By V. S. Varadarajan

This article bargains a different account of Indian paintings in diophantine equations through the sixth via twelfth centuries and Italian paintings on suggestions of cubic and biquadratic equations from the eleventh via sixteenth centuries. the amount lines the old improvement of algebra and the speculation of equations from precedent days to the start of contemporary algebra, outlining a few smooth subject matters equivalent to the basic theorem of algebra, Clifford algebras, and quarternions. it's aimed at undergraduates who've no history in calculus.

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T o construct th e triangl e w e proceed a s follows. W e draw a line AB o f length {a + b)/2 and a semicircl e abov e i t wit h AB a s a diameter . Wit h A a s cente r w e dra w a circular ar c o f radiu s (a — b)/2 an d le t C b e th e poin t wher e thi s ar c meet s th e semicircle. The n th e angl e < ACB i s a righ t angl e an d AC ha s lengt h ( a — 6)/2 , so tha t BC ha s lengt h y/ab. We hav e see n tha t i n th e wor k o f Archimede s ther e appeare d approximation s to irrationa l numbers .

Suc h question s hav e ha d a dee p interes t fo r mathemati cians always , righ t dow n t o th e moder n era . Hilber t aske d i n hi s famous addres s t o the Internationa l Congres s o f mathematician s i n Pari s i n 1 90 0 whethe r a b i s tran scendental i f a an d b are algebrai c numbers , b is irrational , an d a ^ 0,1 . Thi s wa s settled i n th e affirmativ e b y A . O . Gel'fond , an d b y Th . Schneider , i n 1 934 . Such question s severel y ta x th e resource s o f moder n mathematics .

ALGEBRA I N ANCIENT AN D MODERN 1 TIME S 1 1 120 3456 4800 13500 119 c 169 3367 4601 12709 4825 6649 18541 72 360 65 319 97 481 2700 2291 960 600 799 481 3541 1249 6480 4961 a b 3 769 8161 60 45 75 2400 1679 2929 240 161 289 2700 1771 3229 90 56 106 Actually th e number s i n th e table t wer e i n th e sexagesima l syste m an d th e tabl e looked lik e this : a 2,0 57,36 1,20,0 3,45,0 1,12 6,0 45,0 16,0 10,0 1,48,0 b c 1,59 56,7 1,16,41 3,31,49 2,49 1,20,25 1,50,49 5,9,1 1,37 1,5 5,19 38,11 13,19 8,1 1,22,41 1,0 45 40,0 27,59 2,41 29,31 4,0 45,0 1,30 56 8,1 59,1 20,49 12,49 2,16,1 1,15 48,49 4,49 53,49 1,46 The larg e size s o f thes e number s ar e clearl y a n indicatio n tha t th e Babylonian s knew ho w t o generat e suc h triplet s i n a systemati c manner .