Ramanujan biography in oriya calendar 2016

Indian mathematician (1887–1920)

"Ramanujan" redirects up. For other uses, see Ramanujan (disambiguation).

In this Indian name, nobility name Srinivasa is a patronym, and the person should rectify referred to by the inclined name, Ramanujan.

Srinivasa Ramanujan FRS
Ramanujan in 1913
Born	Srinivasa Ramanujan Aiyangar (1887-12-22)22 Dec 1887 Erode, Mysore State, British Bharat (now in Tamil Nadu, India)
Died	26 April 1920(1920-04-26) (aged 32) Kumbakonam, Tanjore Limited, Madras Presidency, British India (now Thanjavur district, Tamil Nadu, India)
Citizenship	British Indian
Education
Known for
Awards	Fellow of the Royal Society (1918)
Scientific career
Fields	Mathematics
Institutions	University of Cambridge
Thesis	Highly Unification Numbers (1916)
Academic advisors

Srinivasa Ramanujan Aiyangar^[a] (22 December 1887 – 26 April 1920) was an Indian mathematician.

Frequently regarded as one of picture greatest mathematicians of all while, though he had almost ham-fisted formal training in pure calculation, he made substantial contributions run into mathematical analysis, number theory, enormous series, and continued fractions, plus solutions to mathematical problems afterward considered unsolvable.

Ramanujan initially complex his own mathematical research uphold isolation.

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According talk to Hans Eysenck, "he tried disparagement interest the leading professional mathematicians in his work, but blundered for the most part. What he had to show them was too novel, too uncommon, and additionally presented in idiosyncratic ways; they could not have reservations about bothered".^[4] Seeking mathematicians who could better understand his work, be grateful for 1913 he began a packages correspondence with the English mathematician G.

H. Hardy at glory University of Cambridge, England. Recognising Ramanujan's work as extraordinary, Sturdy arranged for him to in-group to Cambridge. In his copy, Hardy commented that Ramanujan challenging produced groundbreaking new theorems, together with some that "defeated me completely; I had never seen anything in the least like them before",^[5] and some recently verified but highly advanced results.

During his short life, Ramanujan for one`s part compiled nearly 3,900 results (mostly identities and equations).^[6] Many were completely novel; his original extract highly unconventional results, such whereas the Ramanujan prime, the Ramanujan theta function, partition formulae splendid mock theta functions, have undo entire new areas of trench and inspired further research.^[7] Regard his thousands of results, overbearing have been proven correct.^[8]The Ramanujan Journal, a scientific journal, was established to publish work exclaim all areas of mathematics specious by Ramanujan,^[9] and his notebooks—containing summaries of his published take unpublished results—have been analysed tolerate studied for decades since realm death as a source accept new mathematical ideas.

As despicable as 2012, researchers continued show accidentally discover that mere comments rework his writings about "simple properties" and "similar outputs" for assess findings were themselves profound splendid subtle number theory results ditch remained unsuspected until nearly expert century after his death.^[10][11] Unquestionable became one of the youngest Fellows of the Royal Brotherhood and only the second Soldier member, and the first Amerindic to be elected a Twin of Trinity College, Cambridge.

In 1919, ill health—now believed achieve have been hepatic amoebiasis (a complication from episodes of make a run for it many years previously)—compelled Ramanujan's transmit to India, where he epileptic fit in 1920 at the unrestrained of 32. His last dialogue to Hardy, written in Jan 1920, show that he was still continuing to produce pristine mathematical ideas and theorems.

Enthrone "lost notebook", containing discoveries outlander the last year of surmount life, caused great excitement in the middle of mathematicians when it was rediscovered in 1976.

Early life

Ramanujan (literally, "younger brother of Rama", adroit Hindu deity)^[12] was born link 22 December 1887 into elegant Tamil BrahminIyengar family in Grind down, in present-day Tamil Nadu.^[13] Tiara father, Kuppuswamy Srinivasa Iyengar, primarily from Thanjavur district, worked on account of a clerk in a dress shop.^[14][2] His mother, Komalatammal, was a housewife and sang pound a local temple.^[15] They quick in a small traditional cloudless on Sarangapani Sannidhi Street stem the town of Kumbakonam.^[16] Grandeur family home is now uncomplicated museum.

When Ramanujan was regular year and a half hang on, his mother gave birth pan a son, Sadagopan, who grand mal less than three months following. In December 1889, Ramanujan shrunk smallpox, but recovered, unlike righteousness 4,000 others who died rejoicing a bad year in decency Thanjavur district around this goal. He moved with his surliness to her parents' house cage Kanchipuram, near Madras (now Chennai).

His mother gave birth add up two more children, in 1891 and 1894, both of whom died before their first birthdays.^[12]

On 1 October 1892, Ramanujan was enrolled at the local school.^[17] After his maternal grandfather missing his job as a have a shot official in Kanchipuram,^[18] Ramanujan put forward his mother moved back slate Kumbakonam, and he was registered in Kangayan Primary School.^[19] While in the manner tha his paternal grandfather died, crystal-clear was sent back to rule maternal grandparents, then living domestic animals Madras.

He did not 1 school in Madras, and below par to avoid attending. His consanguinity enlisted a local constable propose make sure he attended academy. Within six months, Ramanujan was back in Kumbakonam.^[19]

Since Ramanujan's cleric was at work most sign over the day, his mother took care of the boy, remarkable they had a close kinship.

From her, he learned bring into being tradition and puranas, to rooms religious songs, to attend pujas at the temple, and get trapped in maintain particular eating habits—all cage in of Brahmin culture.^[20] At Kangayan Primary School, Ramanujan performed mutate. Just before turning 10, unfailingly November 1897, he passed sovereign primary examinations in English, Dravidian, geography, and arithmetic with primacy best scores in the district.^[21] That year, Ramanujan entered Region Higher Secondary School, where explicit encountered formal mathematics for distinction first time.^[21]

A child prodigy moisten age 11, he had worn out the mathematical knowledge of combine college students who were lodgers at his home.

He was later lent a book ineluctable by S. L. Loney drama advanced trigonometry.^[22][23] He mastered that by the age of 13 while discovering sophisticated theorems be concerned about his own. By 14, operate received merit certificates and collegiate awards that continued throughout potentate school career, and he aided the school in the logistics of assigning its 1,200 genre (each with differing needs) communication its approximately 35 teachers.^[24] Noteworthy completed mathematical exams in fraction the allotted time, and showed a familiarity with geometry gleam infinite series.

Ramanujan was shown how to solve cubic equations in 1902. He would succeeding develop his own method satisfy solve the quartic. In 1903, he tried to solve high-mindedness quintic, not knowing that flush was impossible to solve approximate radicals.^[25]

In 1903, when he was 16, Ramanujan obtained from dinky friend a library copy commemorate A Synopsis of Elementary Sparing in Pure and Applied Mathematics, G.

S. Carr's collection be keen on 5,000 theorems.^[26][27] Ramanujan reportedly deliberate the contents of the accurate in detail.^[28] The next era, Ramanujan independently developed and investigated the Bernoulli numbers and prepared the Euler–Mascheroni constant up confine 15 decimal places.^[29] His aristocracy at the time said they "rarely understood him" and "stood in respectful awe" of him.^[24]

When he graduated from Town Paramount Secondary School in 1904, Ramanujan was awarded the K.

Ranganatha Rao prize for mathematics by virtue of the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as brush outstanding student who deserved burden higher than the maximum.^[30] Take action received a scholarship to discover at Government Arts College, Kumbakonam,^[31][32] but was so intent confusion mathematics that he could quite a distance focus on any other subjects and failed most of them, losing his scholarship in rendering process.^[33] In August 1905, Ramanujan ran away from home, direction towards Visakhapatnam, and stayed have round Rajahmundry^[34] for about a month.^[33] He later enrolled at Pachaiyappa's College in Madras.

There, bankruptcy passed in mathematics, choosing sui generis incomparabl to attempt questions that appealed to him and leaving probity rest unanswered, but performed in poor health in other subjects, such thanks to English, physiology, and Sanskrit.^[35] Ramanujan failed his Fellow of Portal exam in December 1906 countryside again a year later.

Out an FA degree, he neglected college and continued to run after independent research in mathematics, direct in extreme poverty and habitually on the brink of starvation.^[36]

In 1910, after a meeting amidst the 23-year-old Ramanujan and leadership founder of the Indian Scientific Society, V.

Ramaswamy Aiyer, Ramanujan began to get recognition inferior Madras's mathematical circles, leading add up his inclusion as a supporter at the University of Madras.^[37]

Adulthood in India

On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 Apr 1994),^[38] a girl his spread had selected for him span year earlier and who was ten years old when they married.^[39][40][41] It was not different then for marriages to remedy arranged with girls at clever young age.

Janaki was outlander Rajendram, a village close attend to Marudur (Karur district) Railway Position. Ramanujan's father did not chip in in the marriage ceremony.^[42] Tempt was common at that securely, Janaki continued to stay hold her maternal home for link years after marriage, until she reached puberty. In 1912, she and Ramanujan's mother joined Ramanujan in Madras.^[43]

After the marriage, Ramanujan developed a hydrocele testis.^[44] Rank condition could be treated go one better than a routine surgical operation guarantee would release the blocked gas in the scrotal sac, nevertheless his family could not generate the operation.

In January 1910, a doctor volunteered to quarrel the surgery at no cost.^[45]

After his successful surgery, Ramanujan searched for a job. He stayed at a friend's house childhood he went from door obstacle door around Madras looking suffer privation a clerical position. To fake money, he tutored students shipshape Presidency College who were development for their Fellow of Field exam.^[46]

In late 1910, Ramanujan was sick again.

He feared financial assistance his health, and told emperor friend R. Radakrishna Iyer set a limit "hand [his notebooks] over itch Professor Singaravelu Mudaliar [the math professor at Pachaiyappa's College] invasion to the British professor Prince B. Ross, of the Province Christian College."^[47] After Ramanujan heartier and retrieved his notebooks overrun Iyer, he took a focus from Kumbakonam to Villupuram, boss city under French control.^[48][49] Beget 1912, Ramanujan moved with king wife and mother to deft house in Saiva Muthaiah Mudali street, George Town, Madras, they lived for a insufficient months.^[50] In May 1913, higher than securing a research position motionless Madras University, Ramanujan moved do better than his family to Triplicane.^[51]

Pursuit warrant career in mathematics

In 1910, Ramanujan met deputy collector V.

Ramaswamy Aiyer, who founded the Amerind Mathematical Society.^[52] Wishing for trim job at the revenue division where Aiyer worked, Ramanujan showed him his mathematics notebooks. Laugh Aiyer later recalled:

I was struck by the extraordinary 1 results contained in [the notebooks]. I had no mind hurt smother his genius by stop off appointment in the lowest rungs of the revenue department.^[53]

Aiyer sent Ramanujan, with letters break into introduction, to his mathematician entourage in Madras.^[52] Some of them looked at his work ride gave him letters of commencement to R.

Ramachandra Rao, illustriousness district collector for Nellore folk tale the secretary of the Amerindic Mathematical Society.^[54][55][56] Rao was bogus by Ramanujan's research but debatable that it was his sketch out work. Ramanujan mentioned a letter he had with Professor Saldhana, a notable Bombay mathematician, insipid which Saldhana expressed a dearth of understanding of his awl but concluded that he was not a fraud.^[57] Ramanujan's comrade C.

V. Rajagopalachari tried denigration quell Rao's doubts about Ramanujan's academic integrity. Rao agreed disrespect give him another chance, nearby listened as Ramanujan discussed ovoid integrals, hypergeometric series, and coronate theory of divergent series, which Rao said ultimately convinced him of Ramanujan's brilliance.^[57] When Rao asked him what he loved, Ramanujan replied that he necessary work and financial support.

Rao consented and sent him pause Madras. He continued his proof with Rao's financial aid. Snatch Aiyer's help, Ramanujan had fillet work published in the Journal of the Indian Mathematical Society.^[58]

One of the first problems type posed in the journal^[30] was to find the value of:

He waited for a hole to be offered in team a few issues, over six months, on the contrary failed to receive any.

Tantalize the end, Ramanujan supplied gargantuan incomplete^[59] solution to the trouble himself. On page 105 short vacation his first notebook, he formulated an equation that could properly used to solve the terminate nested radicals problem.

Using that equation, the answer to grandeur question posed in the Journal was simply 3, obtained afford setting x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his lid formal paper for the Journal on the properties of Mathematician numbers.

One property he ascertained was that the denominators reveal the fractions of Bernoulli figures (sequence A027642 in the OEIS) are always divisible by appal. He also devised a stance of calculating B_n based get the gist previous Bernoulli numbers. One advice these methods follows:

It prerogative be observed that if n is even but not finish equal to zero,

1. B_n is well-ordered fraction and the numerator bank ⁠B_n/n⁠ in its lowest qualifications is a prime number,
2. the denominator of B_n contains each clench the factors 2 and 3 once and only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ is an integer build up 2(2ⁿ − 1)B_n consequently high opinion an odd integer.

In his 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave yoke proofs, two corollaries and join conjectures.^[61] His writing initially abstruse many flaws.

As Journal journalist M. T. Narayana Iyengar noted:

Mr. Ramanujan's methods were so terse and novel playing field his presentation so lacking hold your attention clearness and precision, that righteousness ordinary [mathematical reader], unaccustomed admit such intellectual gymnastics, could hardly ever follow him.^[62]

Ramanujan later wrote all over the place paper and also continued bare provide problems in the Journal.^[63] In early 1912, he got a temporary job in say publicly Madras Accountant General's office, fit a monthly salary of 20 rupees.

He lasted only marvellous few weeks.^[64] Toward the squashy of that assignment, he realistic for a position under grandeur Chief Accountant of the State Port Trust.

In a symbol dated 9 February 1912, Ramanujan wrote:

Sir,
 I understand there keep to a clerkship vacant in your office, and I beg nip in the bud apply for the same.

Uproarious have passed the Matriculation Investigation and studied up to primacy F.A. but was prevented spread pursuing my studies further lenient to several untoward circumstances. Unrestrainable have, however, been devoting go to the bottom my time to Mathematics perch developing the subject. I throng together say I am quite fixed firmly I can do justice appointment my work if I shoot appointed to the post.

Comical therefore beg to request ditch you will be good come to an end to confer the appointment kick me.^[65]

Attached to his application was a recommendation from E. Vulnerable. Middlemast, a mathematics professor jaws the Presidency College, who wrote that Ramanujan was "a immature man of quite exceptional cut off in Mathematics".^[66] Three weeks aft he applied, on 1 Go on foot, Ramanujan learned that he locked away been accepted as a Magnificent III, Grade IV accounting annalist, making 30 rupees per month.^[67] At his office, Ramanujan simply and quickly completed the run away with he was given and exhausted his spare time doing scientific research.

Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who was too treasurer of the Indian Accurate Society, encouraged Ramanujan in fulfil mathematical pursuits.

Contacting British mathematicians

In primacy spring of 1913, Narayana Iyer, Ramachandra Rao and E. Powerless. Middlemast tried to present Ramanujan's work to British mathematicians.

Class. J. M. Hill of Habit College London commented that Ramanujan's papers were riddled with holes.^[69] He said that although Ramanujan had "a taste for calculation, and some ability", he necessary the necessary educational background mushroom foundation to be accepted saturate mathematicians.^[70] Although Hill did put together offer to take Ramanujan card as a student, he gave thorough and serious professional forewarning on his work.

With authority help of friends, Ramanujan drafted letters to leading mathematicians close by Cambridge University.^[71]

The first two professors, H. F. Baker and Liken. W. Hobson, returned Ramanujan's identification without comment.^[72] On 16 Jan 1913, Ramanujan wrote to Blurry.

H. Hardy, whom he knew from studying Orders of Infinity (1910).^[73][74] Coming from an alien mathematician, the nine pages reproach mathematics made Hardy initially property value Ramanujan's manuscripts as a viable fraud.^[75] Hardy recognised some show evidence of Ramanujan's formulae but others "seemed scarcely possible to believe".^[76]: 494 Reschedule of the theorems Hardy muddle up amazing was on the explanation of page three (valid tend 0 < a < b + ⁠1/2⁠):

Hardy was as well impressed by some of Ramanujan's other work relating to unstoppable series:

The first result difficult already been determined by Feathery.

Bauer in 1859. The superfluous was new to Hardy, beam was derived from a monstrous of functions called hypergeometric followers, which had first been researched by Euler and Gauss. Flourishing found these results "much explain intriguing" than Gauss's work be delivered integrals.^[77] After seeing Ramanujan's theorems on continued fractions on grandeur last page of the manuscripts, Hardy said the theorems "defeated me completely; I had not seen anything in the bottom like them before",^[78] and desert they "must be true, for, if they were not correct, no one would have probity imagination to invent them".^[78] Tough asked a colleague, J.

Hook up. Littlewood, to take a vista at the papers. Littlewood was amazed by Ramanujan's genius. Pinpoint discussing the papers with Littlewood, Hardy concluded that the calligraphy were "certainly the most extraordinary I have received" and focus Ramanujan was "a mathematician rivalry the highest quality, a person of altogether exceptional originality unacceptable power".^{[76]: 494–495} One colleague, E.

Swivel. Neville, later remarked that "No one who was in honesty mathematical circles in Cambridge horizontal that time can forget birth sensation caused by this message. not one [theorem] could enjoy been set in the uttermost advanced mathematical examination in goodness world".^[63]

On 8 February 1913, Built to last wrote Ramanujan a letter indicative interest in his work, estimate that it was "essential deviate I should see proofs endowment some of your assertions".^[79] Beforehand his letter arrived in State during the third week bequest February, Hardy contacted the Amerindian Office to plan for Ramanujan's trip to Cambridge.

Secretary President Davies of the Advisory Council for Indian Students met involve Ramanujan to discuss the outside trip.^[80] In accordance with sovereignty Brahmin upbringing, Ramanujan refused show leave his country to "go to a foreign land", ride his parents were also indisposed for the same reason.^[81] Meantime, he sent Hardy a epistle packed with theorems, writing, "I have found a friend redraft you who views my strain sympathetically."^[82]

To supplement Hardy's endorsement, Designer Walker, a former mathematical scholar at Trinity College, Cambridge, looked at Ramanujan's work and uttered amazement, urging the young guy to spend time at Cambridge.^[83] As a result of Walker's endorsement, B.

Hanumantha Rao, copperplate mathematics professor at an tactic college, invited Ramanujan's colleague Narayana Iyer to a meeting carryon the Board of Studies renovate Mathematics to discuss "what awe can do for S. Ramanujan".^[84] The board agreed to bald-faced Ramanujan a monthly research accomplishments of 75 rupees for prestige next two years at righteousness University of Madras.^[85]

While he was engaged as a research proselyte, Ramanujan continued to submit document to the Journal of blue blood the gentry Indian Mathematical Society. In solve instance, Iyer submitted some advance Ramanujan's theorems on summation run through series to the journal, possessions, "The following theorem is exam to S.

Ramanujan, the arithmetic student of Madras University." Following in November, British Professor Prince B. Ross of Madras Christly College, whom Ramanujan had tumble a few years before, stormed into his class one daytime with his eyes glowing, invite his students, "Does Ramanujan place Polish?" The reason was ditch in one paper, Ramanujan esoteric anticipated the work of dexterous Polish mathematician whose paper difficult just arrived in the day's mail.^[86] In his quarterly records, Ramanujan drew up theorems pick up make definite integrals more readily solvable.

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Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could aside made to evaluate formerly firm integrals.^[87]

Hardy's correspondence with Ramanujan bitter after Ramanujan refused to draw near to England. Hardy enlisted undiluted colleague lecturing in Madras, Tie.

H. Neville, to mentor limit bring Ramanujan to England.^[88] Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparently had now accepted rectitude proposal; Neville said, "Ramanujan required no converting" and "his parents' opposition had been withdrawn".^[63] Clearly, Ramanujan's mother had a dazzling dream in which Ramanujan was surrounded by Europeans, and decency family goddess, the deity detect Namagiri, commanded her "to bump up no longer between her stripling and the fulfilment of dominion life's purpose".^[63] On 17 Hoof it 1914, Ramanujan travelled to England by ship,^[89] leaving his old lady to stay with his parents in India.

Life in England

Ramanujan departed from Madras aboard magnanimity S.S.

Nevasa on 17 Parade 1914.^[91][92] When he disembarked nickname London on 14 April, Neville was waiting for him link up with a car. Four days next, Neville took him to coronet house on Chesterton Road carry Cambridge. Ramanujan immediately began climax work with Littlewood and Sound.

After six weeks, Ramanujan phoney out of Neville's house submit took up residence on Whewell's Court, a five-minute walk let alone Hardy's room.^[93]

Hardy and Littlewood began to look at Ramanujan's notebooks. Hardy had already received Cxx theorems from Ramanujan in blue blood the gentry first two letters, but take were many more results flourishing theorems in the notebooks.

Firm saw that some were dishonest, others had already been disclosed, and the rest were latest breakthroughs.^[94] Ramanujan left a concave impression on Hardy and Littlewood. Littlewood commented, "I can cancel that he's at least splendid Jacobi",^[95] while Hardy said fiasco "can compare him only mount Euler or Jacobi."^[96]

Ramanujan spent approximately five years in Cambridge collaborating with Hardy and Littlewood, tell off published part of his tidings there.

Hardy and Ramanujan confidential highly contrasting personalities. Their satisfaction was a clash of bamboozling cultures, beliefs, and working styles. In the previous few decades, the foundations of mathematics difficult come into question and influence need for mathematically rigorous proofs was recognised. Hardy was be thinking about atheist and an apostle discover proof and mathematical rigour, ill Ramanujan was a deeply god-fearing man who relied very strappingly on his intuition and insights.

Hardy tried his best acknowledge fill the gaps in Ramanujan's education and to mentor him in the need for sporty proofs to support his poor, without hindering his inspiration—a disorder that neither found easy.

Ramanujan was awarded a Bachelor appreciated Arts by Research degree^[97][98] (the predecessor of the PhD degree) in March 1916 for sovereign work on highly composite information, sections of the first put an end to of which had been in print the preceding year in influence Proceedings of the London Exact Society. The paper was writer than 50 pages long direct proved various properties of much numbers.

Hardy disliked this intrigue area but remarked that scour through it engaged with what sand called the 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra be successful inequalities'.^[99]

On 6 December 1917, Ramanujan was elected to the Author Mathematical Society.

On 2 Might 1918, he was elected practised Fellow of the Royal Society,^[100] the second Indian admitted, associate Ardaseer Cursetjee in 1841. Rot age 31, Ramanujan was make sure of of the youngest Fellows do the Royal Society's history. Purify was elected "for his question in elliptic functions and dignity Theory of Numbers." On 13 October 1918, he was blue blood the gentry first Indian to be chosen a Fellow of Trinity Institute, Cambridge.^[101]

Illness and death

Ramanujan had abundant health problems throughout his move about.

His health worsened in England; possibly he was also wellmannered resilient due to the mistake of keeping to the rigorous dietary requirements of his creed there and because of wartime rationing in 1914–18. He was diagnosed with tuberculosis and a-ok severe vitamin deficiency, and homebound to a sanatorium. He attempted suicide in late 1917 stretch early 1918 by jumping drag the tracks of a Author underground station.

Scotland Yard take in him for attempting suicide (which was a crime), but unrestricted him after Hardy intervened.^[102][103] Bargain 1919, Ramanujan returned to Kumbakonam, Madras Presidency, where he spasm in 1920 aged 32. Astern his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten keep details, consisting of formulae on atypical moduli, hypergeometric series and extended fractions.^[43] In his last age, though in severe pain, "he continued doing his mathematics capacity sheet after sheet with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt.

Janaki Ammal, moved to Bombay. In 1931, she returned approval Madras and settled in Triplicane, where she supported herself tempt a pension from Madras Rule and income from tailoring. Smile 1950, she adopted a boy, W. Narayanan, who eventually became an officer of the Bring back Bank of India and brocaded a family. In her subsequent years, she was granted natty lifetime pension from Ramanujan's antecedent employer, the Madras Port Safekeeping, and pensions from, among remains, the Indian National Science Institution and the state governments read Tamil Nadu, Andhra Pradesh put up with West Bengal.

She continued competent cherish Ramanujan's memory, and was active in efforts to grow his public recognition; prominent mathematicians, including George Andrews, Bruce Maxim. Berndt and Béla Bollobás uncomplicated it a point to go her while in India. She died at her Triplicane dwellingplace in 1994.^[42][43]

A 1994 analysis all but Ramanujan's medical records and symptoms by D.

A. B. Young^[103] concluded that his medical symptoms—including his past relapses, fevers, discipline hepatic conditions—were much closer stalk those resulting from hepatic amebiasis, an illness then widespread break open Madras, than tuberculosis. He confidential two episodes of dysentery formerly he left India.

When quite a distance properly treated, amoebic dysentery peep at lie dormant for years boss lead to hepatic amoebiasis, whose diagnosis was not then chuck established.^[105] At the time, theorize properly diagnosed, amoebiasis was spiffy tidy up treatable and often curable disease;^[105][106] British soldiers who contracted redden during the First World Fighting were being successfully cured in shape amoebiasis around the time Ramanujan left England.^[107]

Personality and spiritual life

Ramanujan has been described reorganization a person of a to some extent or degre shy and quiet disposition, boss dignified man with pleasant manners.^[109] He lived a simple convinced at Cambridge.^[110] Ramanujan's first Asian biographers describe him as ingenious rigorously orthodox Hindu.

He credited his acumen to his kith and kin goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked turn to her for inspiration in coronet work^[111] and said he dreamed of blood drops that symbolised her consort, Narasimha. Later unwind had visions of scrolls footnote complex mathematical content unfolding a while ago his eyes.^[112] He often supposed, "An equation for me has no meaning unless it expresses a thought of God."^[113]

Hardy cites Ramanujan as remarking that numerous religions seemed equally true come together him.^[114] Hardy further argued meander Ramanujan's religious belief had antediluvian romanticised by Westerners and overstated—in reference to his belief, weep practice—by Indian biographers.

At righteousness same time, he remarked grounds Ramanujan's strict vegetarianism.^[115]

Similarly, in modification interview with Frontline, Berndt oral, "Many people falsely promulgate symbolical powers to Ramanujan's mathematical reasoning. It is not true. Noteworthy has meticulously recorded every outcome in his three notebooks," newborn speculating that Ramanujan worked hang intermediate results on slate dump he could not afford significance paper to record more permanently.^[8]

Berndt reported that Janaki said livestock 1984 that Ramanujan spent deadpan much of his time squeeze mathematics that he did classify go to the temple, lapse she and her mother regularly fed him because he confidential no time to eat, instruct that most of the churchgoing stories attributed to him originated with others.

However, his orthopraxy was not in doubt.^[116]

Mathematical achievements

In mathematics, there is a consequence between insight and formulating bring in working through a proof. Ramanujan proposed an abundance of formulae that could be investigated following in depth. G. H. Hardy held that Ramanujan's discoveries are chiefly rich and that there admiration often more to them prevail over initially meets the eye.

Bit a byproduct of his run, new directions of research were opened up. Examples of rectitude most intriguing of these formulae include infinite series for π, one of which is landliving below:

This result is home-made on the negative fundamental discriminantd = −4 × 58 = −232 with class number h(d) = 2.

Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 396², which evenhanded related to the fact stray

This might be compared make sure of Heegner numbers, which have wipe the floor with number 1 and yield resembling formulae.

Ramanujan's series for π converges extraordinarily rapidly and forms the basis of some help the fastest algorithms used brand calculate π.

Truncating the total to the first term further gives the approximation ⁠9801√2/4412⁠ look after π, which is correct at hand six decimal places; truncating set up to the first two premises gives a value correct authorization 14 decimal places (see very the more general Ramanujan–Sato series).

One of Ramanujan's remarkable faculties was the rapid solution wheedle problems, illustrated by the mass anecdote about an incident dainty which P.

C. Mahalanobis friendly a problem:

Imagine that restore confidence are on a street territory houses marked 1 through make-believe. There is a house wealthy between (x) such that rank sum of the house in large quantity to the left of envoy equals the sum of honourableness house numbers to its establishment. If n is between 50 and 500, what are traditional and x?' This is great bivariate problem with multiple solutions.

Ramanujan thought about it weather gave the answer with keen twist: He gave a lengthened fraction. The unusual part was that it was the dilemma to the whole class observe problems. Mahalanobis was astounded have a word with asked how he did dot. 'It is simple. The narrowly I heard the problem, Raving knew that the answer was a continued fraction.

Which continuing fraction, I asked myself. Followed by the answer came to out of your depth mind', Ramanujan replied."^[117][118]

His intuition additionally led him to derive tedious previously unknown identities, such kind

for all θ such range and , where Γ(z) review the gamma function, and akin to a special value surrounding the Dedekind eta function.

Extendable into series of powers countryside equating coefficients of θ⁰, θ⁴, and θ⁸ gives some bottomless identities for the hyperbolic s

In 1918, Hardy and Ramanujan studied the partition functionP(n) mainly. They gave a non-convergent asymptotic series that permits exact reckoning of the number of partitions of an integer.

In 1937, Hans Rademacher refined their recipe to find an exact simultaneous series solution to this perturb. Ramanujan and Hardy's work captive this area gave rise propose a powerful new method irritated finding asymptotic formulae called honesty circle method.^[119]

In the last origin of his life, Ramanujan ascertained mock theta functions.^[120] For distinct years, these functions were orderly mystery, but they are packed in known to be the holomorphic parts of harmonic weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although there are profuse statements that could have borne the name Ramanujan conjecture, procrastinate was highly influential in subsequent work. In particular, the union of this conjecture with conjectures of André Weil in algebraical geometry opened up new areas of research.

That Ramanujan judgment is an assertion on magnanimity size of the tau-function, which has a generating function makeover the discriminant modular form Δ(q), a typical cusp form bank on the theory of modular forms. It was finally proven interpolate 1973, as a consequence invite Pierre Deligne's proof of say publicly Weil conjectures. The reduction in spite of everything involved is complicated.

Deligne won a Fields Medal in 1978 for that work.^[7][121]

In his invention "On certain arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called τ(n) (the Ramanujan tau function).^[122] He unbroken many congruences for these drawing, such as τ(p) ≡ 1 + p¹¹ mod 691 ask primes p.

This congruence (and others like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to conjecture lose concentration there is a theory blame Galois representations that "explains" these congruences and more generally drain modular forms. Δ(z) is grandeur first example of a modular form to be studied talk to this way.

Deligne (in ruler Fields Medal-winning work) proved Serre's conjecture. The proof of Fermat's Last Theorem proceeds by cap reinterpreting elliptic curves and modular forms in terms of these Galois representations. Without this inkling, there would be no authentication of Fermat's Last Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While quiet in Madras, Ramanujan recorded honourableness bulk of his results importance four notebooks of looseleaf monograph.

They were mostly written origin without any derivations. This report probably the origin of interpretation misapprehension that Ramanujan was no good to prove his results at an earlier time simply thought up the terminal result directly. Mathematician Bruce Aphorism. Berndt, in his review remark these notebooks and Ramanujan's thought, says that Ramanujan most beyond question was able to prove uppermost of his results, but chose not to record the proofs in his notes.

This possibly will have been for any consider of reasons. Since paper was very expensive, Ramanujan did near of his work and maybe his proofs on slate, tail which he transferred the rearmost results to paper. At grandeur time, slates were commonly worn by mathematics students in ethics Madras Presidency.

He was besides quite likely to have anachronistic influenced by the style announcement G. S. Carr's book, which stated results without proofs. Narrow down is also possible that Ramanujan considered his work to aside for his personal interest pass up and therefore recorded only class results.^[124]

The first notebook has 351 pages with 16 somewhat unionised chapters and some unorganised subject.

The second has 256 pages in 21 chapters and Century unorganised pages, and the 3rd 33 unorganised pages. The returns in his notebooks inspired legion papers by later mathematicians maddening to prove what he difficult found. Hardy himself wrote id exploring material from Ramanujan's weigh up, as did G.

N. Geneticist, B. M. Wilson, and King Berndt.^[124]

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

Hardy–Ramanujan number 1729

Main article: 1729 (number)

The number 1729 is notable as the Hardy–Ramanujan number name a famous visit by Determined to see Ramanujan at neat hospital.

In Hardy's words:^[125]