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ByV.Lavanaya

IX E

BHASKARA II(BHASKARACHARYA)

Background InformationFew important contributions of Bhaskara II

to mathematics are as followsSimple mathematical methods AlgebraTrigonometrySiddhanta siromaniFurther Information in SiddhantaFormulasAfter Bhaskara II

CONTENT:

One of most famous Indian mathematician.

He was born in Vijayapura in the year 1114 AD near Bijjada Bida which is in present day Bijapur district, Karnataka, India.

Father was a Brahman named Mahesvara and he was an astrologer.

Nicknamed as Bhaskaracharya “Bhaskara the Teacher”.

Studied inVarahamihira and Brahmagupta at Uijain.

Background Information

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 (Kuttaka). 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.

Few important contributions of Bhaskar II to mathematics are as follows: 

Solutions of  Diophantine equations  of the second order, such as 61x² + 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. Solved quadratic equations with more than one unknown, and found negative and irrational solutions. Preliminary concept of  mathematical analysis.Preliminary concept of  mathematical analysis, along with notable contributions towards  integral calculus. Conceived  differential calculus, after discovering the  derivative and differential  coefficient.Stated  Rolle's theorem, a special case of one of the most important theorems in analysis, the  mean value theorem. Traces of the general mean value theorem are also found in his works. Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.) In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results. (See Trigonometry section below.)

Bhaskara II suggested simple methods to calculate the squares, square roots, cube, and cube roots of big numbers. The Pythagoras theorem was proved by him in only two lines. Bhaskara's 'Khandameru'is the famous Pascal Triangle. Pascal, the European mathematician was born 500 years after Bhaskara. In Lilawati, he solved several problems on permutations and combinations and called the method as 'ankapaash'. He even gave an approximate value of PI as 22/7, which is 3.1416. He was even familiar with the concept of infinity and called it as 'khahar rashi', which means 'anant'. 

Simple mathematical methods 

Bhaskara's  arithmetic text Lilavati  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. 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. Problems involving interest and interest computation. Indeterminate equations (Kuttaka), integer solutions (first

and second order). His contributions to this topic are particularly important, since the rules he gives are (in effect) the same as those given by the  renaissance  European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work of  Aryabhata  and subsequent mathematicians.

Arithmetic

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. Zero. The 'unknown' (includes determining unknown quantities). Determining unknown quantities. Surds  (includes evaluating surds). Kuttaka (for solving  indeterminate equations and Diophantine

equations). Simple equations (indeterminate of second, third and fourth

degree). Simple equations with more than one unknown. Indeterminate quadratic equations  (of the type ax² + b = y²). Solutions of indeterminate equations of the second, third and fourth

degree. Quadratic equations. Quadratic equations with more than one unknown. Operations with products of several unknowns.

Algebra

The Siddhanta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also discovered  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, discoveries first found in his works include the now well known results for  and :

Trigonometry

He wrote Siddhanta Siromani in the year 1150 ADLeelavati (arithmetic)Bijaganita (algebra)Goladhayaya (spheres, celestial globes)Grahaganita (mathematics of the planets)

SIDDHANTA SIROMANI

First time trigonometry was studied as it’s own entity, rather than how it related to other calculations. sin(a + b) = sin a cos b + cos a sin b

sin(a - b) = sin a cos b - cos a sin b.

Further Information in Siddhanta

FORMULAS• Given that he was building on the knowledge and understanding of Brahmagupta it is not surprising that Bhaskaracharya understood about zero and negative numbers. However his understanding went further even than that of Brahmagupta. To give some examples before we examine his work in a little more detail we note that he knew that x2 = 9 had two solutions. He also gave the formula

• Bhaskaracharya studied Pell's equation px2 + 1 = y2 for p = 8, 11, 32, 61 and 67. When p = 61 he found the solutions x = 226153980, y = 1776319049. When p= 67 he found the solutions x = 5967, y = 48842. He studied many Diophantine problems

Bhaskara II dies in 1185A HUGE scientific lull after invasion by

muslims1727, next important Hindu mathematician

Sawai Jai Singh IISeveral of Bhaskara’s findings were not

explored heavily after his death, and ended up being “discovered” later by European mathematicians.

After Bhaskara II