By V. S. Varadarajan
This article deals a distinct account of Indian paintings in diophantine equations in the course of the sixth via twelfth centuries and Italian paintings on options of cubic and biquadratic equations from the eleventh via sixteenth centuries. the quantity lines the historic improvement of algebra and the speculation of equations from precedent days to the start of contemporary algebra, outlining a few glossy subject matters comparable to the elemental theorem of algebra, Clifford algebras, and quarternions. it truly is aimed toward undergraduates who've no history in calculus.
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Extra resources for Algebra in Ancient and Modern Times
B y switchin g a an d b we may assum e tha t a i s even . We no w us e th e identit y (2st)2 + (s 2-t2)2 = (s 2 + t 2)2 which was known to the ancients , t o conclud e that i f we give s an d t intege r values , then a = 2st, b = s 2 -t 2,c = s 2 + t 2 (* ) is a pythagorea n triplet . Fo r b to b e positive , w e must requir e tha t s > t. It s an d t ar e chose n t o hav e n o commo n facto r an d t o b e o f opposit e parity , namely , tha t one o f them i s odd an d th e othe r i s even, the n on e ca n sho w tha t (* ) i s a primitiv e Pythagorean triple t wit h a even .
Ther e i s als o a mor e subtle indication . Notic e tha t i f (a , 6, c) i s a tripl e an d w e multipl y al l o f the m b y the sam e numbe r k s o tha t w e ge t (ka,kb,kc), the n thi s i s agai n a pythagorea n triplet, becaus e k2a2+k2b2 = k 2(a2 + b 2) = k 2c2 Thus onc e w e obtai n on e triple t w e ca n generat e al l thei r multiples . Thi s show s that i t i s enoug h t o obtai n al l triplet s (a , 6, c) wit h th e propert y tha t th e number s a,b,c d o no t hav e a commo n factor .
Tha t peopl e of very ancien t time s possessed technique s fo r writin g dow n thes e triplets i s a remarkabl e fact . Fo r thos e reader s wh o ar e a littl e mor e verse d i n th e elements of prime factorizations o f numbers, I have indicated i n the exercise s belo w how th e proof s o f the statement s mad e abov e ca n b e developed . NOTES AN D EXERCISE S Several ne w concept s hav e bee n mentione d i n th e text , an d i t i s a goo d ide a t o wor k throug h a fe w simpl e exercise s t o becom e familia r wit h them .