Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan (Tamil: சீனிவாச இராமானுஜன் or ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions.

Born and raised in Erode, Tamil Nadu, India, Ramanujan first encountered formal mathematics at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S L Loney. He had mastered them by age 12, and even discovered theorems of his own. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant. He received a scholarship to study at Government College in Kumbakonam, but lost it 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. In 1912 – 1913, he sent samples of his theorems to three academics at the University of Cambridge. Only G.H. Hardy recognized the brilliance of his work, subsequently inviting Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge, dying 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). Although a small number of these results were actually false and some were already known, most of his claims have now been proven correct. 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. However, some of his major discoveries have been rather slow to enter the mathematical mainstream. Recently, Ramanujan's formulae have found applications in crystallography and string theory. The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.

Ramanujan was born on 22 December 1887 in Erode, Tamil Nadu, India, 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 or Komal Ammal was a housewife and also sang at a local temple. 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 theThanjavur district who succumbed to the disease that year. 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, but both children died before their first birthdays.

On 1 October 1892, Ramanujan was enrolled at the local school. In March 1894, he was moved to a Telugu medium school. After his maternal grandfather lost his job as a court official in Kanchipuram, Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School. After 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 going to school. His family enlisted a local constable to make sure he attended school. Within six months, Ramanujan was back in Kumbakonam. 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. 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 finished first in the district. That year, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the first time.

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. 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. He completed mathematical exams in half the allotted time, and showed a familiarity with infinite series. When he was 16, Ramanujan came across the book A Synopsis of Elementary Results in Pure and Applied Mathematics by George S. Carr. This book was a collection of 5000 theorems, and it introduced Ramanujan to the world of mathematics. The next year, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal places. His peers of the time commented that they "rarely understood him" and "stood in respectful awe" of him.

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. He received a scholarship to study at Government College in Kumbakonam. 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. In August 1905, he ran away from home, heading towards Visakhapatnam. 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 Fine Arts degree 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.

On 14 July 1909, Ramanujan was married to a nine-year old bride, Janaki Ammal – in the branch of Hinduism to which Ramanujan belonged, marriage was a formal engagement that was consummated only after the bride turned 17 or 18, as per the traditional calendar. After the marriage, Ramanujan developed a hydrocele testis, an abnormal swelling of the tunica vaginalis, an internal membrane in the testicle. The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac. His family did not have the money for the operation, but in January 1910, a doctor volunteered to do the surgery for free. After his successful surgery, Ramanujan searched for a job. He stayed at friends' houses while he went door to door around the city of Madras (now Chennai) looking for a clerical position. To make some money, he tutored some students at Presidency College who were preparing for their F.A. exam. In late 1910, Ramanujan was sick again, possibly as a result of the surgery earlier in the year. He feared for his health, and even told his friend, R. Radakrishna Iyer, to "hand these [my mathematical notebooks] over to Professor Singaravelu Mudaliar [mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the Madras Christian College." After Ramanujan recovered and got back his notebooks from Iyer, he took a northbound train from Kumbakonam to Villupuram, a coastal city under French control.

He met deputy collector V. Ramaswami Iyer, who had recently founded the Indian Mathematical Society. Ramanujan, wishing for a job at the revenue department where Iyer worked, showed him his mathematics notebooks. As Iyer 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.

Iyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras. 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. 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. 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. 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 Ramaswami Iyer, had his work published in the Journal of Indian Mathematical Society.

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. Ramanujan wrote his first formal paper for the Journal on the properties of Bernoulli numbers. One property he discovered was that the denominators 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. 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.

Ramanujan later wrote another paper and also continued to provide problems in the Journal. In early 1912, he got a temporary job in the Madras Accountant General's office, with a 20 rupee a month salary. He lasted for only a few weeks. 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.

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". 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. 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.

Spring, Narayana Iyeru, 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. 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. 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. The first two professors, H.F. Baker and E.W. Hobson, returned Ramanujan's papers without comment. On 16 January 1913, Ramanujan wrote to G.H. Hardy. Coming from an unknown mathematician, the nine pages of mathematical wonder made Hardy originally view Ramanujan's manuscripts as a possible "fraud". Hardy recognized some of Ramanujan's formulae but others "seemed scarcely possible to believe." 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. It 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". After he saw Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy commented that the "[theorems] defeated me completely; I had never seen anything in the least like them before." He figured that Ramanujan's theorems "must be true, because, if they were not true, no one would have the imagination to invent them." 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." One colleague, E.H. Neville, later commented that "not one [theorem] could have been set in the most advanced mathematical examination in the world."

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." 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. In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land." Meanwhile, Ramanujan sent a letter packed with theorems to Hardy, writing, "I have found a friend in you who views my labour sympathetically."

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. 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. While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one paper, Ramanujan anticipated the work of a Polish mathematician who had published his work shortly after. 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 generalizations 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. 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 Namagiri commanded her "to stand no longer between her son and the fulfillment of his life's purpose."

Ramanujan boarded the S.S. Nevasa on 17 March 1914, and at 10 o'clock in the morning, the ship departed from Madras. He arrived in London on 14 April, with E.H. Neville waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took up residence on Whewell's Court, just a five-minute walk from Hardy's room. Hardy and Ramanujan began to take a look at Ramanujan's notebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems to be found in the notebooks. Hardy saw that some were wrong, some had already been discovered, while the rest were new breakthroughs. Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented "I can believe that he's at least a Jacobi", while Hardy said he "can compare him only with [Leonhard] Euler or Jacobi."

Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood and published a part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs and working styles. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man and relied very strongly on his intuition. While in England, Hardy tried his best to fill the gaps in Ramanujan's education without interrupting his spell of inspiration. Ramanujan was awarded a B.A. degree by research (this degree was later renamed PhD) in March 1916 for his work on highly composite numbers, which was published as a paper in the Journal of the London Mathematical Society. The paper was over 50 pages with different properties of such numbers proven. Hardy remarked that this was one of the most unusual papers seen in mathematical research at that time and that Ramanujan showed extraordinary ingenuity in handling it. On 6 December 1917, he was elected to the London Mathematical Society. He became a Fellow of the Royal Society in 1918, becoming the second Indian to do so, following Ardaseer Cursetjee in 1841, and he was the youngest Fellow in the entire history of the Royal Society. He was elected "for his investigation in Elliptic functions and the Theory of Numbers." On 13 October 1918, he became the first Indian to be elected a Fellow of Trinity College, Cambridge.

Plagued by health problems all throughout his life, living in a country far away from home, and obsessively involved with his mathematics, Ramanujan's health worsened in England, perhaps exacerbated by stress and by the scarcity of vegetarian food during the First World War. He was diagnosed with tuberculosis and a severe vitamin deficiency and was confined to a sanatorium.

Ramanujan returned to Kumbakonam, India in 1919 and died soon thereafter at the age of 32. His widow, S. Janaki Ammal, lived in Chennai (formerly Madras) until her death in 1994.

A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D.A.B. Young concluded that it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver. This is supported by the fact that Ramanujan had spent time in Madras, where the disease was widespread. He had two episodes of dysentery before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis. It was a difficult disease to diagnose, but once diagnosed, could have been readily cured.

Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man with pleasant manners. He lived a rather spartan life while at Cambridge. Ramanujan's first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to his family Goddess, Namagiri of Namakkal, and looked to her for inspiration in his work. He often said, "An equation for me has no meaning, unless it represents a thought of God." G.H. Hardy cites Ramanujan as remarking that all religions seemed equally true to him. Hardy further argued that Ramanujan's religiousness had been romanticised by Westerners and overstated — in reference to his belief, not practice — by Indian biographers. At the same time, he remarked on Ramanujan's strict observance of vegetarianism.