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New PDF release: Algebra in Ancient and Modern Times

By V. S. Varadarajan

ISBN-10: 082180989X

ISBN-13: 9780821809891

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 old improvement of algebra and the idea of equations from precedent days to the start of recent algebra, outlining a few sleek subject matters similar to the elemental theorem of algebra, Clifford algebras, and quarternions. it's aimed toward undergraduates who've no heritage in calculus.

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Extra resources for Algebra in Ancient and Modern Times

Example text

Th e appearanc e o f the squar e roo t i n the secon d o f these formula e make s it clea r tha t t o get rational approximation s w e mus t replac e th e squar e root s by rational fraction s clos e to their value . T o us e thi s techniqu e systematicall y an d in an iterativ e fashio n w e therefor e nee d t o have inequalitie s i n place o f the abov e equalities. Th e chang e fro m 0 to 0/2 is then give n b y the following: I f we star t with we ge t a + 0< ^ef2

It i s easy to see that thes e are all distinct. Similarl y i f we have in addition a solutio n (r,s) t o (P±m) an d defin e (r k,sk) b y (ro, so) = (r , s), (r fe, s fc) = (p , g) * (rfc _i, s k-i) then thes e ar e al l solution s t o (P± m ) whic h ar e distinc t (se e exercis e 3 below). N O T E S A N D EXERCISE S 1. Verif y Brahmagupta' s identity . 2. Verif y tha t i f p , g , r, 5 ar e al l positiv e integer s an d ( P , Q ) = ( p , g) * ( r , s ) , the n P>p,r, Q> q,s and henc e sho w tha t ( P , Q ) i s distinc t fro m th e origina l two .

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. 1090/mawrld/012/04 4 IRRATIONAL NUMBER S : CONSTRUCTIO N AND APPROXIMATIO N The Greek s alread y kne w tha t no t al l segment s tha t aris e i n geometrica l con structions hav e length s whic h ar e rationa l fractions . Thu s y/2 i s no t rational , al though i t i s the lengt h o f th e diagona l o f a squar e o f sid e 1 .

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


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