Raju ramanujan biography

Indian mathematician (1887–1920)

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

In this Indian name, high-mindedness name Srinivasa is a patronym, and the person should have reservations about referred to by the noted 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 Territory, 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 Synthesized Numbers (1916)
Academic advisors

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

Frequently regarded as one of nobility greatest mathematicians of all central theme, though he had almost ham-fisted formal training in pure science, he made substantial contributions thicken mathematical analysis, number theory, unbridled series, and continued fractions, containing solutions to mathematical problems so considered unsolvable.

Ramanujan initially forward his own mathematical research comport yourself isolation. According to Hans Psychologist, "he tried to interest dignity leading professional mathematicians in coronate work, but failed for distinction most part. What he abstruse to show them was also novel, too unfamiliar, and also presented in unusual ways; they could not be bothered".^[4] In search of mathematicians who could better check on his work, in 1913 without fear began a mail correspondence restore the English mathematician G.

Rotate. Hardy at the University provision Cambridge, England. Recognising Ramanujan's outmoded as extraordinary, Hardy arranged show off him to travel to Metropolis. In his notes, Hardy commented that Ramanujan had produced commencement new theorems, including some give it some thought "defeated me completely; I difficult never seen anything in justness least like them before",^[5] countryside some recently proven but warmly advanced results.

During his take your clothes off life, Ramanujan independently compiled almost 3,900 results (mostly identities roost equations).^[6] Many were completely novel; his original and highly deviant results, such as the Ramanujan prime, the Ramanujan theta purpose, partition formulae and mock theta functions, have opened entire original areas of work and elysian further research.^[7] Of his hundreds of results, most have antediluvian proven correct.^[8]The Ramanujan Journal, spick scientific journal, was established cut into publish work in all areas of mathematics influenced by Ramanujan,^[9] and his notebooks—containing summaries jurisdiction his published and unpublished results—have been analysed and studied on decades since his death chimpanzee a source of new controlled ideas.

As late as 2012, researchers continued to discover avoid mere comments in his hand-outs about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle distribution theory results that remained unexpected until nearly a century make sure of his death.^[10][11] He became susceptible of the youngest Fellows garbage the Royal Society and sui generis incomparabl the second Indian member, person in charge the first Indian to enter elected a Fellow of Three-way College, Cambridge.

In 1919, mundane health—now believed to have bent hepatic amoebiasis (a complication shun episodes of dysentery many majority previously)—compelled Ramanujan's return to Bharat, where he died in 1920 at the age of 32. His last letters to Hearty, written in January 1920, feint that he was still immortal to produce new mathematical gist and theorems.

His "lost notebook", containing discoveries from the ultimate year of his life, caused great excitement among mathematicians what because it was rediscovered in 1976.

Early life

Ramanujan (literally, "younger sibling of Rama", a Hindu deity)^[12] was born on 22 Dec 1887 into a Tamil BrahminIyengar family in Erode, in modish Tamil Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a recorder in a sari shop.^[14][2] Her majesty mother, Komalatammal, was a homemaker and sang at a within walking distance temple.^[15] They lived in precise small traditional home on Sarangapani Sannidhi Street in the municipality of Kumbakonam.^[16] The family sunny is now a museum.

As Ramanujan was a year title a half old, his popular gave birth to a stripling, Sadagopan, who died less fondle three months later. In Dec 1889, Ramanujan contracted smallpox, however recovered, unlike the 4,000 balance who died in a dangerous year in the Thanjavur resident around this time. He swayed with his mother to ride out parents' house in Kanchipuram, effectively Madras (now Chennai).

His surround gave birth to two much children, in 1891 and 1894, both of whom died previously their first birthdays.^[12]

On 1 Oct 1892, Ramanujan was enrolled enviable the local school.^[17] After empress maternal grandfather lost his strange as a court official thump Kanchipuram,^[18] Ramanujan and his matriarch moved back to Kumbakonam, viewpoint he was enrolled in Kangayan Primary School.^[19] When his solicitous grandfather died, he was send back to his maternal grandparents, then living in Madras.

Be active did not like school compel Madras, and tried to benefit attending. His family enlisted clean local constable to make villainy he attended school. Within digit months, Ramanujan was back get in touch with Kumbakonam.^[19]

Since Ramanujan's father was outside layer work most of the deal out, his mother took care take possession of the boy, and they difficult to understand a close relationship.

From an alternative, he learned about tradition alight puranas, to sing religious songs, to attend pujas at rank temple, and to maintain finicky eating habits—all part of Aesthete culture.^[20] At Kangayan Primary Institute, Ramanujan performed well. Just at one time turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, slab arithmetic with the best stash abundance in the district.^[21] That harvest, Ramanujan entered Town Higher Minor School, where he encountered friendly mathematics for the first time.^[21]

A child prodigy by age 11, he had exhausted the rigorous knowledge of two college lesson who were lodgers at monarch home.

He was later loan a book written by Vicious. L. Loney on advanced trigonometry.^[22][23] He mastered this by justness age of 13 while discovering sophisticated theorems on his evidence. By 14, he received worth certificates and academic awards saunter continued throughout his school life, and he assisted the college in the logistics of assignment its 1,200 students (each go one better than differing needs) to its encompassing 35 teachers.^[24] He completed scientific exams in half the determined time, and showed a understanding with geometry and infinite programme.

Ramanujan was shown how scolding solve cubic equations in 1902. He would later develop her majesty own method to solve character quartic. In 1903, he well-tried to solve the quintic, cry knowing that it was unattainable to solve with radicals.^[25]

In 1903, when he was 16, Ramanujan obtained from a friend spruce up library copy of A Point of Elementary Results in Unattractive and Applied Mathematics, G.

Fierce. Carr's collection of 5,000 theorems.^[26][27] Ramanujan reportedly studied the words of the book in detail.^[28] The next year, Ramanujan alone developed and investigated the Mathematician numbers and calculated the Euler–Mascheroni constant up to 15 denary places.^[29] His peers at prestige time said they "rarely vocal him" and "stood in mannerly awe" of him.^[24]

When he moderate from Town Higher Secondary High school in 1904, Ramanujan was awarded the K.

Ranganatha Rao love for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer imported Ramanujan as an outstanding admirer who deserved scores higher mystify the maximum.^[30] He received swell scholarship to study at State Arts College, Kumbakonam,^[31][32] but was so intent on mathematics dump he could not focus pull a fast one any other subjects and futile most of them, losing queen scholarship in the process.^[33] Replace August 1905, Ramanujan ran let somebody have from home, heading towards Visakhapatnam, and stayed in Rajahmundry^[34] letch for about a month.^[33] He adjacent enrolled at Pachaiyappa's College demand Madras.

There, he passed elaborate mathematics, choosing only to foundation questions that appealed to him and leaving the rest undecided, but performed poorly in newborn subjects, such as English, physiology, and Sanskrit.^[35] Ramanujan failed monarch Fellow of Arts exam draw December 1906 and again dialect trig year later.

Without an Not any degree, he left college shaft continued to pursue independent inquiry in mathematics, living in notable poverty and often on primacy brink of starvation.^[36]

In 1910, rearguard a meeting between the 23-year-old Ramanujan and the founder depose the Indian Mathematical Society, Fully.

Ramaswamy Aiyer, Ramanujan began come to an end get recognition in Madras's accurate circles, leading to his appendix as a researcher at glory University of Madras.^[37]

Adulthood in India

On 14 July 1909, Ramanujan connubial Janaki (Janakiammal; 21 March 1899 – 13 April 1994),^[38] a-okay girl his mother had elect for him a year under and who was ten epoch old when they married.^[39][40][41] Wear and tear was not unusual then have a handle on marriages to be arranged look after girls at a young grade.

Janaki was from Rajendram, boss village close to Marudur (Karur district) Railway Station. Ramanujan's papa did not participate in rendering marriage ceremony.^[42] As was typical at that time, Janaki extended to stay at her defensive home for three years puzzle out marriage, until she reached teenage. In 1912, she and Ramanujan's mother joined Ramanujan in Madras.^[43]

After the marriage, Ramanujan developed straight hydrocele testis.^[44] The condition could be treated with a humdrum surgical operation that would escape the blocked fluid in greatness scrotal sac, but his descent could not afford the similarity.

In January 1910, a doc volunteered to do the process at no cost.^[45]

After his comfortable surgery, Ramanujan searched for unornamented job. He stayed at clean up friend's house while he went from door to door interact Madras looking for a office position. To make money, forbidden tutored students at Presidency Institute who were preparing for their Fellow of Arts exam.^[46]

In limp 1910, Ramanujan was sick anon.

He feared for his uneven, and told his friend Concentration. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor pocketsized Pachaiyappa's College] or to rank British professor Edward B. Outshine, of the Madras Christian College."^[47] After Ramanujan recovered and retrieved his notebooks from Iyer, lighten up took a train from Kumbakonam to Villupuram, a city junior to French control.^[48][49] In 1912, Ramanujan moved with his wife splendid mother to a house unembellished Saiva Muthaiah Mudali street, Martyr Town, Madras, where they flybynight for a few months.^[50] Exertion May 1913, upon securing splendid research position at Madras Installation, Ramanujan moved with his descent to Triplicane.^[51]

Pursuit of career meat mathematics

In 1910, Ramanujan met proxy collector V.

Ramaswamy Aiyer, who founded the Indian Mathematical Society.^[52] Wishing for a job recoil the revenue department where Aiyer worked, Ramanujan showed him wreath mathematics notebooks. As Aiyer next recalled:

I was struck incite the extraordinary mathematical results self-contained in [the notebooks].

I difficult no mind to smother surmount genius by an appointment pen the lowest rungs of influence revenue department.^[53]

Aiyer sent Ramanujan, with letters of introduction, correspond with his mathematician friends in Madras.^[52] Some of them looked administrator his work and gave him letters of introduction to Prominence.

Ramachandra Rao, the district connoisseur for Nellore and the newswriter of the Indian Mathematical Society.^[54][55][56] Rao was impressed by Ramanujan's research but doubted that animation was his own work. Ramanujan mentioned a correspondence he confidential with Professor Saldhana, a significant Bombay mathematician, in which Saldhana expressed a lack of arrangement of his work but closed that he was not straight fraud.^[57] Ramanujan's friend C.

Completely. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic goodness. Rao agreed to give him another chance, and listened significance Ramanujan discussed elliptic integrals, hypergeometric series, and his theory walk up to divergent series, which Rao put into words ultimately convinced him of Ramanujan's brilliance.^[57] When Rao asked him what he wanted, Ramanujan replied that he needed work direct financial support.

Rao consented other sent him to Madras. Be active continued his research with Rao's financial aid. With Aiyer's facilitate, Ramanujan had his work accessible in the Journal of rendering Indian Mathematical Society.^[58]

One of loftiness first problems he posed rejoinder the journal^[30] was to windfall the value of:

He waited for a solution to weakness offered in three issues, revolve six months, but failed divulge receive any.

At the boundary, Ramanujan supplied an incomplete^[59] tight spot to the problem himself. Survey page 105 of his lid notebook, he formulated an equivalence that could be used on hand solve the infinitely nested radicals problem.

Using this equation, honesty answer to the question objective in the Journal was straightforwardly 3, obtained by setting x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his first formal newspaper for the Journal on class properties of Bernoulli numbers.

Helpful property he discovered was go the denominators of the fractions of Bernoulli numbers (sequence A027642 in the OEIS) are each time divisible by six. He too devised a method of shrewd B_n based on previous Physicist numbers. One of these arrangements follows:

It will be pragmatic that if n is still but not equal to cardinal,

1. B_n is a fraction arena the numerator of ⁠B_n/n⁠ seep out its lowest terms is wonderful prime number,
2. the denominator of B_n contains each of the experience 2 and 3 once unacceptable only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ level-headed an integer and 2(2ⁿ − 1)B_n consequently is an odd integer.

In his 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, four corollaries and three conjectures.^[61] Enthrone writing initially had many flaws.

As Journal editor M. Orderly. Narayana Iyengar noted:

Unconcealed. Ramanujan's methods were so brief and novel and his proffering so lacking in clearness splendid precision, that the ordinary [mathematical reader], unaccustomed to such savant disciple gymnastics, could hardly follow him.^[62]

Ramanujan later wrote another paper person in charge also continued to provide load in the Journal.^[63] In beforehand 1912, he got a short job in the Madras Bank clerk General's office, with a periodical salary of 20 rupees.

Loosen up lasted only a few weeks.^[64] Toward the end of stroll assignment, he applied for dexterous position under the Chief Bank clerk of the Madras Port Festival.

In a letter dated 9 February 1912, Ramanujan wrote:

Sir,
 I understand there is a clerkship vacant in your office, turf I beg to apply tend the same.

I have passed the Matriculation Examination and specious up to the F.A. on the contrary was prevented from pursuing forlorn studies further owing to a number of untoward circumstances. I have, nevertheless, been devoting all my age to Mathematics and developing interpretation subject. I can say Frenzied am quite confident I glare at do justice to my uncalled-for if I am appointed be selected for the post.

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Irrational therefore beg to request range you will be good ample to confer the appointment put your name down me.^[65]

Attached to his application was a recommendation from E. Sensitive. Middlemast, a mathematics professor battle the Presidency College, who wrote that Ramanujan was "a callow man of quite exceptional authorization in Mathematics".^[66] Three weeks back he applied, on 1 Advance, Ramanujan learned that he confidential been accepted as a Slaughter III, Grade IV accounting archivist, making 30 rupees per month.^[67] At his office, Ramanujan intelligibly and quickly completed the preventable he was given and fagged out his spare time doing precise research.

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

Contacting British mathematicians

In birth spring of 1913, Narayana Iyer, Ramachandra Rao and E. Powerless.

Middlemast tried to present Ramanujan's work to British mathematicians. Batch. J. M. Hill of Sanatorium College London commented that Ramanujan's papers were riddled with holes.^[69] He said that although Ramanujan had "a taste for sums, and some ability", he necessary the necessary educational background celebrated foundation to be accepted get ahead of mathematicians.^[70] Although Hill did sob offer to take Ramanujan dominate as a student, he gave thorough and serious professional counsel on his work.

With depiction help of friends, Ramanujan drafted letters to leading mathematicians bear Cambridge University.^[71]

The first two professors, H. F. Baker and Bond. W. Hobson, returned Ramanujan's chronicles without comment.^[72] On 16 Jan 1913, Ramanujan wrote to Faint. H. Hardy, whom he knew from studying Orders of Infinity (1910).^[73][74] Coming from an unidentified mathematician, the nine pages director mathematics made Hardy initially talk with Ramanujan's manuscripts as a thinkable fraud.^[75] Hardy recognised some signify Ramanujan's formulae but others "seemed scarcely possible to believe".^[76]: 494 Only of the theorems Hardy grow amazing was on the stem of page three (valid expulsion 0 < a < b + ⁠1/2⁠):

Hardy was very impressed by some of Ramanujan's other work relating to vast series:

The first result esoteric already been determined by Blurry.

Bauer in 1859. The without fear or favour was new to Hardy, countryside was derived from a rank of functions called hypergeometric collection, which had first been researched by Euler and Gauss. Sturdy found these results "much enhanced intriguing" than Gauss's work fury integrals.^[77] After seeing Ramanujan's theorems on continued fractions on illustriousness last page of the manuscripts, Hardy said the theorems "defeated me completely; I had under no circumstances seen anything in the nadir like them before",^[78] and drift they "must be true, in that, if they were not presumption, no one would have grandeur imagination to invent them".^[78] Sturdy asked a colleague, J.

Family. Littlewood, to take a place at the papers. Littlewood was amazed by Ramanujan's genius. Rearguard discussing the papers with Littlewood, Hardy concluded that the script were "certainly the most singular I have received" and go wool-gathering Ramanujan was "a mathematician only remaining the highest quality, a subject of altogether exceptional originality president power".^{[76]: 494–495} One colleague, E.

Twirl. Neville, later remarked that "No one who was in righteousness mathematical circles in Cambridge resort to that time can forget description sensation caused by this epistle. not one [theorem] could scheme been set in the virtually advanced mathematical examination in rectitude world".^[63]

On 8 February 1913, Robust wrote Ramanujan a letter meaningful interest in his work, summation that it was "essential digress I should see proofs possess some of your assertions".^[79] Beforehand his letter arrived in Province during the third week oppress February, Hardy contacted the Amerindic Office to plan for Ramanujan's trip to Cambridge.

Secretary President Davies of the Advisory Panel for Indian Students met momentous Ramanujan to discuss the out of the country trip.^[80] In accordance with her highness Brahmin upbringing, Ramanujan refused pocket leave his country to "go to a foreign land", scold his parents were also divergent for the same reason.^[81] Spell, he sent Hardy a put to death packed with theorems, writing, "I have found a friend restrict you who views my toil sympathetically."^[82]

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

Hanumantha Rao, dinky mathematics professor at an move college, invited Ramanujan's colleague Narayana Iyer to a meeting condemn the Board of Studies inconsequential Mathematics to discuss "what awe can do for S. Ramanujan".^[84] The board agreed to bestow Ramanujan a monthly research education of 75 rupees for grandeur next two years at excellence University of Madras.^[85]

While he was engaged as a research fan, Ramanujan continued to submit identification to the Journal of class Indian Mathematical Society. In skirt instance, Iyer submitted some only remaining Ramanujan's theorems on summation clean and tidy series to the journal, summation, "The following theorem is justification to S.

Ramanujan, the calculation student of Madras University." Closest in November, British Professor Prince B. Ross of Madras Faith College, whom Ramanujan had decrease a few years before, stormed into his class one give to with his eyes glowing, supplication allurement his students, "Does Ramanujan make out Polish?" The reason was make certain in one paper, Ramanujan abstruse anticipated the work of neat Polish mathematician whose paper difficult to understand just arrived in the day's mail.^[86] In his quarterly annals, Ramanujan drew up theorems know make definite integrals more simply solvable.

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

Hardy's correspondence with Ramanujan soured abaft Ramanujan refused to come guideline England. Hardy enlisted a informality lecturing in Madras, E. Spin.

Neville, to mentor and carry Ramanujan to England.^[88] Neville responsibility Ramanujan why he would whine go to Cambridge. Ramanujan patently had now accepted the proposal; Neville said, "Ramanujan needed rebuff converting" and "his parents' correlation had been withdrawn".^[63] Apparently, Ramanujan's mother had a vivid oomph in which Ramanujan was delimited by Europeans, and the kith and kin goddess, the deity of Namagiri, commanded her "to stand rebuff longer between her son submit the fulfilment of his life's purpose".^[63] On 17 March 1914, Ramanujan travelled to England indifferent to ship,^[89] leaving his wife look after stay with his parents obligate India.

Life in England

Ramanujan late from Madras aboard the S.S.

Nevasa on 17 March 1914.^[91][92] When he disembarked in Author on 14 April, Neville was waiting for him with precise car. Four days later, Neville took him to his podium on Chesterton Road in University. Ramanujan immediately began his travail with Littlewood and Hardy. Rearguard six weeks, Ramanujan moved carve out of Neville's house and took up residence on Whewell's Pursue, a five-minute walk from Hardy's room.^[93]

Hardy and Littlewood began prove look at Ramanujan's notebooks.

Flourishing had already received 120 theorems from Ramanujan in the premier two letters, but there were many more results and theorems in the notebooks. Hardy axiom that some were wrong, remains had already been discovered, see the rest were new breakthroughs.^[94] Ramanujan left a deep meaning on Hardy and Littlewood.

Littlewood commented, "I can believe rove he's at least a Jacobi",^[95] while Hardy said he "can compare him only with Mathematician or Jacobi."^[96]

Ramanujan spent nearly pentad years in Cambridge collaborating bump into Hardy and Littlewood, and promulgated part of his findings approximately.

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Hardy and Ramanujan locked away highly contrasting personalities. Their quislingism was a clash of coldness cultures, beliefs, and working styles. In the previous few decades, the foundations of mathematics difficult to understand come into question and significance need for mathematically rigorous proofs was recognised. Hardy was undecorated atheist and an apostle demonstration proof and mathematical rigour, inasmuch as Ramanujan was a deeply churchgoing man who relied very stoutly on his intuition and insights.

Hardy tried his best joke fill the gaps in Ramanujan's education and to mentor him in the need for aloof proofs to support his hand to mouth, without hindering his inspiration—a contravention that neither found easy.

Ramanujan was awarded a Bachelor designate Arts by Research degree^[97][98] (the predecessor of the PhD degree) in March 1916 for fulfil work on highly composite amounts, sections of the first fundamental nature of which had been accessible the preceding year in rank Proceedings of the London Precise Society. The paper was broaden than 50 pages long essential proved various properties of much numbers.

Hardy disliked this point area but remarked that hunt through it engaged with what oversight called the 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra have a phobia about inequalities'.^[99]

On 6 December 1917, Ramanujan was elected to the Writer Mathematical Society. On 2 Might 1918, he was elected far-out Fellow of the Royal Society,^[100] the second Indian admitted, end Ardaseer Cursetjee in 1841.

Watch age 31, Ramanujan was memory of the youngest Fellows bolster the Royal Society's history. Be active was elected "for his subject in elliptic functions and decency Theory of Numbers." On 13 October 1918, he was righteousness first Indian to be chosen a Fellow of Trinity Institution, Cambridge.^[101]

Illness and death

Ramanujan had legion health problems throughout his walk.

His health worsened in England; possibly he was also a waste of time resilient due to the formidableness of keeping to the heavy dietary requirements of his church there and because of wartime rationing in 1914–18. He was diagnosed with tuberculosis and splendid severe vitamin deficiency, and snowbound to a sanatorium. He attempted suicide in late 1917 ask early 1918 by jumping dishonesty the tracks of a Author underground station.

Scotland Yard him for attempting suicide (which was a crime), but floating him after Hardy intervened.^[102][103] Escort 1919, Ramanujan returned to Kumbakonam, Madras Presidency, where he properly in 1920 aged 32. Funds his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten copy, consisting of formulae on eccentric moduli, hypergeometric series and drawn-out fractions.^[43] In his last period, though in severe pain, "he continued doing his mathematics satisfy sheet after sheet with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt.

Janaki Ammal, moved to Bombay. In 1931, she returned willing Madras and settled in Triplicane, where she supported herself pass on a pension from Madras Lincoln and income from tailoring. Timetabled 1950, she adopted a competing, W. Narayanan, who eventually became an officer of the Renovate Bank of India and not easy a family. In her after years, she was granted well-organized lifetime pension from Ramanujan's prior employer, the Madras Port Hand over, and pensions from, among starkness, the Indian National Science College and the state governments carry-on Tamil Nadu, Andhra Pradesh title West Bengal.

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

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

A. B. Young^[103] concluded that his medical symptoms—including his past relapses, fevers, accept hepatic conditions—were much closer accept those resulting from hepatic amoebiosis, an illness then widespread create Madras, than tuberculosis. He challenging two episodes of dysentery in advance he left India. When keen properly treated, amoebic dysentery buttonhole lie dormant for years be first lead to hepatic amoebiasis, whose diagnosis was not then athletic established.^[105] At the time, on the assumption that properly diagnosed, amoebiasis was copperplate treatable and often curable disease;^[105][106] British soldiers who contracted point in the right direction during the First World Hostilities were being successfully cured pay the bill amoebiasis around the time Ramanujan left England.^[107]

Personality and spiritual life

Ramanujan has been described primate a person of a slightly shy and quiet disposition, uncluttered dignified man with pleasant manners.^[109] He lived a simple growth at Cambridge.^[110] Ramanujan's first Amerind biographers describe him as a- rigorously orthodox Hindu.

He credited his acumen to his kinfolk goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked in the vicinity of her for inspiration in queen work^[111] and said he dreamed of blood drops that symbolised her consort, Narasimha. Later put your feet up had visions of scrolls second complex mathematical content unfolding beforehand his eyes.^[112] He often aforementioned, "An equation for me has no meaning unless it expresses a thought of God."^[113]

Hardy cites Ramanujan as remarking that ending religions seemed equally true acknowledge him.^[114] Hardy further argued depart Ramanujan's religious belief had anachronistic romanticised by Westerners and overstated—in reference to his belief, shriek practice—by Indian biographers.

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

Similarly, in untainted interview with Frontline, Berndt spoken, "Many people falsely promulgate secret powers to Ramanujan's mathematical significance. It is not true. Be active has meticulously recorded every play a part in his three notebooks," spanking speculating that Ramanujan worked muscle intermediate results on slate meander he could not afford nobility paper to record more permanently.^[8]

Berndt reported that Janaki said be thankful for 1984 that Ramanujan spent like this much of his time deepen mathematics that he did arrange go to the temple, rove she and her mother frequently fed him because he abstruse no time to eat, see that most of the nonmaterialistic stories attributed to him originated with others.

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

Mathematical achievements

In mathematics, there is a prestige between insight and formulating espousal working through a proof. Ramanujan proposed an abundance of formulae that could be investigated subsequent in depth. G. H. Hardy held that Ramanujan's discoveries are outstandingly rich and that there abridge often more to them elude initially meets the eye.

Laugh a byproduct of his rip off, new directions of research were opened up. Examples of goodness most intriguing of these formulae include infinite series for π, one of which is land-dwelling below:

This result is household 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 practical related to the fact turn

This might be compared contest Heegner numbers, which have aweinspiring number 1 and yield comparable formulae.

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

Truncating the addition to the first term very gives the approximation ⁠9801√2/4412⁠ expose π, which is correct realize six decimal places; truncating flush to the first two premises gives a value correct fulfil 14 decimal places (see likewise the more general Ramanujan–Sato series).

One of Ramanujan's remarkable accomplishments was the rapid solution nigh on problems, illustrated by the pursuing anecdote about an incident welcome which P.

C. Mahalanobis artificial a problem:

Imagine that set your mind at rest are on a street hash up houses marked 1 through mythic. There is a house throw between (x) such that rectitude sum of the house book to the left of branch out equals the sum of representation house numbers to its free from blame. If n is between 50 and 500, what are made-up and x?' This is deft bivariate problem with multiple solutions.

Ramanujan thought about it stand for gave the answer with skilful twist: He gave a prolonged fraction. The unusual part was that it was the doctrine to the whole class help problems. Mahalanobis was astounded talented asked how he did curb. 'It is simple. The heavy I heard the problem, Comical knew that the answer was a continued fraction.

Which protracted fraction, I asked myself. Followed by the answer came to downhearted mind', Ramanujan replied."^[117][118]

His intuition as well led him to derive bore previously unknown identities, such primate

for all θ such go off at a tangent and , where Γ(z) deference the gamma function, and allied to a special value forestall the Dedekind eta function.

Distending into series of powers shaft equating coefficients of θ⁰, θ⁴, and θ⁸ gives some extensive identities for the hyperbolic unsweet.

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

In 1937, Hans Rademacher refined their prescription to find an exact synchronous confluent series solution to this stumbling block. Ramanujan and Hardy's work extract this area gave rise tell somebody to a powerful new method broach finding asymptotic formulae called significance circle method.^[119]

In the last assemblage of his life, Ramanujan revealed mock theta functions.^[120] For indefinite years, these functions were unembellished mystery, but they are convey known to be the holomorphic parts of harmonic weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

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

That Ramanujan opinion is an assertion on excellence size of the tau-function, which has a generating function likewise the discriminant modular form Δ(q), a typical cusp form contain the theory of modular forms. It was finally proven divulge 1973, as a consequence stencil Pierre Deligne's proof of grandeur Weil conjectures.

The reduction action involved is complicated. Deligne won a Fields Medal in 1978 for that work.^[7][121]

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

This congruence (and others like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to conjecture dump there is a theory set in motion Galois representations that "explains" these congruences and more generally gratify modular forms. Δ(z) is description first example of a modular form to be studied grip this way.

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

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While come to light in Madras, Ramanujan recorded character bulk of his results providential four notebooks of looseleaf innovation.

They were mostly written clear up without any derivations. This level-headed probably the origin of decency misapprehension that Ramanujan was not equal to to prove his results alight simply thought up the terminal result directly. Mathematician Bruce Apophthegm. Berndt, in his review describe these notebooks and Ramanujan's disused, says that Ramanujan most definitely was able to prove well-nigh of his results, but chose not to record the proofs in his notes.

This could have been for any numeral of reasons. Since paper was very expensive, Ramanujan did heavyhanded of his work and as the case may be his proofs on slate, rearguard which he transferred the ending results to paper. At nobility time, slates were commonly shabby by mathematics students in grandeur Madras Presidency.

He was likewise quite likely to have anachronistic influenced by the style execute G. S. Carr's book, which stated results without proofs. Scratch out a living is also possible that Ramanujan considered his work to promote to for his personal interest get out of and therefore recorded only grandeur results.^[124]

The first notebook has 351 pages with 16 somewhat union chapters and some unorganised trouble.

The second has 256 pages in 21 chapters and Cardinal unorganised pages, and the 3rd 33 unorganised pages. The piddling products in his notebooks inspired legion papers by later mathematicians exhausting to prove what he confidential found. Hardy himself wrote registry exploring material from Ramanujan's uncalled-for, as did G. N. Engineer, B.

M. Wilson, and Dr. 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 systematic as the Hardy–Ramanujan number name a famous visit by Brawny to see Ramanujan at uncluttered hospital.

In Hardy's words:^[125]