Introduction
The teacher was asking some simple questions in arithmetic. The class was learning the simple operation of division. When the teacher asked how many bananas would each boy get if three bananas were divided equally among three boys, someone had an answer. One each. Thousand bananas divided equally among thousand boys? The answer was still the same. One. The class was progressing thus, question being asked by the teacher and answers being provided by the student. But there was a boy who had a question. If none of the bananas was divided among no boys, how much would each boy get?
The whole class burst into laughter at what the students thought was a fast one or a silly question. But the teacher seemed to have been impressed. He took it upon himself to explain to the boys that what the student asked was not a silly question. But the teacher seemed to have been impressed. He took it upon himself to explain to the boys that what the student had asked was not a silly question but rather a profound one. He was questioning the teacher about the concept of infinity. A concept that had baffled mathematicians for centuries, until the Indian scientist Bhaskara had provided some light. He had proved that zero divided by zero nor one, but infinity. The student was Srinivasa Ramanujan, the genius who introduced the concept of zero to the world.
LIFE OF SRINIVASA RAMANUJAN
Srinivasa Iyengar Ramanujan, popularly known as S. Ramanujan was a great mathematician from India. He was born December 22, 1887 in Erode, Madras Presidency at the residence of his maternal grandparents. His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and hailed from the district of Thanjavur. His mother, Komalatammal, was a housewife and also sang at a local temple. They lived in Sarangapani Street in a traditional home in the town of Kumbakonam. When he was nearly five years old, Ramanujan entered the primary school in Kumbakonam although he would attend several different primary schools before entering the Town High School in Kumbakonam in January 1898. At the Town High School, Ramanujan was to do well in his entire school subject and showed himself an able all round scholar. In 1900 he began to work on his own on mathematics summing geometric and arithmetic series. It was in the Town High School that Ramanujan came across a mathematics book by G S Carr called Synopsis of elementary results in pure mathematics.
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This book, with its very concise style, allowed Ramanujan to teach himself mathematics, but the style of the book was to have a rather unfortunate effect on the way Ramanujan was later to write down mathematics since it provided the only model that he had of written mathematical arguments. The book contained theorems, formulae and short proofs. It also contained an index to papers on pure mathematics which had been published in the European Journals of Learned Societies during the first half of the 19th century. The book, published in 1856, was of course well out of date by the time Ramanujan used it. By 1904 Ramanujan had begun to undertake deep research. He investigated the series ∑(1/n) and calculated Euler’s constant to 15 decimal places.
He began to study the Bernoulli numbers, although this was entirely his own independent discovery. Ramanujan, on the strength of his good school work, was given a scholarship to the Government College in Kumbakonam which he entered in 1904. However the following year his scholarship was not renewed because Ramanujan devoted more and more of his time to mathematics and neglected his other subjects. Without money he was soon in difficulties and, without telling his parents, he ran away to the town of Vizagapatnam about 650 km north of Madras. He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions. In 1906 Ramanujan went to Madras where he entered Pachaiyappa’s College. His aim was to pass the First Arts examination which would allow him to be admitted to the University of Madras.
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He attended lectures at Pachaiyappa’s College but became ill after three months study. He took the First Arts examination after having left the course. He passed in mathematics but failed all his other subjects and therefore failed the examination. This meant that he could not enter the University of Madras. In the following years he worked on mathematics developing his own ideas without any help and without any real idea of the then current research topics other than that provided by Carr’s book. Continuing his mathematical work Ramanujan studied continued fractions and divergent series in 1908. At this stage he became seriously ill again and underwent an operation in April 1909 after which he took him some considerable time to recover. He married on 14 July 1909 when his mother arranged for him to marry a ten year old girl S Janaki Ammal.
Ramanujan did not live with his wife, however, until she was twelve years old. Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He devoloped relations between elliptic modular equations in 1910. After publication of a brilliant research paper on Bernoulli numbers in 1911 in the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius. In 1911 Ramanujan approached the founder of the Indian Mathematical Society for advice on a job. After this he was appointed to his first job, a temporary post in the Accountant General’s Office in Madras.
It was then suggested that he approach Ramachandra Rao who was a Collector at Nellore. Ramachandra Rao was a founder member of the Indian Mathematical Society who had helped start the mathematics library. Ramachandra Rao told him to return to Madras and he tried, unsuccessfully, to arrange a scholarship for Ramanujan. In 1912 Ramanujan applied for the post of clerk in the accounts section of the Madras Port Trust. Despite the fact that he had no university education, Ramanujan was clearly well known to the university mathematicians in Madras for, with his letter of application, Ramanujan included a reference from E W Middlemast who was the Professor of Mathematics at The Presidency College in Madras. Indeed the University of Madras did give Ramanujan a scholarship in May 1913 for two years and, in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration.
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Setting this up was not an easy matter. Ramanujan was an orthodox Brahmin and so was a strict vegetarian. His religion should have prevented him from travelling but this difficulty was overcome, partly by the work of E H Neville who was a colleague of Hardy’s at Trinity College and who met with Ramanujan while lecturing in India. Ramanujan sailed from India on 17 March 1914. It was a calm voyage except for three days on which Ramanujan was seasick. He arrived in London on 14 April 1914 and was met by Neville.
After four days in London they went to Cambridge and Ramanujan spent a couple of weeks in Neville’s home before moving into rooms in Trinity College on 30th April. Right from the beginning, however, he had problems with his diet. The outbreak of World War I made obtaining special items of food harder and it was not long before Ramanujan had health problems. Right from the start Ramanujan’s collaboration with Hardy led to important results. Hardy was, however, unsure how to approach the problem of Ramanujan’s lack of formal education.
The war soon took Littlewood away on war duty but Hardy remained in Cambridge to work with Ramanujan. Even in his first winter in England, Ramanujan was ill and he wrote in March 1915 that he had been ill due to the winter weather and had not been able to publish anything for five months. What he did publish was the work he did in England, the decision having been made that the results he had obtained while in India, many of which he had communicated to Hardy in his letters, would not be published until the war had ended. On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research (the degree was called a Ph.D. from 1920).
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He had been allowed to enroll in June 1914 despite not having the proper qualifications.
Ramanujan’s dissertation was on Highly composite numbers and consisted of seven of his papers published in England. Ramanujan fell seriously ill in 1917 and his doctors feared that he would die. He did improve a little by September but spent most of his time in various nursing homes. On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London. He had been proposed by an impressive list of mathematicians, namely Hardy, MacMahon, Grace, Larmor, Bromwich, Hobson, Baker, Littlewood, Nicholson, Young, Whittaker, Forsyth and Whitehead.
His election as a fellow of the Royal Society was confirmed on 2 May 1918, and then on 10 October 1918 he was elected a Fellow of Trinity College Cambridge, the fellowship to run for six years. The honors which were bestowed on Ramanujan seemed to help his health improve a little and he renewed his efforts at producing mathematics. By the end of November 1918 Ramanujan’s health had greatly improved. Ramanujan sailed to India on 27 February 1919 arriving on 13 March. However his health was very poor and, despite medical treatment, he died there the following year.
CONTRIBUTIONS TO THE DEVELOPMENT OF MATHEMATICS
Ramanujan worked out the Riemann series, the elliptic integrals, hyper geometric series and functional equations of the zeta function. Ramanujan’s own work on partial sums and products of hyper geometric series have led to major development in the topic , Perhaps his most famous work was on the number p(n) of partitions of an integer n into summands. He made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. The number theory is the abstract study of the structure of number systems and properties of positive integers. It includes various theorems about prime numbers (a prime number is an integer greater than one that has not integral factor).
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Number theory includes analytic number theory, originated by Leonhard Euler (1707-89); Geometric theory – which uses such geometrical methods of analysis as Cartesian co-ordinates, vectors and matrices; and probabilistic number theory based on probability theory. What Ramanujan did will be fully understood by a very few.
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