Contribution of Brahma gupta, in mathematics
- Brahmagupta was born in 598 A.D.in Bhinmal city in the state of Rajasthan. He was a mathematician and astronomer, who wrote many important works on mathematics and astronomy. His best known work is the “Brahmasphuta‐siddhanta”, written in 628 AD in Bhinmal.
- He was the first to use zero as a number. He gave rules to compute with zero.
- He gave four methods of multiplication.
- He gave following formulae, used in G.P. series
a+ar+ar2 +ar3 +……….+arn‐1 =a(rn ‐1)/( r‐1)
- He gave the following formulae(Brahmagupta’s formula): Area of a cyclic quadrilateral with side a,b,c,d =9(s‐a)(s‐b)(s‐c)(s‐d), where 2s= a+b+c+d.
Contribution of Bhaskaracharya , in mathematics
- A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a2 + b2 = c2.
- In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations are explained.
- Solutions of indeterminate quadratic equations (of the type ax2 + b = y2).
- Integer solutions of linear and quadratic indeterminate equations (Kuṭṭaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century.
- A cyclic Chakravala method for solving indeterminate equations of the form ax2 + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method.
- The first general method for finding the solutions of the problem x2 − ny2 = 1 (so-called “Pell’s equation”) was given by Bhaskara II.
- Solutions of Diophantine equations of the second order, such as 61×2 + 1 = y2. This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century.
Bhaskara’s arithmetic text Leelavati covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations.
Lilavati is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and measurement. More specifically the contents include:
- Properties of zero (including division, and rules of operations with zero).
- Further extensive numerical work, including use of negative numbers and surds.
- Estimation of π.
- Arithmetical terms, methods of multiplication, and squaring.
- Inverse rule of three, and rules of 3, 5, 7, 9, and 11.
His Bijaganita (“Algebra”) was a work in twelve chapters. It was the first text to recognize that a positive number has two square roots (a positive and negative square root). His work Bijaganita is effectively a treatise on algebra and contains the following topics:
- Positive and negative numbers.
- The ‘unknown’ (includes determining unknown quantities).
- Determining unknown quantities.
- Surds (includes evaluating surds).
- Kuṭṭaka (for solving indeterminate equations and Diophantine equations).
- Simple equations (indeterminate of second, third and fourth degree).
The Siddhānta Shiromani (written in 1150) demonstrates Bhaskara’s knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also developed spherical trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, results found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for sin(a+b) and sin(a-b).
His work, the Siddhānta Shiromani, is an astronomical treatise and contains many theories not found in earlier works. Preliminary concepts of infinitesimal calculus and mathematical analysis, along with a number of results in trigonometry, differential calculus and integral calculus that are found in the work are of particular interest.
Evidence suggests Bhaskara was acquainted with some ideas of differential calculus. Bhaskara also goes deeper into the ‘differential calculus’ and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of ‘infinitesimals’.
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