By V. S. Varadarajan

This article deals a unique 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 quantity strains the historic improvement of algebra and the speculation of equations from precedent days to the start of contemporary algebra, outlining a few sleek topics akin to the elemental theorem of algebra, Clifford algebras, and quarternions. it truly is aimed toward undergraduates who've no heritage in calculus.

**Read or Download Algebra in Ancient and Modern Times PDF**

**Similar algebra & trigonometry books**

Publication by means of Garnett, J.

**An introduction to Gröbner bases**

Because the basic device for doing specific computations in polynomial earrings in lots of variables, Gröbner bases are an enormous portion of all machine algebra structures. also they are vital in computational commutative algebra and algebraic geometry. This publication offers a leisurely and reasonably accomplished advent to Gröbner bases and their purposes.

- Topological Groups: Characters, Dualities, and Minimal Group Topoligies
- Motives (Proceedings of Symposia in Pure Mathematics) (Part 1)
- An Introduction to Operator Algebras
- Noncommutative Rings, Group Rings, Diagram Algebras and Their Applications
- Mathematik 2: Lehrbuch fur ingenieurwissenschaftliche Studiengange
- One Semester of Elliptic Curves (EMS Series of Lectures in Mathematics)

**Extra resources for Algebra in Ancient and Modern Times**

**Sample text**

Pk,Qk) = 1 . Henc e verif y tha t (pk-> qk) (2pfc_ i - f - 3 ^ - i , P f c - i + 2^-i ) Show tha t th e first si x o f thes e solution s ar e (2,1), (7,4) , (26,15), (97,56) , (362,209) , (1 351 ,780 ) Also startin g wit h th e solutio n ( 1 , 1) t o X — 3Y = — 2 verif y tha t (1,1), (5,3)(1 9,1 1 ) , (71 ,41 ) , (265,1 53 ) are als o solution s t o th e sam e equation . Henc e deduc e th e approximation s t o 2 6 5 r- 1 35 1 I53

Halve th e numbe r o f roots , whic h i n thi s cas e yield s five . Thi s yo u multipl y by itself ; th e produc t i s twenty-five. Ad d thi s t o thirty-nine ; th e su m i s sixty-four . Now tak e th e roo t o f this , whic h i s eight , an d subtrac t fro m i t hal f th e numbe r o f roots, whic h i s five. Th e remainde r i s three. Thi s i s the roo t yo u aske d for . This se t o f instruction s applie s t o th e equatio n X2 + 2BX = C giving th e sequenc e o f step s X2 + 2BX + B 2 = C + B 2 (X + B ) 2 = C + B 2 X + B = VC + B2 X = y/C + B 2 - B There i s n o nee d t o dwel l to o muc h o n ho w cumbersom e wer e th e processe s available t o peopl e i n thos e day s i n discoverin g an d communicatin g mathematica l ideas an d calculations .

I f & is a positive intege r sho w tha t (2A : + 1 ) = 4k + 4k + 1 and deduc e tha t th e squar e of a n od d intege r i s not only od d bu t leave s th e remainde r 1 whe n divide d b y 4. Usin g a similar argument , sho w tha t th e squar e of an intege r no t divisibl e b y 3 leaves th e remainde r 1 whe n divide d b y 3. For provin g tha t th e description give n abov e o f all primitive triplet s wit h a eve n i s correct one needs to know tha t ever y intege r i s a product o f primes (repetition s allowed ) an d tha t thi s ca n be done i n a unique manne r if we disregard th e orde r i n which th e primes appea r i n the factorization .