SRINIVASA RAMANUJAN AIYANGAR

 SRINIVASA RAMANUJAN AIYANGAR

S.PONGOMATHI,

 MATHEMATICS

Early Life and Education:

Srinivasa Ramanujan Aiyangar was an Indian Mathematician who was born in Erode, India in 1887 on December 22. He was born into a family that was not very well to do. He went to school at the nearby place, Kumbakonam. Ramanujan is very well known for his efforts on continued fractions and series of hypergeometry. When Ramanujan was thirteen, he could work out Loney’s Trigonometry exercises without any help. At the of fourteen, he was able to acquire the theorems of cosine and sine given by L. Euler. Synopsis of Elementary Results in Pure and Applied Mathematics by George Shoobridge Carr was reached by him in 1903. The book helped him a lot and opened new dimensions to him were opened which helped him introduce about 6,165 theorems for himself. As he had no proper and good books in his reach, he had to figure out on his own the solutions for all the questions. It was in this quest that he discovered many tremendous methods and new algebraic series.

In 1904, he received a merit scholarship in a local college and became more indulgent into mathematics. He lost his interest in all other subjects due to which he lost his scholarship. Even after two attempts, he did not succeed to get a first degree in the field of arts. In 1909, he got married and continued his clerical work and, side by side, his investigations of mathematics. Finally in 1911, he published some of his results.

It was in January 1913 that he sent his work to a Cambridge Professor named G. H. Hardy but he did not appreciate Ramanujan’s work much as he had not really done reached the standard of the mathematicians of the west. But he was given a scholarship in May by the University of Madras

                             

Contributions and Achievements:

Ramanujan went to Cambridge in 1914 and it helped him a lot but by that time his mind worked on the patterns on which it had worked before and he seldom adopted new ways. By then, it was more about intuition than argument. Hardy said Ramanujan could have become an outstanding mathematician if his skills had been recognized earlier. It was said about his talents of continued fractions and hypergeometric series that, “he was unquestionably one of the great masters.” It was due to his sharp memory, calculative mind, patience and insight that he was a great formalist of his days. But it was due to his some methods of working in the work analysis and theories of numbers that did not let him excel that much.

He got elected as the fellow in 1918 at the Trinity College at Cambridge and the Royal Society. He departed from this world on April 26, 1920.

Srinivasa Ramanujan

It is one of the most romantic stories in the history of mathematics: in 1913, the English mathematician G. H. Hardy received a strange letter from an unknown clerk in Madras, India. The ten-page letter contained about 120 statements of theorems on infinite series, improper integrals, continued fractions, and number theory (Here is a .dvi file with a sample of these results). Every prominent mathematician gets letters from cranks, and at first glance Hardy no doubt put this letter in that class. But something about the formulas made him take a second look, and show it to his collaborator J. E. Littlewood. After a few hours, they concluded that the results "must be true because, if they were not true, no one would have had the imagination to invent them".

Thus was Srinivasa Ramanujan (1887-1920) introduced to the mathematical world. Born in South India, Ramanujan was a promising student, winning academic prizes in high school. But at age 16 his life took a decisive turn after he obtained a book titled A Synopsis of Elementary Results in Pure and Applied Mathematics. The book was simply a compilation of thousands of mathematical results, most set down with little or no indication of proof. It was in no sense a mathematical classic; rather, it was written as an aid to coaching English mathematics students facing the notoriously difficult Tripos examination, which involved a great deal of wholesale memorization. But in Ramanujan it inspired a burst of feverish mathematical activity, as he worked through the book's results and beyond. Unfortunately, his total immersion in mathematics was disastrous for Ramanujan's academic career: ignoring all his other subjects, he repeatedly failed his college exams.

As a college dropout from a poor family, Ramanujan's position was precarious. He lived off the charity of friends, filling notebooks with mathematical discoveries and seeking patrons to support his work. Finally he met with modest success when the Indian mathematician Ramachandra Rao provided him with first a modest subsidy, and later a clerkship at the Madras Port Trust. During this period Ramanujan had his first paper published, a 17-page work on Bernoulli numbers that appeared in 1911 in the Journal of the Indian Mathematical Society. Still no one was quite sure if Ramanujan was a real genius or a crank. With the encouragement of friends, he wrote to mathematicians in Cambridge seeking validation of his work. Twice he wrote with no response; on the third try, he found Hardy.

Hardy wrote enthusiastically back to Ramanujan, and Hardy's stamp of approval improved Ramanujan's status almost immediately. Ramanujan was named a research scholar at the University of Madras, receiving double his clerk's salary and required only to submit quarterly reports on his work. But Hardy was determined that Ramanujan be brought to England. Ramanujan's mother resisted at first--high-caste Indians shunned travel to foreign lands--but finally gave in, ostensibly after a vision. In March 1914, Ramanujan boarded a steamer for England.

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 Euler or 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. The problem of finding p(n) was studied by Euler, who found a formula for the generating function of p(n) (that is, for the infinite series whose nth term is p(n)xⁿ). While this allows one to calculate p(n) recursively, it doesn't lead to an explicit formula. Hardy and Ramanujan came up with such a formula (though they only proved it works asymptotically; Rademacher proved it gives the exact value of p(n)).

Ramanujan's years in England were mathematically productive, and he gained the recognition he hoped for. Cambridge granted him a Bachelor of Science degree "by research" in 1916, and he was elected a Fellow of the Royal Society (the first Indian to be so honored) in 1918. But the alien climate and culture took a toll on his health. Ramanujan had always lived in a tropical climate and had his mother (later his wife) to cook for him: now he faced the English winter, and he had to do all his own cooking to adhere to his caste's strict dietary rules. Wartime shortages only made things worse. In 1917 he was hospitalized, his doctors fearing for his life. By late 1918 his health had improved; he returned to India in 1919. But his health failed again, and he died the next year.

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".

Achievements: Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. His most famous work was on the number p(n) of partitions of an integer n into summands.

Srinivasa Ramanujan was a mathematician par excellence. He is widely believed to be the greatest mathematician of the 20th Century. Srinivasa Ramanujan made significant contribution to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infiniteseries.

Srinivasa Aiyangar Ramanujan was born on December 22, 1887 in Erode, Tamil Nadu. His father worked in Kumbakonam as a clerk in a cloth merchant's shop. At the of five Ramanujan went to primary school in Kumbakonam. In 1898 at age 10, he entered the Town High School in Kumbakonam. At the age of eleven he was lent books on advanced trigonometry written by S. L. Loney by two lodgers at his home who studied at the Government college. He mastered them by the age of thirteen. Ramanujan was a bright student, winning academic prizes in high school.

At age of 16 his life took a decisive turn after he obtained a book titled" A Synopsis of Elementary Results in Pure and Applied Mathematics". The book was simply a compilation of thousands of mathematical results, most set down with little or no indication of proof. The book generated Ramanujan's interest in mathematics and he worked through the book's results and beyond. 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. He was given a scholarship to the Government College in Kumbakonam which he entered in 1904. But he neglected his subjects at the cost of mathematics and failed in college examination. He dropped out of the college.

Ramanujan lived off the charity of friends, filling notebooks with mathematical discoveries and seeking patrons to support his work. 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. 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.

On 14 July 1909 Ramanujan marry a ten year old girl S Janaki Ammal. During this period Ramanujan had his first paper published, a 17-page work on Bernoulli numbers that appeared in 1911 in the Journal of the Indian Mathematical Society. In 191,1 Ramanujan approached the founder of the Indian Mathematical Society for advice on a job. He got the job of clerk at the Madras Port Trust with the help of Indian mathematician Ramachandra Rao.

The professor of civil engineering at the Madras Engineering College C L T Griffith was interested in Ramanujan's abilities and, having been educated at University College London, knew the professor of mathematics there, namely M J M Hill. He wrote to Hill on 12 November 1912 sending some of Ramanujan's work and a copy of his 1911 paper on Bernoulli numbers. Hill replied in a fairly encouraging way but showed that he had failed to understand Ramanujan's results on divergent series. In January 1913 Ramanujan wrote to G H Hardy having seen a copy of his 1910 book Orders of infinity. Hardy, together with Littlewood, studied the long list of unproved theorems which Ramanujan enclosed with his letter. Hardy wrote back to Ramanujan a evinced interest in his work.

University of Madras gave Ramanujan a scholarship in May 1913 for two years and, in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration. Right from the start Ramanujan's collaboration with Hardy led to important results. In a joint paper with Hardy, Ramanujan gave an asymptotic formula for p(n). It had the remarkable property that it appeared to give the correct value of p(n), and this was later proved by Rademacher.

Ramanujan had problems settling in London. He was an orthodox Brahmin and right from the beginning 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.

On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research. He had been allowed to enrol 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 February 18, 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and later he was also elected as a fellow of the Royal Society of London. 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 on April 26, 1920.