D.R.Bhagat
India is a land of great mathematicians who gave new concepts and theories and contributed a lot in developing the mathematical sciences. Algebra, Trigonometry and Calculus, all had their roots in India. The ancient Indian Baudhayana who lived about three centuries before Pythagoras was the first to explain the Pythagoras Theorem. His other discoveries included to draw a circle and a square of the same area, calculation of the value of the square root of 2, and calculation of the value of Pi up to five decimals. Aryabhat first deduced that the earth was round. Brahamagupta , another Indian mathematician from Rajasthan who lived in 7th century was the first who explained how to find the cube and cube root of an integer. He also gave the concept of Zero and explained that it was a number in its own right place rather than as simply a placeholder digit. The number zero is also attributed to another 7th century Indian mathematician Bhaskara who was also first to explain the infinity. Sridhara who lived between 870-930 A.D has contributed in practical applications of algebra and was one of the first to give a formula for solving the quadratic equations. He also wrote a mathematical treatise’ Patigenta’ which covers the mathematical operations of addition, subtraction, multiplication, division and finding squares, cubes and their roots. Bhaskarcharya was another genius in mathematics who has written six books on arithmetic, algebra, trigonometry, calculus, geometry and astronomy. His work on calculus predates about 500 years of Newton and Leibnitz who are credited with the discovery of Calculus.
Among the modern mathematicians, the genius of Srinivasa Ramanunjan is unparalleled and unimaginable. He was born at Erode in Tamil Nadu on 22 December, 1887 in a poor family. One day the school teacher was asking some very simple questions of arithmetic to the students regarding operation of division. He asked” how many bananas would each boy get if three bananas are equally divided among three boys?”A student answered “one each”. Then he again asked, “If one thousand bananas are divided among one thousand boys equally, then how much bananas each boy would get?” The other student replied, “One each”. There were some other similar questions which were being asked and answered by the students in the class. Suddenly, little Ramanunjan who was just nine years , raised his hand and asked the teacher, Sir “If none of the bananas was divided among no boys, then how much would each boy get?”The whole class burst into laughter at what the students thought was a silly question. However, the teacher remained serious who was thoroughly impressed by the question. He told the boys that it was not a silly question but rather a profound one. Ramanunjan was indeed asking about the concept of infinity, which had baffled the mathematicians for centuries until the Indian mathematician Bhaskara put some light on its concept. He proved that zero divided by zero is neither zero nor one but infinity. The teacher explained the genius of Ramanunjan to the students who were astounded. Soon the word spread around about the intellect of Ramanunjan and students much senior to him began to contact him for clarification of their doubts in mathematical problems. At the age of 13, he got a book of trigonometry from a college student, which was not even a subject of his class. He studied the book thoroughly and mastered it within a short time. He even discovered some theorems and formulas of his own.
He passed his matriculation examination in first class in 1903 from a school in Kumbakonam and was awarded the scholarship for further studies. However, he failed in the first year of his college examination because due to his extreme love for mathematics, he ignored all other subjects. Hence, his scholarship was discontinued. His life took a decisive turn at the age of 15 when he got a book titled “A Synopsis of Elementary Results in Pure and Applied mathematics” by G.S. Carr. The book contained more than six thousand theorems without their solutions. This book created much of his interest in mathematics. He solved most of the problems on his own and also undertook deep research in mathematics. He discovered Bernaulli numbers which are extremely important in number theory and analysis. Although these were already discovered by Bernaulli, a great mathematician but his discovery was totally independent and he was not aware of this discovery. He also calculated the value of Euler Constant up to 15 decimals. This constant also recurs in analysis and number theory. He used to note down his results and jottings on loose paper sheets and at one time, he had two thousand of these sheets. He got married to a 10 year old girl S. Janki Ammal in 1909. His first paper was published in the Journal of Indian Mathematical Society in 1911. He was financially poor and was unable to purchase papers for his use. After marriage, an additional liability to support his wife also came upon him. Now he needed regular income to support his family and his research work. He approached several offices for a clerical job but was unable to do so. Luckily, one day he approached Francis Spring, the director of Madras Port Trust. He displayed his notebooks with theorems and jottings. He got him appointed in his office as a clerk on a monthly salary of Rs.25
In 1913, he came to know about G.H. Hardy, a renowned mathematician and a professor of mathematics at Cambridge University in England. He immediately wrote a letter to him and dispatched his 120 theorems along with the letter. When Hardy studied those theorems, he could not decide whether the sender was a crank person or a genius. He then took the help of Prof. J.E.Littlewood, another Professor of mathematics and then they came to know that the sender was indeed a genius. He wrote back to Ramanunjan asking him to come to Cambridge. He started his journey from Madras on 17 March 1914 and reached London on 14 April. He joined Hardy and both of them started collaboration in their research work. The notable works of this collaboration were the’ Partition Theory’, that counts the number of ways a natural number can be decomposed into smaller parts and the ‘Normal Order’ Method to analyse the behavior of additive arithmetical functions. They also worked on Divergent Theory of numbers. Prof. Hardy was much impressed by his work and the intrinsic intellect he possessed. He remarked that Ramanunjan played with mathematics like a child played with toys. Ramanunjan was awarded the B.A. degree by research, now known as Ph.D in 1916 for his work on highly composite numbers. This paper was also published in the Journal of the London Mathematical Society. He was elected as member to the London Mathematical Society in December, 1917. He was also elected as the Fellow of Royal Society of London in 1918 and was the second Indian who was bestowed with this honor and in the same year, the Fellow of the Trinity College Cambridge.
This self- taught prodigy was diagnosed with tuberculosis in 1917 and despite best treatment at England, he could not recover. In 1919, he returned to India and in spite of his illness continued his work on mathematics. In 1920, he wrote a last letter to Prof. Hardy incorporating there in a new theory of ‘Mock Theta Functions’ which is quite helpful in developing the 21st century mathematics. It was most unfortunate for India and the mathematical world at large that this most brilliant mathematician of 20th century departed from this world at very young age of 32 on April 26, 1920. His birthday is celebrated as National Mathematics Day. A film “The Man Who Knew Infinity” is dedicated to his life and his achievements which serves an inspiration young students of our country.
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