Srinivasa Ramanujan

Srinivasa Ramanujan (22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. Living in India with no access to the larger mathematical community, which was centred in Europe at the time, Ramanujan developed his own mathematical research in isolation. As a result, he rediscovered known theorems in addition to producing new work. Ramanujan was said to be a natural genius by the English mathematician G. H. Hardy, in the same league as mathematicians such as Euler and Gauss.^[1] He died at the age of 32.

Ramanujan was born at Erode, Madras Presidency (now Tamil Nadu) in a Tamil Brahmin family of Thenkalai Iyengar sect.^[2][3][4] His introduction to formal mathematics began at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney that he mastered by the age of 12; he even discovered theorems of his own, and re-discovered Euler's identity independently.^[5] He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan had conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant.

Ramanujan received a scholarship to study at Government College in Kumbakonam, which was later rescinded when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself.^[6] In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge. G. H. Hardy, recognizing the brilliance of his work, invited Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge. Ramanujan died of illness, malnutrition, and possibly liver infection in 1920 at the age of 32.

During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identities and equations).^[7] Nearly all his claims have now been proven correct, although a small number of these results were actually false and some were already known.^[8] He stated results that were both original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research.^[9] However, the mathematical mainstream has been rather slow in absorbing some of his major discoveries.^{[citation needed]} The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.^[10]

In December 2011, in recognition of his contribution to mathematics, the Government of India declared that Ramanujan's birthday (22 December) should be celebrated every year as National Mathematics Day, and also declared 2012 the National Mathematics Year.

Ramanujan was born on 22 December 1887 in Erode, Madras Presidency (now Tamil Nadu), at the residence of his maternal grandparents.^[13] His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and hailed from the district of Thanjavur.^[14] His mother, Komalatammal, was a housewife and also sang at a local temple.^[15] They lived in Sarangapani Street in a traditional home in the town of Kumbakonam. The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son named Sadagopan, who died less than three months later. In December 1889, Ramanujan had smallpox and recovered, unlike thousands in the Thanjavur District who died from the disease that year.^[16] He moved with his mother to her parents' house in Kanchipuram, near Madras (now Chennai). In November 1891, and again in 1894, his mother gave birth to two children, but both children died in infancy.

On 1 October 1892, Ramanujan was enrolled at the local school.^[17] In March 1894, he was moved to a Telugu medium school. After his maternal grandfather lost his job as a court official in Kanchipuram,^[18] Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School.^[19] When his paternal grandfather died, he was sent back to his maternal grandparents, who were now living in Madras. He did not like school in Madras, and he tried to avoid attending. His family enlisted a local constable to make sure he attended school. Within six months, Ramanujan was back in Kumbakonam.^[19]

Since Ramanujan's father was at work most of the day, his mother took care of him as a child. He had a close relationship with her. From her, he learned about tradition and puranas. He learned to sing religious songs, to attend pujas at the temple and particular eating habits – all of which are part of Brahmin culture.^[20] At the Kangayan Primary School, Ramanujan performed well. Just before the age of 10, in November 1897, he passed his primary examinations in English, Tamil, geography and arithmetic. With his scores, he stood first in the district.^[21] That year, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the first time.^[21]

By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book on advanced trigonometry written by S. L. Loney.^[5][22] He completely mastered this book by the age of 13 and discovered sophisticated theorems on his own. By 14, he was receiving merit certificates and academic awards which continued throughout his school career and also assisted the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers.^[23] He completed mathematical exams in half the allotted time, and showed a familiarity with geometry and infinite series. Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.

In 1903 when he was 16, Ramanujan obtained from a friend a library-loaned copy of a book by G. S. Carr.^[24][25] The book was titled A Synopsis of Elementary Results in Pure and Applied Mathematics and was a collection of 5000 theorems. Ramanujan reportedly studied the contents of the book in detail.^[26] The book is generally acknowledged as a key element in awakening the genius of Ramanujan.^[26] The next year, he had independently developed and investigated the Bernoulli numbers and had calculated the Euler–Mascheroni constant up to 15 decimal places.^[27] His peers at the time commented that they "rarely understood him" and "stood in respectful awe" of him.^[23]

When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than the maximum possible marks.^[23] He received a scholarship to study at Government Arts College, Kumbakonam,^[28][29] However, Ramanujan was so intent on studying mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process.^[30] In August 1905, he ran away from home, heading towards Visakhapatnam and stayed in Rajahmundry ^[31] for about a month.^[32] He later enrolled at Pachaiyappa's College in Madras. He again excelled in mathematics but performed poorly in other subjects such as physiology. Ramanujan failed his Fellow of Arts exam in December 1906 and again a year later. Without a degree, he left college and continued to pursue independent research in mathematics. At this point in his life, he lived in extreme poverty and was often on the brink of starvation

Attention towards mathematics

Ramanujan met deputy collector V. Ramaswamy Aiyer, who had recently founded the Indian Mathematical Society.^[42] Ramanujan, wishing for a job at the revenue department where Ramaswamy Aiyer worked, showed him his mathematics notebooks. As Ramaswamy Aiyer later recalled:

I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.^[43]

Ramaswamy Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras.^[42] Some of these friends looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Mathematical Society.^[44][45][46] Ramachandra Rao was impressed by Ramanujan's research but doubted that it was actually his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understanding for his work but concluded that he was not a phony.^[47] Ramanujan's friend, C. V. Rajagopalachari, persisted with Ramachandra Rao and tried to quell any doubts over Ramanujan's academic integrity. Rao agreed to give him another chance, and he listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, which Rao said ultimately "converted" him to a belief in Ramanujan's mathematical brilliance.^[47] When Rao asked him what he wanted, Ramanujan replied that he needed some work and financial support. Rao consented and sent him to Madras. He continued his mathematical research with Rao's financial aid taking care of his daily needs. Ramanujan, with the help of Ramaswamy Aiyer, had his work published in the Journal of the Indian Mathematical Society.^[48]

One of the first problems he posed in the journal was:

He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied the solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely nested radicals problem.

Using this equation, the answer to the question posed in the Journal was simply 3.^[49] Ramanujan wrote his first formal paper for the Journal on the properties of Bernoulli numbers. One property he discovered was that the denominators (sequence A027642 in OEIS) of the fractions of Bernoulli numbers were always divisible by six. He also devised a method of calculating B_n based on previous Bernoulli numbers. One of these methods went as follows:

It will be observed that if n is even but not equal to zero,
(i) B_n is a fraction and the numerator of in its lowest terms is a prime number,
(ii) the denominator of B_n contains each of the factors 2 and 3 once and only once,
(iii) is an integer and consequently is an odd integer.

In his 17-page paper, "Some Properties of Bernoulli's Numbers", Ramanujan gave three proofs, two corollaries and three conjectures.^[50] Ramanujan's writing initially had many flaws. As Journal editor M. T. Narayana Iyengar noted:

Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.^[51]

Ramanujan later wrote another paper and also continued to provide problems in the Journal.^[52] In early 1912, he got a temporary job in the Madras Accountant General's office, with a salary of 20 rupees per month. He lasted for only a few weeks.^[53] Toward the end of that assignment he applied for a position under the Chief Accountant of the Madras Port Trust. In a letter dated 9 February 1912, Ramanujan wrote:

Sir,
I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.^[54]

Attached to his application was a recommendation from E. W. Middlemast, a mathematics professor at the Presidency College, who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics".^[55] Three weeks after he had applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making 30 rupees per month.^[56] At his office, Ramanujan easily and quickly completed the work he was given, so he spent his spare time doing mathematical research. Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits.

Contacting English mathematicians

In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. One mathematician, M. J. M. Hill of University College London, commented that Ramanujan's papers were riddled with holes.^[57] He said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the educational background and foundation needed to be accepted by mathematicians.^[58] Although Hill did not offer to take Ramanujan on as a student, he did give thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.^[59]

The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without comment.^[60] On 16 January 1913, Ramanujan wrote to G. H. Hardy. Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible "fraud".^[61] Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe".^[62] One of the theorems Hardy found so incredible was found on the bottom of page three (valid for 0 < a < b + 1/2):

Hardy was also impressed by some of Ramanujan's other work relating to infinite series:

The first result had already been determined by a mathematician named Bauer. The second one was new to Hardy, and was derived from a class of functions called a hypergeometric series which had first been researched by Leonhard Euler and Carl Friedrich Gauss. Compared to Ramanujan's work on integrals, Hardy found these results "much more intriguing".^[63] After he saw Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy commented that "they [theorems] defeated me completely; I had never seen anything in the least like them before".^[64] He figured that Ramanujan's theorems "must be true, because, if they were not true, no one would have the imagination to invent them".^[64] Hardy asked a colleague, J. E. Littlewood, to take a look at the papers. Littlewood was amazed by the mathematical genius of Ramanujan. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and commented that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power".^[65] One colleague, E. H. Neville, later commented that "not one [theorem] could have been set in the most advanced mathematical examination in the world".^[66]

On 8 February 1913, Hardy wrote a letter to Ramanujan, expressing his interest for his work. Hardy also added that it was "essential that I should see proofs of some of your assertions".^[67] Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to plan for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip.^[68] In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land".^[69] Meanwhile, Ramanujan sent a letter packed with theorems to Hardy, writing, "I have found a friend in you who views my labour sympathetically."^[70]

To supplement Hardy's endorsement, a former mathematical lecturer at Trinity College, Cambridge, Gilbert Walker, looked at Ramanujan's work and expressed amazement, urging him to spend time at Cambridge.^[71] As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan". The board agreed to grant Ramanujan a research scholarship of 75 rupees per month for the next two years at the University of Madras.^[73] While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one instance, Narayana Iyer submitted some theorems of Ramanujan on summation of series to the above mathematical journal adding “The following theorem is due to S. Ramanujan, the mathematics student of Madras University”. Later in November, British Professor Edward B. Ross of Madras Christian College, whom Ramanujan had met a few years before, stormed into his class one day with his eyes glowing, asking his students, “Does Ramanujan know Polish?” The reason was that in one paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had just arrived by the day’s mail. In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to evaluate formerly unyielding integrals.

Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England.^[76] Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparently had now accepted the proposal; as Neville put it, "Ramanujan needed no converting and that his parents' opposition had been withdrawn". Apparently, Ramanujan's mother had a vivid dream in which the family Goddess, the deity of Namagiri, commanded her "to stand no longer between her son and the fulfilment of his life's purpose". Ramanujan then set sail for England, leaving his wife to stay with his parents in India.