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

This article bargains a different account of Indian paintings in diophantine equations throughout 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 speculation of equations from precedent days to the start of contemporary algebra, outlining a few sleek topics resembling the basic theorem of algebra, Clifford algebras, and quarternions. it really is aimed at undergraduates who've no heritage in calculus.

Best algebra & trigonometry books

A Course in Universal Algebra

Common algebra has loved a very explosive development within the final two decades, and a pupil coming into the topic now will discover a bewildering quantity of fabric to digest. this article isn't really meant to be encyclopedic; relatively, a couple of subject matters vital to common algebra were built sufficiently to convey the reader to the edge of present learn.

Calculus: Concepts and Applications

The acclaimed Calculus: recommendations and purposes is now to be had in a brand new version, revised to mirror very important alterations within the complicated Placement curriculum, and up-to-date to include suggestions from teachers in the course of the U. S. With over forty years of expertise instructing AP Calculus, Paul Foerster constructed Calculus: suggestions and functions with the highschool scholar in brain, yet with all of the content material of a college-level direction.

Extra resources for Algebra in Ancient and Modern Times

Sample text

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. 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 .

Prov e tha t i f 771 and n hav e n o common factor , the n n and 77 1 canno t hav e a common factor . Deduc e tha t i f n > 1 and 77 1 and Tl hav e n o commo n factor, 7 7 cannot divid e 77 1 . 16 V. S . VARADARAJA N 10. 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 .

S o P2 = 7 , q 2 = 2 , ra 2 = - 3 giving us (7 , 2; —3). Proceedin g i n thi s manne r w e get i n successio n (11,3; 4), 1( 8,5 ; - 1), 1 1 ( 9,33 (256,71; 3), (393, 1 09 ; 4), 1( 37,38 ; - 4 ), (649, 1 80 ; -3) ; 1) We hav e alread y see n tha t (649,180) is a solutio n i n thi s case . W e ca n als o d o thi s i n th e manne r o f Bhaskara , wit h th e Xi chose n differentl y a s explained . NOTES A N D EXERCISE S 1. Trea t th e cas e N = 6 1 b y th e cakravala without usin g shor t cuts , proceedin g a s i n th e example above , an d sho w that on e gets the solutio n X = 1 76631 9049 , Y in either method .