By V. S. Varadarajan
This article deals a different account of Indian paintings in diophantine equations in the course of 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 historic improvement of algebra and the speculation of equations from precedent days to the start of recent algebra, outlining a few glossy issues corresponding to the basic theorem of algebra, Clifford algebras, and quarternions. it's aimed toward undergraduates who've no heritage in calculus.
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Additional resources for Algebra in Ancient and Modern Times
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 . Number s which satisf y polynomia l equation s wit h rationa l coefficient s ar e calle d algebraic numbers; al l othe r number s ar e know n a s transcendental numbers.
W e shall discus s thi s i n greate r detai l below . If w e have a solutio n (x , y) o f (P m ), then , writin g xA y yx — _1 x 2 - Ny 2 _ 1 m + yy/N Vx + yy/N ALGEBRA I N ANCIENT AND 1 MODERN TIMES 9 we see tha t x/y i s a goo d rationa l approximatio n t o y/N wheneve r x an d y ar e large i n compariso n t o m . Thus , th e easil y verifie d calculation s 265 2 - 3 x 1 53 2 = -1 2, 1 35 2 - 3 x 780 2 = 1 give the approximation s 1 265 35 1 113' 78 0 to \/ 3 which go back to Archimede s an d hi s work on approximation s t o 7r .
3 . 5388 3 1 . 5. 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 .
Algebra in Ancient and Modern Times by V. S. Varadarajan