By V. S. Varadarajan
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 options 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 topics reminiscent of the basic theorem of algebra, Clifford algebras, and quarternions. it truly is aimed at undergraduates who've no historical past in calculus.
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Extra resources for Algebra in Ancient and Modern Times
Have bee n found , find Xi+i suc h tha t Pi 4- Xi+iqi = 0 (mo d |ra;|) , x < \Zi V < Xi+i + |rn^ | i1 + 5. Define Pi£;+i + AT^ +i p Pi+1 = j j > * H- ftXi+i tfi+ 1 = j j , |ra;| |mt |m x? ™>i+l 1 + -A T = * 6. Keep goin g til l yo u reac h mk=p2k- Nql = 1 Of cours e ther e i s n o nee d t o sa y tha t thi s i s a n astonishin g algorithm . I t i s absolutely no t obviou s tha t thi s procedur e eve r terminate s wit h som e m k = 1 , o r even that th e p; and qi are all positive integers . I f the algorithm leads to an m k = 1 , we shall hav e th e solutio n X=pk, Y = qk for X2 - NY 2 = 1 The cakravdla a s th e Indian s use d it , differe d fro m th e abov e i n on e respect ; the choic e o f Xi+\ i n th e congruenc e clas s o f — Xi modul o \rrii\ was require d t o b e made i n such a way as to minimize N — x2+l (se e [DS] , pp.
It follow s fro m thi s exercis e tha t the a% are precisely the denominators in fraction expansion of VN. ] To se e this , le t VN = [6o > &1J&2> • • •]• Sinc e & o = [VN] i t i s clea r tha t bo = Clu - Further , 6X = [l/(VN—po)] = [(VN+po)/\mo\)] s o that i t i s the intege r t define d b y — P o + £ | m o | < VN < —po 4 - £|wio | + |wio| » showin g tha t £ = a i = q\. Thu s & i = d i - Suppos e no w tha t bi = a^, Z < 71 . The n VN = [ a o , a i , . . , a n - i , a n + ry J = • tfn + W n -1 where = [flOjfll j • • • jfln ] Then _ Vffig w ~ Pn Pn-l - VNqn-1 and s o bn+1 = — ^ri+ 1 from th e las t exercise .
1 _ 11 0 2TTy/2 ~ ~99 3 • 2749 3 1 1 . 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.
Algebra in Ancient and Modern Times by V. S. Varadarajan