the remainde r 1 whe n divide d b y 4 s o tha t c 2 mus t leav e th e remainde r 2 whe n divide d b y 4 , a contradiction, becaus e a s c i s even , c 2 i s divisibl e b y 4 !

Suppos e tha t (a , 6, c) i s a Pythagorea n triple t whic h i s primitive . Suppos e first tha t bot h a and b are even. The n c 2 = a 2 + b 2 is even, whic h make s c even. S o 2 divide s al l thre e o f them , violatin g ou r assumptio n o f primitivity . O n th e othe r hand, suppos e tha t a and b are bot h odd . The n a 2 an d b 2 ar e bot h odd , s o that c 2 is even, showin g that c must b e even. Bu t a 2 an d b 2 must bot h leav e the remainde r 1 whe n divide d b y 4 s o tha t c 2 mus t leav e th e remainde r 2 whe n divide d b y 4 , a contradiction, becaus e a s c i s even , c 2 i s divisibl e b y 4 !

Ar e solution s ALGEBRA I N ANCIEN T AN D MODER N TIME S 21 5. Tr y t o prov e th e theorem s o f Brahmagupt a state d i n th e tex t o n cycli c quadrilateral s (fo r more details , se e [E] , problem studie s 7. 1 0) . CAKRAVALA The cakravala is an algorith m fo r solvin g (P±i) develope d b y the India n schoo l of mathematician s an d astronomer s betwee n th e 6 t h an d 1 2 th centurie s ([W] , [DS ] (Part II)) . We shall describ e thi s algorith m a little late r an d illustrat e i t wit h som e examples.

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


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