Download e-book for kindle: Algebra in Ancient and Modern Times by V. S. Varadarajan

By V. S. Varadarajan

ISBN-10: 082180989X

ISBN-13: 9780821809891

This article deals a unique account of Indian paintings in diophantine equations through the sixth via twelfth centuries and Italian paintings on ideas of cubic and biquadratic equations from the eleventh via sixteenth centuries. the amount lines the ancient improvement of algebra and the idea of equations from precedent days to the start of contemporary algebra, outlining a few sleek issues corresponding to the basic theorem of algebra, Clifford algebras, and quarternions. it really is aimed at undergraduates who've no heritage in calculus.

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On e o f th e remarkabl e aspect s o f thi s equation i s that unlik e an y equatio n encountere d hithert o i t ha s an infinite number of solutions. Brahmagupt a alread y kne w thi s fac t an d wa s i n fac t familia r wit h what i s nowadays understoo d a s the multiplicativ e structur e o f the se t o f solutions, allowing hi m t o construc t a n infinit y o f solution s a s soo n a s a singl e solutio n i s known. Ferma t wa s howeve r th e firs t t o insis t o n a proo f tha t thi s equatio n always ha s a n infinit y o f solution s an d t o hav e aske d fo r a method that generates all the solutions.

7 "" " " 9 9 s - 2 4 2 T W 1 5 " 2. 4 4 2 . 1 1 253953678.. 1 1 253953951 .. Ramanujan gav e many othe r approximation s t o n. Her e ar e two which ough t t o b e compared wit h the true value. Th e first one is an improvement o f the approximatio n Yj| whic h i s quit e old . 1 41 5926535897932384626434.. 1 41 59265358979432.. 1 41 59265265262.. From classica l time s th e proble m o f squarin g th e circl e occupie d mathemati cians. , A n ar e rationa l numbers . Remarkably , Car l Lindeman n (1852—1939) prove d tha t IT cannot satisf y an y suc h algebrai c equation .

Wa s 1 98 ° 4 0 ' 2 " . Th e difference , 1 5 ° 5 5 ' 2 " is due t o th e annua l precessio n o f th e sta r whic h i s taken t o b e 4 7 " , correspondin g roughl y t o th e period o f revolutio n o f 26,00 0 year s fo r th e orbi t o f eart h aroun d th e distan t star . Thi s differenc e thus give s a laps e o f 1 21 6 years , thu s indicatin g tha t th e boo k wa s writte n aroun d 58 4 A . D . For mor e detail s se e th e accoun t give n i n th e translatio n b y H . T . Colebrook e o f th e work s o f Brahmagupta an d Bhaskar a : Algebra, with arithmetic and mensuration, from the Sanscrit of Brahmagupta and Bhascara, London, 1 81 7.

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

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