The most fundamental contribution of ancient India in mathematics is the invention of decimal system of enumeration, including the invention of zero. The decimal system uses nine digits (1 to 9) and the symbol zero (for nothing) to denote all natural numbers by assigning a place value to the digits. The Arabs carried this system to Africa and Europe.
The Vedas and Valmiki Ramayana used this system, though the exact dates of these works are not known. MohanjoDaro and Harappa excavations (which may be around 3000 B.C. old) also give specimens of writing in India. Aryans came 1000 years later, around 2000 B.C. Being very religious people, they were deeply interested in planetary positions to calculate auspicious times, and they developed astronomy and mathematics towards this end. They identified various nakshatras (constellations) and named the months after them. They could count up to 1012, while the Greeks could count up to 104 and Romans up to 108. Values of irrational numbers such as and were also known to them to a high degree of approximation. Pythagoras Theorem can be also traced to the Aryan's Sulbasutras. These Sutras, estimated to be between 800 B.C. and 500 B.C., cover a large number of geometric principles. Jaina religious works (dating from 500 B.C. to 100 B.C.) show they knew how to solve quadratic equations (though ancient Chinese and Babylonians also knew this prior to 2000 B.C.). Jainas used as the value of (circumference = x Diameter). They were very fond of large numbers, and they classified numbers as enumerable, unenumerable and infinite. The Jainas also worked out formulae for permutations and combinations though this knowledge may have existed in Vedic times. Sushruta Samhita (famous medicinal work, around 6th century B.C.) mentions that 63 combinations can be made out of 6 different rasas (tastes -bitter, sour, sweet, salty, astringent and hot).
In the year 1881 A.D., at a village named Bakhshali near Peshawar, a farmer found a manuscript during excavation. About 70 leaves were found, and are now famous as the Bakhshali Manuscript. Western scholars estimate its date as about third or fourth century A.D. It is devoted mostly to arithmetic and algebra, with a few problems on geometry and mensuration.
With this historical background, we come to the famous Indian mathematicians.
Aryabhata (475 A.D. -550 A.D.) is the first well known Indian mathematician. Born in Kerala, he completed his studies at the university of Nalanda. In the section Ganita (calculations) of his astronomical treatise Aryabhatiya (499 A.D.), he made the fundamental advance in finding the lengths of chords of circles, by using the half chord rather than the full chord method used by Greeks. He gave the value of as 3.1416, claiming, for the first time, that it was an approximation. (He gave it in the form that the approximate circumference of a circle of diameter 20000 is 62832.) He also gave methods for extracting square roots, summing arithmetic series, solving indeterminate equations of the type ax -by = c, and also gave what later came to be known as the table of Sines. He also wrote a text book for astronomical calculations, Aryabhatasiddhanta. Even today, this data is used in preparing Hindu calendars (Panchangs). In recognition to his contributions to astronomy and mathematics, India's first satellite was named Aryabhata.
Brahmagupta (598 A.D. -665 A.D.) is renowned for introduction of negative numbers and operations on zero into arithmetic. His main work was Brahmasphutasiddhanta, which was a corrected version of old astronomical treatise Brahmasiddhanta. This work was later translated into Arabic as Sind Hind. He formulated the rule of three and proposed rules for the solution of quadratic and simultaneous equations. He gave the formula for the area of a cyclic quadrilateral as where s is the semi perimeter. He was the first mathematician to treat algebra and arithmetic as two different branches of mathematics. He gave the solution of the indeterminate equation Nx²+1 = y². He is also the founder of the branch of higher mathematics known as "Numerical Analysis".
After Brahmagupta, the mathematician of some consequence was Sridhara, who wrote Patiganita Sara, a book on algebra, in 750 A.D. Even Bhaskara refers to his works. After Sridhara, the most celebrated mathematician was Mahaviracharaya or Mahavira. He wrote Ganita Sara Sangraha in 850 A.D., which is the first text book on arithmetic in present day form. He is the only Indian mathematician who has briefly referred to the ellipse (which he called Ayatvrit). The Greeks, by contrast, had studied conic sections in great detail.
Bhaskara (1114 A.D. -1185 A.D.) or Bhaskaracharaya is the most well known ancient Indian mathematician. He was born in 1114 A.D. at Bijjada Bida (Bijapur, Karnataka) in the Sahyadari Hills. He was the first to declare that any number divided by zero is infinity and that the sum of any number and infinity is also infinity. He is famous for his book Siddhanta Siromani (1150 A.D.). It is divided into four sections -Leelavati (a book on arithmetic), Bijaganita (algebra), Goladhayaya (chapter on sphere -celestial globe), and Grahaganita (mathematics of the planets). Leelavati contains many interesting problems and was a very popular text book. Bhaskara introduced chakrawal, or the cyclic method, to solve algebraic equations. Six centuries later, European mathematicians like Galois, Euler and Lagrange rediscovered this method and called it "inverse cyclic". Bhaskara can also be called the founder of differential calculus. He gave an example of what is now called "differential coefficient" and the basic idea of what is now called "Rolle's theorem". Unfortunately, later Indian mathematicians did not take any notice of this. Five centuries later, Newton and Leibniz developed this subject. As an astronomer, Bhaskara is renowned for his concept of Tatkalikagati (instantaneous motion).
After this period, India was repeatedly raided by muslims and other rulers and there was a lull in scientific research. Industrial revolution and Renaissance passed India by. Before Ramanujan, the only noteworthy mathematician was Sawai Jai Singh II, who founded the present city of Jaipur in 1727 A.D. This Hindu king was a great patron of mathematicians and astronomers. He is known for building observatories (Jantar Mantar) at Delhi, Jaipur, Ujjain, Varanasi and Mathura. Among the instruments he designed himself are Samrat Yantra, Ram Yantra and Jai Parkash.
Well known Indian mathematicians of 20th century are:
Srinivasa Aaiyangar Ramanujan is undoubtedly the most celebrated
Indian Mathematical genius. He was born in a poor family at Erode in Tamil Nadu
on December 22, 1887. Largely self taught, he feasted on Loney's Trigonometry
at the age of 13, and at the age of 15, his senior friends gave him Synopsis
of Elementary Results in Pure and Applied Mathematics by George Carr. He
used to write his ideas and results on loose sheets. His three filled notebooks
are now famous as Ramanujan's Frayed Notebooks. Though he had no
qualifying degree, the University of Madras granted him a monthly scholarship of
Rs. 75 in 1913. A few months earlier, he had sent a letter to great
mathematician G.H. Hardy, in which he mentioned 120 theorems and formulae. Hardy
and his colleague at Cambridge University, J.E. Littlewood immediately
recognised his genius. Ramanujan sailed for Britain on March 17, 1914. Between
1914 and 1917, Ramanujan published 21 papers, some in collaboration with Hardy.
His achievements include Hardy-Ramanujan-Littlewood circle method in number
theory, Roger-Ramanujan's identities in partition of numbers, work on algebra of
inequalities, elliptic functions, continued fractions, partial sums and products
of hypergeometric series, etc. He was the second Indian to be elected Fellow of
the Royal Society in February, 1918. Later that year, he became the first Indian
to be elected Fellow of Trinity College, Cambridge. Ramanujan had an intimate
familiarity with numbers. During an illness in England, Hardy visited Ramanujan
in the hospital. When Hardy remarked that he had taken taxi number 1729, a
singularly unexceptional number, Ramanujan immediately responded that this
number was actually quite remarkable: it is the smallest integer that can be
represented in two ways by the sum of two cubes: 1729=1³+12³=9³+10³.
Unfortunately, Ramanujan's health deteriorated due to tuberculosis, and he returnted to India in 1919. He died in Madras on April 26, 1920.
P.C. Mahalanobis : He founded the Indian Statistical Research Institute in Calcutta. In 1958, he started the National Sample Surveys which gained international fame. He died in 1972 at the age of 79.
C.R. Rao : A well known statistician, famous for his "theory of estimation"(1945). His formulae and theory include "Cramer -Rao inequality", "Fischer -Rao theorem" and "Rao - Blackwellisation".
D.R. Kaprekar (1905-1988) : Fond of numbers. Well known for "Kaprekar Constant" 6174. Take any four digit number in which all digits are not alike. Arrange its digits in descending order and subtract from it the number formed by arranging the digits in ascending order. If this process is repeated with reminders, ultimately number 6174 is obtained, which then generates itself.
Harish Chandra (1923-1983) : Greatly developed the branch of higher mathematics known as the infinite dimensional group representation theory.
Narendra Karmarkar : India born Narendra Karmarkar, working at Bell Labs USA, stunned the world in 1984 with his new algorithm to solve linear programming problems. This made the complex calculations much faster, and had immediate applications in airports, warehouses, communication networks etc.