__Great Indian Mathematician __

__Aryabhata__

Aryabhata, also called Aryabhata I or Aryabhata the Elder, (born 476, possibly Ashmaka or Kusumapura near Patalipurta Patna, India), astronomer and the earliest Indian mathematician, whose work and history are available to modern scholars. He is also known as Aryabhata I or Aryabhata the Elder. He flourished in Kusumapura—near Patalipurta (Patna), then the capital of the Gupta dynasty—where he composed at least two works, Aryabhatiya (c. 499) and the now lost Aryabhatasiddhanta.

Aryabhatiya was particularly popular in South India, where numerous mathematicians over the ensuing millennium wrote commentaries. The work is divided into three sections: Ganita (“Mathematics”), Kala-kriya (“Time Calculations”), and Gola (“Sphere”).

In Ganita Aryabhata names the first 10 decimal places and gives algorithms

for obtaining square and cubic roots, using the decimal number system. Then he treats geometric measurements—employing 62,832/20,000 (= 3.1416) for Ï€, very close to the actual value 3.14159—and develops properties of similar right-angled triangles and of two intersecting circles.

Using the Pythagorean theorem, he obtained one of the two methods for constructing his table of sines. He also realized that second-order sine difference is proportional to sine. Mathematical series, quadratic equations, compound interest (involving a quadratic equation), proportions (ratios), and the solution of various linear equations are among the arithmetic and algebraic topics included. Aryabhata’s general solution for linear indeterminate equations, which Bhaskara I called kuttakara (“pulverizer”), consisted of breaking the problem down into new problems with successively smaller coefficients—essentially the Euclidean algorithm and related to the method of continued fractions. With Kala-kriya Aryabhata turned to astronomy—in particular, treating planetary motion along the ecliptic. The topics include definitions of various units of time, eccentric and epicyclic models of planetary motion, planetary longitude corrections for different terrestrial locations.

Aryabhatiya ends with spherical astronomy in Gola, where he applied plane trigonometry to spherical geometry by projecting points and lines on the surface of a sphere onto appropriate planes. Topics include prediction of solar and lunar eclipses and an explicit statement that the apparent westward motion of the stars is due to the spherical Earth’s rotation about its axis. Aryabhata also correctly ascribed the luminosity of the Moon and planets to reflected sunlight.

The Indian government named its first satellite Aryabhata (launched 1975) in his honor.

**BRAHMAGUPTA**** ****[ 598 AD- 668 AD ]**

Brahmagupta was an Indian mathematician and astronomer . He was the native of Bhinmal town , Jalore District , Rajasthan . He is the author of two early works on Mathematics and astronomy - ' BRAHMASPHUTASIDDHANTA ' and 'KHANDAKHADYAKA'. He was the first to give rules to compute with zero and He was the first to invent the number ZERO. The texts composed by Brahmagupta was composed in elliptical verse in Sanskrit , as was common practice in mathematics. He gave the solution of general linear equation. He gave many formula’s related to arithmetic and trigonometry. He gave the solution of sum of the squares of first n natural numbers and sum of cubes of first n natural numbers.

**MAHAVIRA ****[9**^{th}
century]

Mahavira was a , jain mathematician from Karnataka , India. He was the author of the book 'GANITA SARA SAMGRAHA'. It is the earliest Indian text entirely devoted to mathematics . He is highly respected among Indian mathematicians because of his establishment of terminology for concepts such as equilateral , isosceles triangle , rhombus , circle and semi circle . He discovered the algebraic identities. He also found out the formulas for combinations. He devised a formula which approximated the area and perimeters of ellipse and found methods to calculate the square and cube roots of a number.

**BHASKARA - II ****[1114 - 1185]**

Bhaskara - II was an Indian mathematician . He was born in Bijapur , Karnataka. He was one of the greatest mathematician of medieval India . His contributions made a remarkable sight to the great Indian mathematics . Some of the important contributions made by Bhaskara - II was

He gave more defined solution to the equations like Pythagoras theorem i. e

a^{2} + b^{2} = c^{2}.

He stated the Rolle's theorem , which is a special case of one of the most important theorems on analysis, the mean value theorem .

He developed the spherical trigonometry.

His well known formulas for sine functions are sin ( a + b) and sin(a - b) . On 20 November 1981 , the ISRO launched the BHASKARA- II satellite honoring the great mathematician and astronomer.

**MADHAVA OF SANGAMAGRAMA ****[ 1340 AD- 1425 AD]**

He was a great mathematician and astrologer from the town of Sangamagrama , Kerala , India . He was the first to use the infinite series approximations for a range of trigonometric functions . He made a pioneer contributions to the study of infinite series , algebra , calculus , trigonometric functions . The infinite series for trigonometric functions.

**SRINIVASA RAMANUJAN****[ 22 December 1887- 26 April 1920]**

He was an Indian mathematician who lived during the British rule in India . He made substantial contributions to Mathematical analysis , Number Theory , Infinite Series and Continued Fractions, including solutions to Mathematical problems considered to be unsolvable.

When he was 15 years old, he obtained a copy of George Shoobridge Carr’s Synopsis of Elementary Results in Pure and Applied Mathematics. This collection of thousands of theorems, many presented with only the briefest of proofs and with no material newer than 1860, aroused his genius. Having verified the results in Carr’s book, Ramanujan went beyond it, developing his own theorems and ideas. In 1903 he secured a scholarship to the University of Madras but lost it the following year because he neglected all other studies in pursuit of mathematics.

Ramanujan continued his work, without employment and living in the poorest circumstances. After marrying in 1909 he began a search for permanent employment that culminated in an interview with a government official, Ramachandra Rao. Impressed by Ramanujan’s mathematical prowess, Rao supported his research for a time, but Ramanujan, unwilling to exist on charity, obtained a clerical post with the Madras Port Trust.

In 1911 Ramanujan published the first of his papers in the Journal of the Indian Mathematical Society. His genius slowly gained recognition, and in 1913 he began a correspondence with the British mathematician Godfrey H. Hardy that led to a special scholarship from the University of Madras and a grant from Trinity College, Cambridge. Overcoming his religious objections, Ramanujan traveled to England in 1914, where Hardy tutored him and collaborated with him in some research.

Ramanujan’s knowledge of mathematics (most of which he had worked out for himself) was startling. Although he was almost completely unaware of modern developments in mathematics, his mastery of continued fractions was unequaled by any living mathematician. He worked out the Riemann series, the elliptic integrals, hyper-geometric series, the functional equations of the zeta function, and his own theory of divergent series, in which he found a value for the sum of such series using a technique he invented that came to be called Ramanujan summation. On the other hand, he knew nothing of doubly periodic functions, the classical theory of quadratic forms, or Cauchy’s theorem, and he had only the most nebulous idea of what constitutes a mathematical proof. Though brilliant, many of his theorems on the theory of prime numbers were wrong.

In England Ramanujan made further advances, especially in the partition of numbers (the number of ways that a positive integer can be expressed as the sum of positive integers; e.g., 4 can be expressed as 4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1). His papers were published in English and European journals, and in 1918 he was elected to the Royal Society of London. In 1917 Ramanujan had contracted tuberculosis, but his condition improved sufficiently for him to return to India in 1919. He died the following year, generally unknown to the world at large but recognized by mathematicians as a phenomenal genius. Ramanujan left behind three notebooks and a sheaf of pages (also called the “lost notebook”) containing many unpublished results that mathematicians continued to verify long after his death.