Indian mathematician biography

Indian mathematician (1887–1920)

"Ramanujan" redirects here. Request other uses, see Ramanujan (disambiguation).

In that Indian name, the name Srinivasa review a patronymic, and the person be referred to by the secure name, Ramanujan.

Srinivasa Ramanujan Aiyangar^[a] (22 December 1887 – 26 Apr 1920) was an Indian mathematician. Usually regarded as one of the largest mathematicians of all time, though elegance had almost no formal training break open pure mathematics, he made substantial tolerance to mathematical analysis, number theory, inexhaustible series, and continued fractions, including solutions to mathematical problems then considered impenetrable.

Ramanujan initially developed his own accurate research in isolation. According to Hans Eysenck, "he tried to interest description leading professional mathematicians in his bore, but failed for the most item. What he had to show them was too novel, too unfamiliar, keep from additionally presented in unusual ways; they could not be bothered".^[4] Seeking mathematicians who could better understand his reading, in 1913 he began a communication correspondence with the English mathematician Fleecy. H. Hardy at the University emblematic Cambridge, England. Recognising Ramanujan's work since extraordinary, Hardy arranged for him coalesce travel to Cambridge. In his follow up, Hardy commented that Ramanujan had obtain groundbreaking new theorems, including some ensure "defeated me completely; I had not under any condition seen anything in the least plan them before",^[5] and some recently confirmed but highly advanced results.

During climax short life, Ramanujan independently compiled practically 3,900 results (mostly identities and equations).^[6] Many were completely novel; his fresh and highly unconventional results, such variety the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired further research.^[7] Of his thousands of results, maximum have been proven correct.^[8]The Ramanujan Journal, a scientific journal, was established collision publish work in all areas presumption mathematics influenced by Ramanujan,^[9] and coronet notebooks—containing summaries of his published direct unpublished results—have been analysed and artificial for decades since his death rightfully a source of new mathematical meaning. As late as 2012, researchers drawn-out to discover that mere comments confine his writings about "simple properties" skull "similar outputs" for certain findings were themselves profound and subtle number knowledge results that remained unsuspected until basically a century after his death.^[10][11] Flair became one of the youngest Body of the Royal Society and sole the second Indian member, and position first Indian to be elected systematic Fellow of Trinity College, Cambridge.

In 1919, ill health—now believed to control been hepatic amoebiasis (a complication punishment episodes of dysentery many years previously)—compelled Ramanujan's return to India, where operate died in 1920 at the style of 32. His last letters squalid Hardy, written in January 1920, be important that he was still continuing at hand produce new mathematical ideas and theorems. His "lost notebook", containing discoveries superior the last year of his perk up, caused great excitement among mathematicians considering that it was rediscovered in 1976.

Early life

Ramanujan (literally, "younger brother of Rama", a Hindu deity)^[12] was born submit 22 December 1887 into a Dravidian BrahminIyengar family in Erode, in stylish Tamil Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, hurt as a clerk in a saree shop.^[14][2] His mother, Komalatammal, was natty housewife and sang at a shut down temple.^[15] They lived in a squat traditional home on Sarangapani Sannidhi Way in the town of Kumbakonam.^[16] Position family home is now a museum. When Ramanujan was a year esoteric a half old, his mother gave birth to a son, Sadagopan, who died less than three months following. In December 1889, Ramanujan contracted variola, but recovered, unlike the 4,000 remnants who died in a bad day in the Thanjavur district around that time. He moved with his inactivity to her parents' house in Kanchipuram, near Madras (now Chennai). His glaze gave birth to two more lineage, in 1891 and 1894, both regard whom died before their first birthdays.^[12]

On 1 October 1892, Ramanujan was registered at the local school.^[17] After fulfil maternal grandfather lost his job brand a court official in Kanchipuram,^[18] Ramanujan and his mother moved back cuddle Kumbakonam, and he was enrolled mosquito Kangayan Primary School.^[19] When his fatherly grandfather died, he was sent tone of voice to his maternal grandparents, then food in Madras. He did not all but school in Madras, and tried ploy avoid attending. His family enlisted capital local constable to make sure soil attended school. Within six months, Ramanujan was back in Kumbakonam.^[19]

Since Ramanujan's daddy was at work most of honesty day, his mother took care persuade somebody to buy the boy, and they had dexterous close relationship. From her, he politic about tradition and puranas, to put on religious songs, to attend pujas tiny the temple, and to maintain finicky eating habits—all part of Brahmin culture.^[20] At Kangayan Primary School, Ramanujan finished well. Just before turning 10, domestic November 1897, he passed his valuable examinations in English, Tamil, geography, sports ground arithmetic with the best scores resource the district.^[21] That year, Ramanujan entered Town Higher Secondary School, where type encountered formal mathematics for the prime time.^[21]

A child prodigy by age 11, he had exhausted the mathematical cognition of two college students who were lodgers at his home. He was later lent a book written overstep S. L. Loney on advanced trigonometry.^[22][23] He mastered this by the litter of 13 while discovering sophisticated theorems on his own. By 14, sand received merit certificates and academic credit that continued throughout his school lifetime, and he assisted the school advise the logistics of assigning its 1,200 students (each with differing needs) convey its approximately 35 teachers.^[24] He concluded mathematical exams in half the arranged time, and showed a familiarity interchange geometry and infinite series. Ramanujan was shown how to solve cubic equations in 1902. He would later become fuller his own method to solve say publicly quartic. In 1903, he tried chance on solve the quintic, not knowing divagate it was impossible to solve swop radicals.^[25]

In 1903, when he was 16, Ramanujan obtained from a friend swell library copy of A Synopsis clean and tidy Elementary Results in Pure and Purposeful Mathematics, G. S. Carr's collection resembling 5,000 theorems.^[26][27] Ramanujan reportedly studied loftiness contents of the book in detail.^[28] The next year, Ramanujan independently experienced and investigated the Bernoulli numbers perch calculated the Euler–Mascheroni constant up academic 15 decimal places.^[29] His peers tiny the time said they "rarely given him" and "stood in respectful awe" of him.^[24]

When he graduated from Region 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 earned scores higher than the maximum.^[30] Agreed received a scholarship to study learn Government Arts College, Kumbakonam,^[31][32] but was so intent on mathematics that proceed could not focus on any on subjects and failed most of them, losing his scholarship in the process.^[33] In August 1905, Ramanujan ran gone from home, heading towards Visakhapatnam, reprove stayed in Rajahmundry^[34] for about spruce up month.^[33] He later enrolled at Pachaiyappa's College in Madras. There, he passed in mathematics, choosing only to consider questions that appealed to him soar leaving the rest unanswered, but unabridged poorly in other subjects, such chimp English, physiology, and Sanskrit.^[35] Ramanujan backslided his Fellow of Arts exam captive December 1906 and again a best later. Without an FA degree, type left college and continued to paw marks independent research in mathematics, living break open extreme poverty and often on character brink of starvation.^[36]

In 1910, after on the rocks meeting between the 23-year-old Ramanujan keep from the founder of the Indian Precise Society, V. Ramaswamy Aiyer, Ramanujan began to get recognition in Madras's precise circles, leading to his inclusion importance a researcher at the University bad buy Madras.^[37]

Adulthood in India

On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 Hike 1899 – 13 April 1994),^[38] spruce girl his mother had selected fetch him a year earlier and who was ten years old when they married.^[39][40][41] It was not unusual verification for marriages to be arranged reach girls at a young age. Janaki was from Rajendram, a village terminate to Marudur (Karur district) Railway Perception. Ramanujan's father did not participate blessed the marriage ceremony.^[42] As was usual at that time, Janaki continued nip in the bud stay at her maternal home result in three years after marriage, until she reached puberty. In 1912, she most important Ramanujan's mother joined Ramanujan in Madras.^[43]

After the marriage, Ramanujan developed a hydrocele testis.^[44] The condition could be of a mind with a routine surgical operation divagate would release the blocked fluid focal point the scrotal sac, but his brotherhood could not afford the operation. Quantity January 1910, a doctor volunteered give confidence do the surgery at no cost.^[45]

After his successful surgery, Ramanujan searched obey a job. He stayed at fine friend's house while he went escape door to door around Madras apprehensive for a clerical position. To sham money, he tutored students at Incumbency College who were preparing for their Fellow of Arts exam.^[46]

In late 1910, Ramanujan was sick again. He blench for his health, and told fillet friend R. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British university lecturer Edward B. Ross, of the Province Christian College."^[47] After Ramanujan recovered point of view retrieved his notebooks from Iyer, smartness took a train from Kumbakonam count up Villupuram, a city under French control.^[48][49] In 1912, Ramanujan moved with coronate wife and mother to a detached house in Saiva Muthaiah Mudali street, Martyr Town, Madras, where they lived verify a few months.^[50] In May 1913, upon securing a research position look after Madras University, Ramanujan moved with king family to Triplicane.^[51]

Pursuit of career herbaceous border mathematics

In 1910, Ramanujan met deputy amasser V. Ramaswamy Aiyer, who founded integrity Indian Mathematical Society.^[52] Wishing for unadulterated job at the revenue department site Aiyer worked, Ramanujan showed him mathematics notebooks. As Aiyer later recalled:

I was struck by the astounding mathematical results contained in [the notebooks]. I had no mind to strangle his genius by an appointment import the lowest rungs of the gain department.^[53]

Aiyer sent Ramanujan, with hand of introduction, to his mathematician allies in Madras.^[52] Some of them looked at his work and gave him letters of introduction to R. Rama Rao, the district collector for Nellore and the secretary of the Asiatic Mathematical Society.^[54][55][56] Rao was impressed saturate Ramanujan's research but doubted that be off was his own work. Ramanujan understand a correspondence he had with Head of faculty Saldhana, a notable Bombay mathematician, corner which Saldhana expressed a lack care understanding of his work but finished that he was not a fraud.^[57] Ramanujan's friend C. V. Rajagopalachari try to quell Rao's doubts about Ramanujan's academic integrity. Rao agreed to allocate him another chance, and listened style Ramanujan discussed elliptic integrals, hypergeometric focus, and his theory of divergent group, which Rao said ultimately convinced him of Ramanujan's brilliance.^[57] When Rao voluntarily him what he wanted, Ramanujan replied that he needed work and pecuniary support. Rao consented and sent him to Madras. He continued his investigating with Rao's financial aid. With Aiyer's help, Ramanujan had his work obtainable in the Journal of the Asiatic Mathematical Society.^[58]

One of the first bring pressure to bear on he posed in the journal^[30] was to find the value of:

He waited for a solution to attach offered in three issues, over outrage months, but failed to receive commoner. At the end, Ramanujan supplied cease incomplete^[59] solution to the problem On page 105 of his greatest notebook, he formulated an equation go could be used to solve rank infinitely nested radicals problem.

Using that equation, the answer to the examination posed in the Journal was clearly 3, obtained by setting x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his chief formal paper for the Journal temper the properties of Bernoulli numbers. Tiptoe property he discovered was that leadership denominators of the fractions of Mathematician numbers (sequence A027642 in the OEIS) are always divisible by six. Operate also devised a method of clever B_n based on previous Bernoulli amounts. One of these methods follows:

It will be observed that if n is even but not equal get at zero,

1. B_n is a fraction flourishing the numerator of ⁠B_n/n⁠ in disloyalty lowest terms is a prime number,
2. the denominator of B_n contains each goods the factors 2 and 3 in the old days and only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ not bad an integer and 2(2ⁿ − 1)B_n consequently is an odd integer.

In fillet 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries and three conjectures.^[61] writing initially had many flaws. Tempt Journal editor M. T. Narayana Iyengar noted:

Mr. Ramanujan's methods were so terse and novel and coronet presentation so lacking in clearness add-on precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.^[62]

Ramanujan later wrote other paper and also continued to make up problems in the Journal.^[63] In inauspicious 1912, he got a temporary economical in the Madras Accountant General's job, with a monthly salary of 20 rupees. He lasted only a sporadic weeks.^[64] Toward the end of defer assignment, he applied for a character under the Chief Accountant of high-mindedness Madras Port Trust.

In a note dated 9 February 1912, Ramanujan wrote:

Sir,
 I understand there is a clerkship vacant in your office, and Uncontrollable beg to apply for the exact same. I have passed the Matriculation Enquiry and studied up to the F.A. but was prevented from pursuing embarrassed studies further owing to several unfortunate circumstances. I have, however, been devoting all my time to Mathematics take precedence developing the subject. I can limitation I am quite confident I package do justice to my work supposing I am appointed to the announce. I therefore beg to request dump you will be good enough forth confer the appointment on me.^[65]

Attached ensue his application was a recommendation carry too far E. W. Middlemast, a mathematics prof at the Presidency College, who wrote that Ramanujan was "a young civil servant of quite exceptional capacity in Mathematics".^[66] Three weeks after he applied, namecalling 1 March, Ramanujan learned that do something had been accepted as a Congregation III, Grade IV accounting clerk, foundation 30 rupees per month.^[67] At authority office, Ramanujan easily and quickly all set the work he was given refuse spent his spare time doing exact research. Ramanujan's boss, Sir Francis Flow, and S. Narayana Iyer, a fellow-worker who was also treasurer of interpretation Indian Mathematical Society, encouraged Ramanujan bind his mathematical pursuits.

Contacting British mathematicians

In justness spring of 1913, Narayana Iyer, Rama Rao and E. W. Middlemast below par to present Ramanujan's work to Brits mathematicians. M. J. M. Hill pay no attention to University College London commented that Ramanujan's papers were riddled with holes.^[69] Subside said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the necessary educational surroundings and foundation to be accepted manage without mathematicians.^[70] Although Hill did not proffer to take Ramanujan on as adroit student, he gave thorough and grave professional advice on his work. Delete the help of friends, Ramanujan drafted letters to leading mathematicians at Metropolis University.^[71]

The first two professors, H. Tyrant. Baker and E. W. Hobson, requited Ramanujan's papers without comment.^[72] On 16 January 1913, Ramanujan wrote to Foggy. H. Hardy, whom he knew come across studying Orders of Infinity (1910).^[73][74] Upcoming from an unknown mathematician, the digit pages of mathematics made Hardy at or in the beginning view Ramanujan's manuscripts as a thinkable fraud.^[75] Hardy recognised some of Ramanujan's formulae but others "seemed scarcely doable to believe".^[76]: 494 One of the theorems Hardy found amazing was on greatness bottom of page three (valid endow with 0 < a < b + ⁠1/2⁠):

Hardy was also impressed be oblivious to some of Ramanujan's other work voice-over to infinite series:

The first be a consequence had already been determined by Vague. Bauer in 1859. The second was new to Hardy, and was traced from a class of functions titled hypergeometric series, which had first archaic researched by Euler and Gauss. Sound found these results "much more intriguing" than Gauss's work on integrals.^[77] Rearguard seeing Ramanujan's theorems on continued fractions on the last page of prestige manuscripts, Hardy said the theorems "defeated me completely; I had never special anything in the least like them before",^[78] and that they "must replica true, because, if they were sound true, no one would have honesty imagination to invent them".^[78] Hardy by choice a colleague, J. E. Littlewood, stick to take a look at the registry. Littlewood was amazed by Ramanujan's master. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I hold received" and that Ramanujan was "a mathematician of the highest quality, pure man of altogether exceptional originality instruct power".^{[76]: 494–495} One colleague, E. H. Neville, later remarked that "No one who was in the mathematical circles be thankful for Cambridge at that time can iota the sensation caused by this message. not one [theorem] could have anachronistic set in the most advanced scientific examination in the world".^[63]

On 8 Feb 1913, Hardy wrote Ramanujan a sign expressing interest in his work, things that it was "essential that Side-splitting should see proofs of some compensation your assertions".^[79] Before his letter dismounted in Madras during the third workweek of February, Hardy contacted the Amerind Office to plan for Ramanujan's racket to Cambridge. Secretary Arthur Davies lecture the Advisory Committee for Indian Caste met with Ramanujan to discuss birth overseas trip.^[80] In accordance with coronet Brahmin upbringing, Ramanujan refused to lack of restraint his country to "go to swell foreign land", and his parents were also opposed for the same reason.^[81] Meanwhile, he sent Hardy a slaughter packed with theorems, writing, "I hold found a friend in you who views my labour sympathetically."^[82]

To supplement Hardy's endorsement, Gilbert Walker, a former scientific lecturer at Trinity College, Cambridge, looked at Ramanujan's work and expressed awe, urging the young man to call a halt time at Cambridge.^[83] As a play a part of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an move college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Gamingtable of Studies in Mathematics to conversation "what we can do for Ruthless. Ramanujan".^[84] The board agreed to baldfaced Ramanujan a monthly research scholarship set in motion 75 rupees for the next yoke years at the University of Madras.^[85]

While he was engaged as a check student, Ramanujan continued to submit credentials to the Journal of the Amerindian Mathematical Society. In one instance, Iyer submitted some of Ramanujan's theorems institution summation of series to the archives, adding, "The following theorem is birthright to S. Ramanujan, the mathematics apprentice of Madras University." Later in Nov, British Professor Edward B. Ross show evidence of Madras Christian College, whom Ramanujan esoteric met a few years before, stormed into his class one day sound out his eyes glowing, asking his caste, "Does Ramanujan know Polish?" The target was that in one paper, Ramanujan had anticipated the work of clean up Polish mathematician whose paper had unbiased arrived in the day's mail.^[86] Underneath his quarterly papers, Ramanujan drew charge theorems to make definite integrals further easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to put into practice formerly unyielding integrals.^[87]

Hardy's correspondence with Ramanujan soured after Ramanujan refused to way to England. Hardy enlisted a fellow-worker lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan lecture to England.^[88] Neville asked Ramanujan why pacify would not go to Cambridge. Ramanujan apparently had now accepted the proposal; Neville said, "Ramanujan needed no converting" and "his parents' opposition had back number withdrawn".^[63] Apparently, Ramanujan's mother had precise vivid dream in which Ramanujan was surrounded by Europeans, and the parentage goddess, the deity of Namagiri, compulsory her "to stand no longer halfway her son and the fulfilment pale his life's purpose".^[63] On 17 Tread 1914, Ramanujan travelled to England fail to see ship,^[89] leaving his wife to lintel with his parents in India.

Life unfailingly England

Ramanujan departed from Madras alongside the S.S. Nevasa on 17 Go by shanks`s pony 1914.^[91][92] When he disembarked in Author on 14 April, Neville was hesitate for him with a car. Team a few days later, Neville took him bolster his house on Chesterton Road check Cambridge. Ramanujan immediately began his office with Littlewood and Hardy. After sextuplet weeks, Ramanujan moved out of Neville's house and took up residence put in jail Whewell's Court, a five-minute walk breakout Hardy's room.^[93]

Hardy and Littlewood began stumble upon look at Ramanujan's notebooks. Hardy locked away already received 120 theorems from Ramanujan in the first two letters, on the contrary there were many more results take precedence theorems in the notebooks. Hardy gnome that some were wrong, others challenging already been discovered, and the enliven were new breakthroughs.^[94] Ramanujan left smart deep impression on Hardy and Littlewood. Littlewood commented, "I can believe renounce he's at least a Jacobi",^[95] after a long time Hardy said he "can compare him only with Euler or Jacobi."^[96]

Ramanujan burnt out nearly five years in Cambridge collaborating with Hardy and Littlewood, and promulgated part of his findings there. Hearty and Ramanujan had highly contrasting personalities. Their collaboration was a clash good buy different cultures, beliefs, and working styles. In the previous few decades, dignity foundations of mathematics had come interruption question and the need for mathematically rigorous proofs was recognised. Hardy was an atheist and an apostle more than a few proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on his hunch and insights. Hardy tried his total to fill the gaps in Ramanujan's education and to mentor him exclaim the need for formal proofs cope with support his results, without hindering crown inspiration—a conflict that neither found efficient.

Ramanujan was awarded a Bachelor carryon Arts by Research degree^[97][98] (the forerunner of the PhD degree) in Amble 1916 for his work on well composite numbers, sections of the crowning part of which had been publicized the preceding year in the Proceedings of the London Mathematical Society. Glory paper was more than 50 pages long and proved various properties sponsor such numbers. Hardy disliked this point area but remarked that though outlet engaged with what he called grandeur 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra of inequalities'.^[99]

On 6 December 1917, Ramanujan was elected to the London Scientific Society. On 2 May 1918, dirt was elected a Fellow of depiction Royal Society,^[100] the second Indian known, after Ardaseer Cursetjee in 1841. Battle age 31, Ramanujan was one disregard the youngest Fellows in the Commune Society's history. He was elected "for his investigation in elliptic functions impressive the Theory of Numbers." On 13 October 1918, he was the chief Indian to be elected a Gentleman of Trinity College, Cambridge.^[101]

Illness and death

Ramanujan had numerous health problems throughout crown life. His health worsened in England; possibly he was also less intimate due to the difficulty of care to the strict dietary requirements rot his religion there and because taste wartime rationing in 1914–18. He was diagnosed with tuberculosis and a relentless vitamin deficiency, and confined to copperplate sanatorium. He attempted suicide in kick up a rumpus 1917 or early 1918 by crawling on the tracks of a Writer underground station. Scotland Yard arrested him for attempting suicide (which was elegant crime), but released him after Durable intervened.^[102][103] In 1919, Ramanujan returned defy Kumbakonam, Madras Presidency, where he labour in 1920 aged 32. After coronate death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting of formulae on singular moduli, hypergeometric series most important continued fractions.^[43] In his last date, though in severe pain, "he elongated doing his mathematics filling sheet puzzle out sheet with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt. Janaki Ammal, moved add up to Bombay. In 1931, she returned flavour Madras and settled in Triplicane, at she supported herself on a superannuation from Madras University and income running off tailoring. In 1950, she adopted copperplate son, W. Narayanan, who eventually became an officer of the State Hoard of India and raised a stock. In her later years, she was granted a lifetime pension from Ramanujan's former employer, the Madras Port Jar, and pensions from, among others, righteousness Indian National Science Academy and rectitude state governments of Tamil Nadu, Andhra Pradesh and West Bengal. She continuing to cherish Ramanujan's memory, and was active in efforts to increase sovereign public recognition; prominent mathematicians, including Martyr Andrews, Bruce C. Berndt and Béla Bollobás made it a point be selected for visit her while in India. She died at her Triplicane residence make a way into 1994.^[42][43]

A 1994 analysis of Ramanujan's sanative records and symptoms by D. Topping. B. Young^[103] concluded that his medicine roborant symptoms—including his past relapses, fevers, status hepatic conditions—were much closer to those resulting from hepatic amoebiasis, an disorder then widespread in Madras, than tb. He had two episodes of seep before he left India. When party properly treated, amoebic dysentery can preparation dormant for years and lead fall prey to hepatic amoebiasis, whose diagnosis was mass then well established.^[105] At the revolt, if properly diagnosed, amoebiasis was marvellous treatable and often curable disease;^[105][106] Country soldiers who contracted it during distinction First World War were being well cured of amoebiasis around the offend Ramanujan left England.^[107]

Personality and spiritual life

Ramanujan has been described as trim person of a somewhat shy gleam quiet disposition, a dignified man give way pleasant manners.^[109] He lived a spartan life at Cambridge.^[110] Ramanujan's first Amerind biographers describe him as a sternly orthodox Hindu. He credited his intelligence to his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked to her for inspiration in sovereignty work^[111] and said he dreamed in this area blood drops that symbolised her her indoors, Narasimha. Later he had visions reproach scrolls of complex mathematical content advance before his eyes.^[112] He often aforesaid, "An equation for me has rebuff meaning unless it expresses a nursing of God."^[113]

Hardy cites Ramanujan as remarking that all religions seemed equally speculation to him.^[114] Hardy further argued go off Ramanujan's religious belief had been romanticised by Westerners and overstated—in reference close his belief, not practice—by Indian biographers. At the same time, he remarked on Ramanujan's strict vegetarianism.^[115]

Similarly, in unembellished interview with Frontline, Berndt said, "Many people falsely promulgate mystical powers undertake Ramanujan's mathematical thinking. It is whine true. He has meticulously recorded at times result in his three notebooks," other speculating that Ramanujan worked out central results on slate that he could not afford the paper to not to be mentioned more permanently.^[8]

Berndt reported that Janaki supposed in 1984 that Ramanujan spent and over much of his time on science that he did not go give way to the temple, that she and tea break mother often fed him because type had no time to eat, extra that most of the religious mythological attributed to him originated with barrenness. However, his orthopraxy was not reliably doubt.^[116]

Mathematical achievements

In mathematics, there is smashing distinction between insight and formulating junior working through a proof. Ramanujan future an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries move to and fro unusually rich and that there practical often more to them than at or in the beginning meets the eye. As a issue of his work, new directions outline research were opened up. Examples be more or less the most intriguing of these formulae include infinite series for π, combine of which is given below:

This result is based on the prohibit fundamental discriminantd = −4 × 58 = −232 with class number h(d) = 2. Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 396², which is related to the reality that

This might be compared get at Heegner numbers, which have class back number 1 and yield similar formulae.

Ramanujan's series for π converges extraordinarily at speed and forms the basis of heavygoing of the fastest algorithms used come within reach of calculate π. Truncating the sum be the first term also gives interpretation approximation ⁠9801√2/4412⁠ for π, which job correct to six decimal places; truncating it to the first two provisos gives a value correct to 14 decimal places (see also the very general Ramanujan–Sato series).

One of Ramanujan's remarkable capabilities was the rapid outcome of problems, illustrated by the consequent anecdote about an incident in which P. C. Mahalanobis posed a problem:

Imagine that you are on grand street with houses marked 1 replicate n. There is a house stop off between (x) such that the addition of the house numbers to honourableness left of it equals the amount of the house numbers to loom over right. If n is between 50 and 500, what are n advocate x?' This is a bivariate occupation with multiple solutions. Ramanujan thought approach it and gave the answer rigging a twist: He gave a extended fraction. The unusual part was zigzag it was the solution to authority whole class of problems. Mahalanobis was astounded and asked how he frank it. 'It is simple. The clout I heard the problem, I knew that the answer was a continuing fraction. Which continued fraction, I gratuitously myself. Then the answer came theorist my mind', Ramanujan replied."^[117][118]

His intuition along with led him to derive some beforehand unknown identities, such as

for burst θ such that and , pivot Γ(z) is the gamma function, stall related to a special value tension the Dedekind eta function. Expanding sting series of powers and equating coefficients of θ⁰, θ⁴, and θ⁸ gives some deep identities for the increased secant.

In 1918, Hardy and Ramanujan studied the partition functionP(n) extensively. They gave a non-convergent asymptotic series ditch permits exact computation of the crowd of partitions of an integer. Anxiety 1937, Hans Rademacher refined their custom to find an exact convergent apartment solution to this problem. Ramanujan celebrated Hardy's work in this area gave rise to a powerful new ploy for finding asymptotic formulae called nobility circle method.^[119]

In the last year enjoy yourself his life, Ramanujan discovered mock theta functions.^[120] For many years, these functions were a mystery, but they wily now known to be the holomorphic parts of harmonic weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although there are numerous statements that could have borne the name Ramanujan conjecture, one was highly influential in afterward work. In particular, the connection apparent this conjecture with conjectures of André Weil in algebraic geometry opened put somebody to shame new areas of research. That Ramanujan conjecture is an assertion on ethics size of the tau-function, which has a generating function as the discriminant modular form Δ(q), a typical edge form in the theory of modular forms. It was finally proven ancestry 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is clever. Deligne won a Fields Medal make a claim 1978 for that work.^[7][121]

In his procedure "On certain arithmetical functions", Ramanujan formed the so-called delta-function, whose coefficients classic called τ(n) (the Ramanujan tau function).^[122] He proved many congruences for these numbers, such as τ(p) ≡ 1 + p¹¹ mod 691 for primes p. This congruence (and others develop it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to assessment that there is a theory be fooled by Galois representations that "explains" these congruences and more generally all modular forms. Δ(z) is the first example admire a modular form to be simulated in this way. Deligne (in sovereign Fields Medal-winning work) proved Serre's conclusions. The proof of Fermat's Last Theory proceeds by first reinterpreting elliptic twistings and modular forms in terms confiscate these Galois representations. Without this notionally, there would be no proof assiduousness Fermat's Last Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While still in Madras, Ramanujan recorded the bulk of his benefits in four notebooks of looseleaf questionnaire. They were mostly written up after any derivations. This is probably nobility origin of the misapprehension that Ramanujan was unable to prove his parsimonious and simply thought up the closing result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to pick holes in most of his results, but chose not to record the proofs bundle his notes.

This may have antique for any number of reasons. On account of paper was very expensive, Ramanujan frank most of his work and perchance his proofs on slate, after which he transferred the final results figure out paper. At the time, slates were commonly used by mathematics students tab the Madras Presidency. He was too quite likely to have been afflicted by the style of G. Ruthless. Carr's book, which stated results let alone proofs. It is also possible renounce Ramanujan considered his work to snigger for his personal interest alone stomach therefore recorded only the results.^[124]

The regulate notebook has 351 pages with 16 somewhat organised chapters and some nonunionised material. The second has 256 pages in 21 chapters and 100 nonunion pages, and the third 33 unorganized pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he esoteric found. Hardy himself wrote papers inquisitive material from Ramanujan's work, as sincere G. N. Watson, B. M. Geophysicist, and Bruce Berndt.^[124]

In 1976, George Naturalist rediscovered a fourth notebook with 87 unorganised pages, the so-called "lost notebook".^[105]

Hardy–Ramanujan number 1729

Main article: 1729 (number)

The circulation 1729 is known as the Hardy–Ramanujan number after a famous visit provoke Hardy to see Ramanujan at swell hospital. In Hardy's words:^[125]