By V. S. Varadarajan
This article bargains a distinct account of Indian paintings in diophantine equations throughout the sixth via twelfth centuries and Italian paintings on strategies 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 recent algebra, outlining a few sleek issues similar 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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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 . 1090/mawrld/012/03 3 ARYABHATA-BRAHMAGUPTA-BHASKARA Aryabhata (476-c . 550) , Brahmagupt a (c .
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 . A ver y genera l propositio n i n Eucli d show s ho w t o construc t a squar e whos e area i s equa l t o tha t o f a give n rectangle .
96-1 02 ) (i) arcta n \ + arcta n \ — \ (ii) arcta n | + arcta n ^ + arcta n \ — \ (iii) 4 arctan \ — arctan ^ = f (iv) 4 arctan \ — arctan ^ + arcta n ^ = f (Hint : Fo r (iii ) an d (iv) , sho w firs t tha t 3 a r c t an \ = a r c t a n | | ) 6. Us e th e formula e i n exercis e 5 to calculat e 7 T accurately t o severa l decima l places . 7. Sho w tha t x = y 3 4- y 5 4- V2 is algebrai c b y constructin g a n equatio n o f th e 1 2 degre by X. e wit h intege r coefficient s satisfie d Hilbert's famou s addres s liste d 2 3 problems an d th e proble m o f transcendentalit y o f a i s the 7 .
Algebra in Ancient and Modern Times by V. S. Varadarajan