Chronology of indian mathematicians

1. Chronology of Indian mathematicians… ….And their contributions

2. Prepared By- Name - Meeran Ali Ahmad Class - X A Roll No. - 07

3. Introduction Indian mathematics emerged in the Indian subcontinent from 1200 BC  until the end of the 18th century. In the classical period of Indian mathematics (400 AD to 1200 AD), important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number,  negative numbers,  arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of mathematics.

4. Some Indian mathematicians Vedic- Baudhayana Katyayana Panini, ca. 5th c. BC, Algebraic grammarian Yajnavalkya, credited with authorship of the Shatapatha Brahmana, which contains calculations related to altar construction.

5. Classical Post-Vedic Sanskrit to Pala period mathematicians (5th c. BC to 11th c. AD) Aryabhata - Astronomer who gave accurate calculations for astronomical constants, 476AD-520AD Aryabhata II Bhaskara I Brahmagupta - Helped bring the concept of zero into arithmetic (598 AD-670 AD) Bhāskara II Mahavira Pavuluri Mallana - the first Telugu Mathematician Varahamihira Shridhara (between 650-850) - Gave a good rule for finding the volume of a sphere.

6. Medieval to Mughal period Narayana Pandit Madhava of Sangamagrama some elements of Calculus hi Parameshvara (1360–1455), discovered drk-ganita, a mode of astronomy based on observations, Madhava's Kerala school Nilakantha Somayaji,1444-1545 - Mathematician and Astronomer, Madhava's Kerala school Mahendra Suri (14th century) Shankara Variyar (c. 1530) Raghunatha Siromani, (1475–1550), Logician, Navadvipa school

7. Aryabhatta

8. 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 Aryabhatta . About -

9. Aryabhatta is the first writer on astronomy to whom the Hindus do not allow the honour of a divine inspiration. Writers on mathematical science distinctly state that he was the earliest uninspired and a merely human writer on astronomy. This is a notice which sufficiently proves his being an historical character. He also ascribed to the epicycles, by which the motion of a planet is represented, a form varying from the circle and nearly elliptic . His text specifies the earth's diameter, 1050 yojanas; and the orbit or circumference of the earth's wind [spiritus vector] 3393 yojanas; which, as the scholiast rightly argues, is no discrepancy. His contributions….

10. Brahmagupta

11. The great 7th Century Indian mathematician and astronomer Brahmagupta wrote some important works on both mathematics and astronomy. He was from the state of Rajasthan of northwest India (he is often referred to as Bhillamalacarya, the teacher from Bhillamala), and later became the head of the astronomical observatory at Ujjain in central India. Most of his works are composed in elliptic verse, a common practice in Indian mathematics at the time, and consequently have something of a poetic ring to them. It seems likely that Brahmagupta's works, especially his most famous text, the “Brahmasphut- asiddhanta”, were brought by the 8th Century Abbasid caliph Al-Mansur to his newly founded centre of learning at Baghdad on the banks of the Tigris, providing an important link between Indian mathematics and astronomy and the nascent upsurge in science and mathematics in the Islamic world. About -

12. His contributions In his work on arithmetic, Brahmagupta explained how to find the cube and cube-root of an integer and gave rules facilitating the computation of squares and square roots.  He also gave rules for dealing with five types of combinations of fractions. He gave the sum of the squares of the first  n natural numbers as  n ( n  + 1)(2 n  + 1) ⁄ 6 and the sum of the cubes of the first  n natural numbers as ( n ( n  + 1) ⁄ 2 ) ² . Furthermore, he pointed out, quadratic equations (of the type  x 2  + 2 = 11, for example) could in theory have two possible solutions, one of which could be negative, because 3 2  = 9 and -3 2  = 9. In addition to his work on solutions to general linear equations and quadratic equations, Brahmagupta went yet further by considering systems of simultaneous equations (set of equations containing multiple variables), and solving quadratic equations with two unknowns, something which was not even considered in the West until a thousand years later, when Fermat was considering similar problems in 1657.

13. Bhaskara II

14. 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 centurieslater, 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). About -

15. His contributions … Terms for numbers In English, the multiples of 1000 are termed as thousand, million, billion, trillion, quadrillion etc. These terms were named recently in English, but Bhaskaracharya gave the terms for numbers in multiples of ten which are as follows: eka(1), dasha(10), shata(100), sahastra(1000), ayuta(10,000), laksha(100,000), prayuta (1,000,000=million), koti(107), Kutarbuda(108), abja(109=billion), kharva (1010), nikharva (1011), mahapadma (1012=trillion), shanku(1013), jaladhi(1014), antya(1015=quadrillion), Madhya (1016) and parardha(1017).  Kuttak Kuttak according to modern mathematics is 'indeterminate equation of first order'. In the western world, the method of solving such equations was called as 'pulverizer'. Bhaskara suggested a generalized solution to get multiple answers for these equations.  Simple mathematical methods  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.

16. Srinivasa Ramanujan

17. Srinivasa Ramanujan (1887-1920) hailed as an all-time great mathematician, like Euler, Gauss or Jacobi, for his  natural genius , has left behind 4000 original theorems, despite his lack of formal education and a short life-span. In his formative years, after having failed in his F.A. (First examination in Arts) class at College, he ran from pillar to post in search of a benefactor. It is during this period, 1903-1914, he kept a record of the final results of his original research work in the form of entries in two large-sized Note Books. These were the ones which he showed to Dewan Bahadur Ramachandra Rao (Collector of Nellore), V. Ramaswamy Iyer (Founder of Indian Mathematical Society), R. Narayana Iyer (Treasurer of IMS and Manager, Madras Port Trust), and to several others to convince them of his abilities as a Mathematician. The orchestrated efforts of his admirers, culminated in the encouragement he received from Prof. G.H. Hardy of Trinity College, Cambridge, whose warm response to the historic letter of Ramanujan which contained about 100 theorems, resulted in inducing the Madras University, to its lasting credit, to rise to the occasion thrice - in offering him the first research scholarship of the University in May 1913 ; then in offering him a scholarship of 250 pounds a year for five years with 100 pounds for passage by ship and for initial outfit to go to England in 1914 ; and finally, by granting Ramanujan 250 pounds a year as an allowance for 5 years commencing from April 1919 soon after his triumphant return from Cambridge ``with a scientific standing and reputation such as no Indian has enjoyed before''. About -

18. Ramanujan's arrival at Cambridge was the beginning of a very successful five-year collaboration with Hardy. In some ways the two made an odd pair: Hardy was a great exponent of rigor in analysis, while Ramanujan's results were (as Hardy put it) "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account". Hardy did his best to fill in the gaps in Ramanujan's education without discouraging him. He was amazed by Ramanujan's uncanny formal intuition in manipulating infinite series, continued fractions, and the like: "I have never met his equal, and can compare him only with Euleror Jacobi."One remarkable result of the Hardy-Ramanujan collaboration was a formula for the number p( n ) of partitions of a number  n . A partition of a positive integer  n  is just an expression for  n  as a sum of positive integers, regardless of order. Thus p(4) = 5 because 4 can be written as 1+1+1+1, 1+1+2, 2+2, 1+3, or 4. Besides his published work, Ramanujan left behind several notebooks, which have been the object of much study. The English mathematician G. N. Watson wrote a long series of papers about them. More recently the American mathematician Bruce C. Berndt has written a multi-volume study of the notebooks. In 1997  The Ramanujan Journal  was launched to publish work "in areas of mathematics influenced by Ramanujan".

19. Calyampudi Radhakrishna Rao

20. Calyampudi Radhakrishna Rao was born to C.D. Naidu and A. Laxmikantamma on 10 September 1920 in Huvvina Hadagalli in present day Karnataka. He was the eighth in a family of 10 children. After his father’sretirement, the family settled down in Vishakapatnam in Andhra Pradesh. From his earliest years, Rao had an interest in mathematics. After completing high school he joined the Mrs. A.V.N. College at Vishakapatnam for the Intermediate course. He received his M.A. in Mathematics with first rank in 1940. Rao decided to pursue a research career in mathematics, but was denied a scholarship on the grounds of late submission of the application. He then went to Kolkata for an interview for a job. He did not get the job, but by chance he visited the Indian Statistical Institute, then located in a couple of rooms in the Physics Department of the Presidency College, Kolkata. He applied for a one-year training course at the Institute and was admitted to the Training Section of the Institute from 1 January 1941. In July 1941 he joined the M.A Statistics program of the Calcutta University. By the time he passed the M.A. exam in 1943, winning the gold medal of the University, he had already published some research papers! About -

21. The living legend and doyen of Indian Statistics, 91 year old Prof. Calyampudi Radhakrishna (C. R.) Rao was awarded the Guy Medal in Gold of the Royal Statistical Society, UK on the 29th of June, 2011 "For his fundamental contributions to statistical theory and methodology, including unbiased estimation, variance reduction by sufficiency, efficiency of estimation, information geometry, as well as the application of matrix theory in linear statistical inference", the announcement stated. The Gold Medal is awarded by the Royal Statistical Society (triennially, except the war period) and named after William Guy. There are Silver and Bronze Medals too, C. R. Rao already obtained the Silver Medal in 1965. Since 1892 he is the 34th recipient of the Gold Medal. Previously, R. A. Fisher (1946), E. S. Pearson (1955), J. Neyman (1966), M. S. Bartlett (1969), H. Cramér (1972), and D. Cox (1973) received this prize, just to mention a few. Among the recipients only H. Cramér and J. Neyman were outside Great Britain. C. R. Rao is the first non-European and non-American to receive the award. I believe that he has long deserved this prize. His formulae and theory include "Cramer -Rao inequality", "Fischer -Rao theorem" and "Rao - Blackwellisation". . In 1980, 18th June she solved the multiplication of 13 digit number 7,686,369,774,870 and 2,465,099,745,779 picked up by the computer science department of imperial college, London. Shakuntala solved the question in a flash and took 28 seconds to solve the entire problem, and her answer was 18,947,668,177,995,426,462,773,730. This amazing incident helped her get a place in the Guinness book of world record

22. Shakuntla devi

23. Shakuntala Devi was born on 4th of November, 1939 in Bengaluru in a well-known Brahmin priest family. She did card tricks with her father when she was only three. Shakuntala Devi received her early lessons in mathematics from her grandfather. By the age of 5, Shakuntala Devi became an expert in complex mental arithmetic and was recognised as a child prodigy. She demonstrated her talents to a large assembly of students and professors at the University of Mysore a year later. And when she was 8 years old, she demonstrated her talents at Annamalai University. Shakuntala Devi has authored a few books. She shares some of the methods of mental calculations in her world famous book, Figuring: The Joy of Numbers. Puzzles to puzzle You, More Puzzles to puzzle you, The Book of Numbers, Mathability: Awaken the Math Genius in Your Child, Astrology for you, Perfect Murder, In the Wonderland of Numbers are some of the popular books written by her. Her book, In the Wonderland of Numbers, talks about a girl Neha, and her fascination for numbers. About -

24. Her contributions … Shakuntala Devi was a genius and once in 1977 she mentally solved the 23rd root of a 201 digit number without any help from mechanical aid. She shares some of the methods of mental calculations in her world famous book, Figuring: The Joy of Numbers. Puzzles to puzzle You, More Puzzles to puzzle you, The Book of Numbers, Mathability: Awaken the Math Genius in Your Child, Astrology for you, Perfect Murder, In the Wonderland of Numbers are some of the popular books written by her. Her book, In the Wonderland of Numbers, talks about a girl Neha, and her fascination for numbers. She has been travelling around the globe performing for the student community, Prime Ministers, Presidents, Politicians and Educationalists.

25. Conclusion 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. Indians have significantly contributed in the field of mathematics and ,if God wills, they will do the same in the near future.

26. Thank You