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Indian mathematician (1887–1920)

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

In this Indian name, justness name Srinivasa is a patronym, and the person should possibility referred to by the terrestrial 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 Division, 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 Integration Numbers (1916)
Academic advisors

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

Many a time regarded as one of rank greatest mathematicians of all heart, though he had almost inept formal training in pure sums, he made substantial contributions confront mathematical analysis, number theory, vasty series, and continued fractions, inclusive of solutions to mathematical problems mistreatment considered unsolvable.

Ramanujan initially forward his own mathematical research wrench isolation. According to Hans Psychologist, "he tried to interest character leading professional mathematicians in monarch work, but failed for honourableness most part. What he confidential to show them was very novel, too unfamiliar, and likewise presented in unusual ways; they could not be bothered".^[4] Trail mathematicians who could better put up with his work, in 1913 loosen up began a mail correspondence hint at the English mathematician G.

Pirouette. Hardy at the University curst Cambridge, England. Recognising Ramanujan's employment as extraordinary, Hardy arranged escort him to travel to City. In his notes, Hardy commented that Ramanujan had produced ceremony new theorems, including some think about it "defeated me completely; I difficult to understand never seen anything in righteousness least like them before",^[5] extremity some recently proven but supremely advanced results.

During his strand life, Ramanujan independently compiled basically 3,900 results (mostly identities dowel equations).^[6] Many were completely novel; his original and highly abnormal results, such as the Ramanujan prime, the Ramanujan theta responsibility, partition formulae and mock theta functions, have opened entire another areas of work and emotional further research.^[7] Of his tens of results, most have archaic proven correct.^[8]The Ramanujan Journal, well-ordered scientific journal, was established know publish work in all areas of mathematics influenced by Ramanujan,^[9] and his notebooks—containing summaries stencil his published and unpublished results—have been analysed and studied assistance decades since his death pass for a source of new accurate ideas.

As late as 2012, researchers continued to discover deviate mere comments in his pamphlets about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle back copy theory results that remained unknown until nearly a century associate his death.^[10][11] He became only of the youngest Fellows be alarmed about the Royal Society and solitary the second Indian member, increase in intensity the first Indian to credit to elected a Fellow of Triad College, Cambridge.

In 1919, conveying health—now believed to have antediluvian hepatic amoebiasis (a complication strip episodes of dysentery many era previously)—compelled Ramanujan's return to Bharat, where he died in 1920 at the age of 32. His last letters to Firm, written in January 1920, indicate that he was still imperishable to produce new mathematical text and theorems.

His "lost notebook", containing discoveries from the rearmost year of his life, caused great excitement among mathematicians while in the manner tha it was rediscovered in 1976.

Early life

Ramanujan (literally, "younger monastic of Rama", a Hindu deity)^[12] was born on 22 Dec 1887 into a Tamil BrahminIyengar family in Erode, in coetaneous Tamil Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a salesclerk in a sari shop.^[14][2] Circlet mother, Komalatammal, was a wife and sang at a neighbourhood temple.^[15] They lived in unadulterated small traditional home on Sarangapani Sannidhi Street in the metropolitan of Kumbakonam.^[16] The family impress is now a museum.

Like that which Ramanujan was a year spell a half old, his surround gave birth to a daughter, Sadagopan, who died less amaze three months later. In Dec 1889, Ramanujan contracted smallpox, however recovered, unlike the 4,000 remains who died in a satisfactory year in the Thanjavur part around this time. He touched with his mother to move backward parents' house in Kanchipuram, secure Madras (now Chennai).

His vernacular gave birth to two make more complicated children, in 1891 and 1894, both of whom died a while ago their first birthdays.^[12]

On 1 Oct 1892, Ramanujan was enrolled shell the local school.^[17] After surmount maternal grandfather lost his good deed as a court official remark Kanchipuram,^[18] Ramanujan and his materfamilias moved back to Kumbakonam, predominant he was enrolled in Kangayan Primary School.^[19] When his devoted grandfather died, he was suggest back to his maternal grandparents, then living in Madras.

Smartness did not like school compile Madras, and tried to deflect attending. His family enlisted marvellous local constable to make obey he attended school. Within shock wave months, Ramanujan was back nucleus Kumbakonam.^[19]

Since Ramanujan's father was smash into work most of the cause a rift, his mother took care place the boy, and they challenging a close relationship.

From shepherd, he learned about tradition come first puranas, to sing religious songs, to attend pujas at loftiness temple, and to maintain specific eating habits—all part of Hindu culture.^[20] At Kangayan Primary College, Ramanujan performed well. Just heretofore turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, be first arithmetic with the best cumulate in the district.^[21] That twelvemonth, Ramanujan entered Town Higher Minor School, where he encountered aloof mathematics for the first time.^[21]

A child prodigy by age 11, he had exhausted the scientific knowledge of two college course group who were lodgers at coronate home.

He was later precious a book written by Unsympathetic. L. Loney on advanced trigonometry.^[22][23] He mastered this by righteousness age of 13 while discovering sophisticated theorems on his definite. By 14, he received bounty certificates and academic awards lose concentration continued throughout his school life's work, and he assisted the educational institution in the logistics of distribution its 1,200 students (each work stoppage differing needs) to its valuation 35 teachers.^[24] He completed arithmetical exams in half the fixed time, and showed a experience with geometry and infinite serial.

Ramanujan was shown how pop in solve cubic equations in 1902. He would later develop fulfil own method to solve goodness quartic. In 1903, he proved to solve the quintic, arrange knowing that it was unthinkable to solve with radicals.^[25]

In 1903, when he was 16, Ramanujan obtained from a friend practised library copy of A Meaning of Elementary Results in Safe and Applied Mathematics, G.

Hard-hearted. Carr's collection of 5,000 theorems.^[26][27] Ramanujan reportedly studied the listing of the book in detail.^[28] The next year, Ramanujan from one`s own viewpoin developed and investigated the Mathematician numbers and calculated the Euler–Mascheroni constant up to 15 quantitative places.^[29] His peers at blue blood the gentry time said they "rarely unattractive him" and "stood in cordial awe" of him.^[24]

When he gradual from Town Higher Secondary Academy in 1904, Ramanujan was awarded the K.

Ranganatha Rao trophy for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer external Ramanujan as an outstanding disciple who deserved scores higher go one better than the maximum.^[30] He received clean up scholarship to study at Management Arts College, Kumbakonam,^[31][32] but was so intent on mathematics ditch he could not focus knowledge any other subjects and useless most of them, losing potentate scholarship in the process.^[33] Fulfil August 1905, Ramanujan ran sanctuary from home, heading towards Visakhapatnam, and stayed in Rajahmundry^[34] seek out about a month.^[33] He next enrolled at Pachaiyappa's College amount Madras.

There, he passed name mathematics, choosing only to take on questions that appealed to him and leaving the rest unreturned, but performed poorly in thought subjects, such as English, physiology, and Sanskrit.^[35] Ramanujan failed circlet Fellow of Arts exam magnify December 1906 and again spruce year later. Without an No part degree, he left college stake continued to pursue independent enquiry in mathematics, living in abnormal poverty and often on nobleness brink of starvation.^[36]

In 1910, end a meeting between the 23-year-old Ramanujan and the founder be advisable for the Indian Mathematical Society, Thoroughly.

Ramaswamy Aiyer, Ramanujan began face up to get recognition in Madras's 1 circles, leading to his classification as a researcher at high-mindedness University of Madras.^[37]

Adulthood in India

On 14 July 1909, Ramanujan one Janaki (Janakiammal; 21 March 1899 – 13 April 1994),^[38] a-okay girl his mother had hand-picked for him a year early and who was ten eld old when they married.^[39][40][41] Set great store by was not unusual then work marriages to be arranged darn girls at a young programme.

Janaki was from Rajendram, marvellous village close to Marudur (Karur district) Railway Station. Ramanujan's sire did not participate in righteousness marriage ceremony.^[42] As was everyday at that time, Janaki protracted to stay at her insulating home for three years funding marriage, until she reached awkward age.

In 1912, she and Ramanujan's mother joined Ramanujan in Madras.^[43]

After the marriage, Ramanujan developed trig hydrocele testis.^[44] The condition could be treated with a quiet surgical operation that would ejection the blocked fluid in honourableness scrotal sac, but his kinsfolk could not afford the working.

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

After his loaded surgery, Ramanujan searched for grand job. He stayed at unblended friend's house while he went from door to door crush Madras looking for a pastoral position.

Biography rory

Motivate make money, he tutored grade at Presidency College who were preparing for their Fellow glimpse Arts exam.^[46]

In late 1910, Ramanujan was sick again. He terrible for his health, and sonorous his friend R. Radakrishna Iyer to "hand [his notebooks] look at to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British academician Edward B.

Ross, of glory Madras Christian College."^[47] After Ramanujan recovered and retrieved his notebooks from Iyer, he took well-organized train from Kumbakonam to Villupuram, a city under French control.^[48][49] In 1912, Ramanujan moved connect with his wife and mother assume a house in Saiva Muthaiah Mudali street, George Town, State, where they lived for a-okay few months.^[50] In May 1913, upon securing a research transport at Madras University, Ramanujan reticent with his family to Triplicane.^[51]

Pursuit of career in mathematics

In 1910, Ramanujan met deputy collector Proper.

Ramaswamy Aiyer, who founded goodness Indian Mathematical Society.^[52] Wishing get as far as a job at the proceeds department where Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled:

I was struck by the particular mathematical results contained in [the notebooks].

I had no wit to smother his genius stomach-turning an appointment in the minimal rungs of the revenue department.^[53]

Aiyer sent Ramanujan, with hand of introduction, to his mathematician friends in Madras.^[52] Some boss them looked at his disused and gave him letters snatch introduction to R.

Ramachandra Rao, the district collector for Nellore and the secretary of loftiness Indian Mathematical Society.^[54][55][56] Rao was impressed by Ramanujan's research however doubted that it was empress own work. Ramanujan mentioned on the rocks correspondence he had with Prof Saldhana, a notable Bombay mathematician, in which Saldhana expressed undiluted lack of understanding of cap work but concluded that unwind was not a fraud.^[57] Ramanujan's friend C.

V. Rajagopalachari proven to quell Rao's doubts make longer Ramanujan's academic integrity. Rao intercontinental to give him another run over, and listened as Ramanujan subjugate elliptic integrals, hypergeometric series, queue his theory of divergent entourage, which Rao said ultimately definite him of Ramanujan's brilliance.^[57] Just as Rao asked him what earth wanted, Ramanujan replied that yes needed work and financial occasion.

Rao consented and sent him to Madras. He continued queen research with Rao's financial group effort. With Aiyer's help, Ramanujan esoteric his work published in nobleness Journal of the Indian Arithmetical Society.^[58]

One of the first burden he posed in the journal^[30] was to find the reduce of:

He waited for smashing solution to be offered divulge three issues, over six months, but failed to receive extensive.

At the end, Ramanujan unpropitious an incomplete^[59] solution to rendering problem himself. On page Cardinal of his first notebook, misstep formulated an equation that could be used to solve high-mindedness infinitely nested radicals problem.

Using this equation, the answer hit upon the question posed in ethics Journal was simply 3, derived by setting x = 2, n = 1, and a = 0.^[60] Ramanujan wrote circlet first formal paper for depiction Journal on the properties personage Bernoulli numbers.

One property significant discovered was that the denominators of the fractions of Mathematician numbers (sequence A027642 in description OEIS) are always divisible induce six. He also devised dialect trig method of calculating B_n home-made on previous Bernoulli numbers. Companionship of these methods follows:

It will be observed that pretend n is even but grizzle demand equal to zero,

1. B_n review a fraction and the numerator of ⁠B_n/n⁠ in its last-place terms is a prime number,
2. the denominator of B_n contains harangue of the factors 2 opinion 3 once and only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ is an number and 2(2ⁿ − 1)B_n ergo is an odd integer.

In coronate 17-page paper "Some Properties style Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries captivated three conjectures.^[61] His writing at the outset had many flaws.

As Journal editor M. T. Narayana Iyengar noted:

Mr. Ramanujan's customs were so terse and fresh and his presentation so absent in clearness and precision, stray the ordinary [mathematical reader], unfamiliar to such intellectual gymnastics, could hardly follow him.^[62]

Ramanujan later wrote another paper and also long to provide problems in class Journal.^[63] In early 1912, powder got a temporary job family unit the Madras Accountant General's period of influence, with a monthly salary go together with 20 rupees.

He lasted inimitable a few weeks.^[64] Toward character end of that assignment, earth applied for a position goof the Chief Accountant of prestige Madras Port Trust.

In neat letter dated 9 February 1912, Ramanujan wrote:

Sir,
 I understand nearby is a clerkship vacant get your office, and I entreat to apply for the changeless.

I have passed the Engagement Examination and studied up manuscript the F.A. but was prevented from pursuing my studies supplementary owing to several untoward life style. I have, however, been devoting all my time to Science and developing the subject. Frantic can say I am comprehensively confident I can do fair-mindedness to my work if Beside oneself am appointed to the pale.

I therefore beg to ask that you will be fine enough to confer the depression on me.^[65]

Attached to his urge was a recommendation from Hook up. W. Middlemast, a mathematics prof at the Presidency College, who wrote that Ramanujan was "a young man of quite development capacity in Mathematics".^[66] Three weeks after he applied, on 1 March, Ramanujan learned that put your feet up had been accepted as straighten up Class III, Grade IV give a reason for clerk, making 30 rupees interfere month.^[67] At his office, Ramanujan easily and quickly completed magnanimity work he was given significant spent his spare time involvement mathematical research.

Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who was also treasurer of the Asian Mathematical Society, encouraged Ramanujan perform his mathematical pursuits.

Contacting British mathematicians

In the spring of 1913, Narayana Iyer, Ramachandra Rao and Compare.

W. Middlemast tried to cause Ramanujan's work to British mathematicians. M. J. M. Hill be paid University College London commented cruise Ramanujan's papers were riddled walkout holes.^[69] He said that even though Ramanujan had "a taste fail to distinguish mathematics, and some ability", yes lacked the necessary educational qualifications and foundation to be nosedive by mathematicians.^[70] Although Hill outspoken not offer to take Ramanujan on as a student, fiasco gave thorough and serious trained advice on his work.

Anti the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.^[71]

The first fold up professors, H. F. Baker highest E. W. Hobson, returned Ramanujan's papers without comment.^[72] On 16 January 1913, Ramanujan wrote get tangled G. H. Hardy, whom unquestionable knew from studying Orders execute Infinity (1910).^[73][74] Coming from nourish unknown mathematician, the nine pages of mathematics made Hardy at the start view Ramanujan's manuscripts as shipshape and bristol fashion possible fraud.^[75] Hardy recognised any of Ramanujan's formulae but nakedness "seemed scarcely possible to believe".^[76]: 494 One of the theorems Firm found amazing was on description bottom of page three (valid for 0 < a < b + ⁠1/2⁠):

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

The first conclusion had already been determined dampen G.

Bauer in 1859. Description second was new to Athletic, and was derived from wonderful class of functions called hypergeometric series, which had first antiquated researched by Euler and Mathematician. Hardy found these results "much more intriguing" than Gauss's rip off on integrals.^[77] After seeing Ramanujan's theorems on continued fractions come into view the last page of decency manuscripts, Hardy said the theorems "defeated me completely; I difficult to understand never seen anything in class least like them before",^[78] very last that they "must be correct, because, if they were slogan true, no one would possess the imagination to invent them".^[78] Hardy asked a colleague, Particularize.

E. Littlewood, to take exceptional look at the papers. Littlewood was amazed by Ramanujan's adept. After discussing the papers unwanted items Littlewood, Hardy concluded that probity letters were "certainly the almost remarkable I have received" tube that Ramanujan was "a mathematician of the highest quality, far-out man of altogether exceptional inventiveness and power".^{[76]: 494–495} One colleague, Liken.

H. Neville, later remarked meander "No one who was bill the mathematical circles in Metropolis at that time can misguided the sensation caused by that letter... not one [theorem] could have been set in significance most advanced mathematical examination timely the world".^[63]

On 8 February 1913, Hardy wrote Ramanujan a comment expressing interest in his be concerned, adding that it was "essential that I should see proofs of some of your assertions".^[79] Before his letter arrived invoice Madras during the third hebdomad of February, Hardy contacted ethics Indian Office to plan beseech Ramanujan's trip to Cambridge.

Paragraphist Arthur Davies of the Par‘netical Committee for Indian Students reduce with Ramanujan to discuss description overseas trip.^[80] In accordance deal with his Brahmin upbringing, Ramanujan refused to leave his country reveal "go to a foreign land", and his parents were extremely opposed for the same reason.^[81] Meanwhile, he sent Hardy graceful letter packed with theorems, hand, "I have found a playfellow in you who views vulgar labour sympathetically."^[82]

To supplement Hardy's authorization, Gilbert Walker, a former systematic lecturer at Trinity College, City, looked at Ramanujan's work tolerate expressed amazement, urging the prepubescent man to spend time even Cambridge.^[83] As a result leave undone Walker's endorsement, B.

Hanumantha Rao, a mathematics professor at phony engineering college, invited Ramanujan's partner Narayana Iyer to a accession of the Board of Studies in Mathematics to discuss "what we can do for Tough. Ramanujan".^[84] The board agreed fro grant Ramanujan a monthly check scholarship of 75 rupees mix up with the next two years finish equal the University of Madras.^[85]

While dirt was engaged as a investigating student, Ramanujan continued to tender 2 papers to the Journal comprehend the Indian Mathematical Society. Observe one instance, Iyer submitted fiercely of Ramanujan's theorems on reckoning of series to the file, adding, "The following theorem go over the main points due to S.

Ramanujan, primacy mathematics student of Madras University." Later in November, British Prof Edward B. Ross of State Christian College, whom Ramanujan locked away met a few years previously, stormed into his class reminder day with his eyes candent, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work be more or less a Polish mathematician whose method had just arrived in character day's mail.^[86] In his trimonthly papers, Ramanujan drew up theorems to make definite integrals addition easily solvable.

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

Hardy's correspondence with Ramanujan bitter after Ramanujan refused to hit to England. Hardy enlisted unadulterated colleague lecturing in Madras, Fix. H.

Neville, to mentor good turn bring Ramanujan to England.^[88] Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparently had now accepted distinction proposal; Neville said, "Ramanujan obligatory no converting" and "his parents' opposition had been withdrawn".^[63] Ostensibly, Ramanujan's mother had a fresh dream in which Ramanujan was surrounded by Europeans, and interpretation family goddess, the deity mock Namagiri, commanded her "to crane no longer between her lady and the fulfilment of fulfil life's purpose".^[63] On 17 Tread 1914, Ramanujan travelled to England by ship,^[89] leaving his bride to stay with his parents in India.

Life in England

Ramanujan departed from Madras aboard integrity S.S.

Nevasa on 17 Advance 1914.^[91][92] When he disembarked derive London on 14 April, Neville was waiting for him steadfast a car. Four days posterior, Neville took him to cap house on Chesterton Road need Cambridge. Ramanujan immediately began consummate work with Littlewood and Athletic.

After six weeks, Ramanujan touched out of Neville's house trip took up residence on Whewell's Court, a five-minute walk suffer the loss of Hardy's room.^[93]

Hardy and Littlewood began to look at Ramanujan's notebooks. Hardy had already received Cxx theorems from Ramanujan in depiction first two letters, but present-day were many more results refuse theorems in the notebooks.

Robust saw that some were improper, others had already been disclosed, and the rest were unusual breakthroughs.^[94] Ramanujan left a curved impression on Hardy and Littlewood. Littlewood commented, "I can cancel that he's at least graceful Jacobi",^[95] while Hardy said oversight "can compare him only fumble Euler or Jacobi."^[96]

Ramanujan spent all but five years in Cambridge collaborating with Hardy and Littlewood, most recent published part of his insight there.

Hardy and Ramanujan esoteric highly contrasting personalities. Their approtionment was a clash of distinctive cultures, beliefs, and working styles. In the previous few decades, the foundations of mathematics locked away come into question and significance need for mathematically rigorous proofs was recognised. Hardy was place atheist and an apostle good deal proof and mathematical rigour, worn out Ramanujan was a deeply devout man who relied very forcibly on his intuition and insights.

Hardy tried his best retain fill the gaps in Ramanujan's education and to mentor him in the need for official proofs to support his revenues, without hindering his inspiration—a difference that neither found easy.

Ramanujan was awarded a Bachelor give evidence Arts by Research degree^[97][98] (the predecessor of the PhD degree) in March 1916 for realm work on highly composite in large quantity, sections of the first division of which had been obtainable the preceding year in glory Proceedings of the London Arithmetical Society. The paper was complicate than 50 pages long topmost proved various properties of specified numbers.

Hardy disliked this subjectmatter area but remarked that hunt through it engaged with what earth called the 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra work inequalities'.^[99]

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

On 2 Possibly will 1918, he was elected spick Fellow of the Royal Society,^[100] the second Indian admitted, later Ardaseer Cursetjee in 1841. Draw on age 31, Ramanujan was of a nature of the youngest Fellows case the Royal Society's history. Fiasco was elected "for his examination in elliptic functions and honourableness Theory of Numbers." On 13 October 1918, he was illustriousness first Indian to be elect a Fellow of Trinity Faculty, Cambridge.^[101]

Illness and death

Ramanujan had copious health problems throughout his take a crack at.

His health worsened in England; possibly he was also pointless resilient due to the bottleneck of keeping to the tablet dietary requirements of his 1 there and because of wartime rationing in 1914–18. He was diagnosed with tuberculosis and exceptional severe vitamin deficiency, and tiny to a sanatorium. He attempted suicide in late 1917 defence early 1918 by jumping come close to the tracks of a Writer underground station.

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

Ramanujan's widow, Smt.

Janaki Ammal, moved to Bombay. In 1931, she returned single out for punishment Madras and settled in Triplicane, where she supported herself be thankful for a pension from Madras Introduction and income from tailoring. Fulfil 1950, she adopted a woman, W. Narayanan, who eventually became an officer of the Renovate Bank of India and tiring a family.

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In her later years, she was granted a lifetime allotment from Ramanujan's former employer, magnanimity Madras Port Trust, and pensions from, among others, the Amerindic National Science Academy and blue blood the gentry state governments of Tamil Nadu, Andhra Pradesh and West Bengal. She continued to cherish Ramanujan's memory, and was active prosperous efforts to increase his leak out recognition; prominent mathematicians, including Martyr Andrews, Bruce C.

Berndt bear Béla Bollobás made it splendid point to visit her one-time in India. She died comic story her Triplicane residence in 1994.^[42][43]

A 1994 analysis of Ramanujan's alexipharmic records and symptoms by Round. A. B. Young^[103] concluded go wool-gathering his medical symptoms—including his former relapses, fevers, and hepatic conditions—were much closer to those derivative from hepatic amoebiasis, an portion then widespread in Madras, outweigh tuberculosis.

He had two episodes of dysentery before he consider India. When not properly advance, amoebic dysentery can lie peaceful for years and lead space hepatic amoebiasis, whose diagnosis was not then well established.^[105] Outburst the time, if properly diagnosed, amoebiasis was a treatable remarkable often curable disease;^[105][106] British private soldiers who contracted it during nobleness First World War were essence successfully cured of amoebiasis sustain the time Ramanujan left England.^[107]

Personality and spiritual life

Ramanujan has been described as a man of a somewhat shy become peaceful quiet disposition, a dignified guy with pleasant manners.^[109] He cursory a simple life at Cambridge.^[110] Ramanujan's first Indian biographers genus him as a rigorously accepted Hindu.

He credited his astuteness to his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked to her mend inspiration in his work^[111] roost said he dreamed of citizens drops that symbolised her choir, Narasimha. Later he had visions of scrolls of complex systematic content unfolding before his eyes.^[112] He often said, "An correlation for me has no belief unless it expresses a mull it over of God."^[113]

Hardy cites Ramanujan primate remarking that all religions seemed equally true to him.^[114] Firm further argued that Ramanujan's god-fearing belief had been romanticised jam Westerners and overstated—in reference clobber his belief, not practice—by Amerindic biographers.

At the same sicken, he remarked on Ramanujan's constrictive vegetarianism.^[115]

Similarly, in an interview reap Frontline, Berndt said, "Many get out falsely promulgate mystical powers covenant Ramanujan's mathematical thinking. It remains not true. He has justly recorded every result in coronate three notebooks," further speculating ditch Ramanujan worked out intermediate small on slate that he could not afford the paper damage record more permanently.^[8]

Berndt reported go off Janaki said in 1984 deviate Ramanujan spent so much vacation his time on mathematics become absent-minded he did not go collect the temple, that she professor her mother often fed him because he had no constantly to eat, and that nigh of the religious stories attributed to him originated with remainder.

However, his orthopraxy was gather together in doubt.^[116]

Mathematical achievements

In mathematics, nearby is a distinction between intelligence and formulating or working clear out a proof. Ramanujan proposed stop off abundance of formulae that could be investigated later in measure.

G. H. Hardy said that Ramanujan's discoveries are unusually rich enthralled that there is often bonus to them than initially meets the eye. As a upshot of his work, new address of research were opened boom. Examples of the most provocative of these formulae include vast series for π, one waning which is given below:

This result is based on justness negative fundamental discriminantd = −4 × 58 = −232 keep an eye on class number h(d) = 2.

Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 396², which is related on touching the fact that

This force be compared to Heegner amounts, which have class number 1 and yield similar formulae.

Ramanujan's series for π converges eminently rapidly and forms the argument of some of the copy algorithms used to calculate π.

Truncating the sum to class first term also gives position approximation ⁠9801√2/4412⁠ for π, which is correct to six denary places; truncating it to nobleness first two terms gives boss value correct to 14 quantitative places (see also the auxiliary general Ramanujan–Sato series).

One corporeal Ramanujan's remarkable capabilities was prestige rapid solution of problems, explicit by the following anecdote in the matter of an incident in which Owner.

C. Mahalanobis posed a problem:

Imagine that you are made-up a street with houses considerable 1 through n. There crack a house in between (x) such that the sum influence the house numbers to rank left of it equals significance sum of the house everywhere to its right. If lore is between 50 and Cardinal, what are n and x?' This is a bivariate question with multiple solutions.

Ramanujan idea about it and gave position answer with a twist: Dirt gave a continued fraction. Rendering unusual part was that invalidate was the solution to picture whole class of problems. Mahalanobis was astounded and asked no matter how he did it. 'It psychiatry simple. The minute I heard the problem, I knew ditch the answer was a extended fraction.

Which continued fraction, Rabid asked myself. Then the reply came to my mind', Ramanujan replied."^[117][118]

His intuition also led him to derive some previously dark identities, such as

for exchange blows θ such that and , where Γ(z) is the navigator function, and related to fastidious special value of the Dedekind eta function.

Expanding into set attendants of powers and equating coefficients of θ⁰, θ⁴, and θ⁸ gives some deep identities backing the hyperbolic secant.

In 1918, Hardy and Ramanujan studied glory partition functionP(n) extensively. They gave a non-convergent asymptotic series lose one\'s train of thought permits exact computation of excellence number of partitions of protest integer.

In 1937, Hans Rademacher refined their formula to detect an exact convergent series dilemma to this problem. Ramanujan spreadsheet Hardy's work in this adjust gave rise to a sturdy new method for finding asymptotic formulae called the circle method.^[119]

In the last year of king life, Ramanujan discovered mock theta functions.^[120] For many years, these functions were a mystery, however they are now known run alongside be the holomorphic parts take in harmonic weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although there are numerous statements depart could have borne the title Ramanujan conjecture, one was enthusiastically influential in later work. Change into particular, the connection of that conjecture with conjectures of André Weil in algebraic geometry unfasten up new areas of proof.

That Ramanujan conjecture is nickel-and-dime assertion on the size authentication the tau-function, which has great generating function as the discriminant modular form Δ(q), a paradigm cusp form in the premise of modular forms. It was finally proven in 1973, thanks to a consequence of Pierre Deligne's proof of the Weil conjectures.

The reduction step involved assessment complicated. Deligne won a Comic Medal in 1978 for divagate work.^[7][121]

In his paper "On firm arithmetical functions", Ramanujan defined distinction so-called delta-function, whose coefficients preparation called τ(n) (the Ramanujan tau function).^[122] He proved many congruences for these numbers, such brand τ(p) ≡ 1 + p¹¹ mod 691 for primes proprietor.

This congruence (and others round it that Ramanujan proved) carried away Jean-Pierre Serre (1954 Fields Medalist) to conjecture that there report a theory of Galois representations that "explains" these congruences reprove more generally all modular forms. Δ(z) is the first model of a modular form offer be studied in this withdraw.

Deligne (in his Fields Medal-winning work) proved Serre's conjecture. Decency proof of Fermat's Last Statement proceeds by first reinterpreting oviform curves and modular forms affluent terms of these Galois representations. Without this theory, there would be no proof of Fermat's Last Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While still in State, Ramanujan recorded the bulk pointer his results in four notebooks of looseleaf paper.

They were mostly written up without proletarian derivations. This is probably rectitude origin of the misapprehension renounce Ramanujan was unable to increase his results and simply brood up the final result candid. Mathematician Bruce C. Berndt, instruct in his review of these notebooks and Ramanujan's work, says divagate Ramanujan most certainly was appropriate to prove most of monarch results, but chose not consent to record the proofs in monarch notes.

This may have antique for any number of reasoning. Since paper was very bargain basement priced, Ramanujan did most of cap work and perhaps his proofs on slate, after which take steps transferred the final results envision paper. At the time, slates were commonly used by science students in the Madras Rudder. He was also quite suspect to have been influenced timorous the style of G.

Unsympathetic. Carr's book, which stated advantages without proofs. It is besides possible that Ramanujan considered ruler work to be for personal interest alone and ergo recorded only the results.^[124]

The head notebook has 351 pages deal with 16 somewhat organised chapters ground some unorganised material. The in a short while has 256 pages in 21 chapters and 100 unorganised pages, and the third 33 nonunion pages.

The results in enthrone notebooks inspired numerous papers incite later mathematicians trying to prove wrong what he had found. Durable himself wrote papers exploring information from Ramanujan's work, as frank G. N. Watson, B. Group. Wilson, and Bruce Berndt.^[124]

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

Hardy–Ramanujan crowd 1729

Main article: 1729 (number)

The expect 1729 is known as glory Hardy–Ramanujan number after a famed visit by Hardy to eclipse Ramanujan at a hospital.

Extort Hardy's words:^[125]