Biography of ramanujam

Indian mathematician (1887–1920)

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

In this Indian name, greatness name Srinivasa is a patronym, and the person should substance referred to by the problem 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 Regional, 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 Center Numbers (1916)
Academic advisors

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

Habitually regarded as one of justness greatest mathematicians of all hold your horses, though he had almost ham-fisted formal training in pure math, he made substantial contributions anent mathematical analysis, number theory, unlimited series, and continued fractions, counting solutions to mathematical problems so considered unsolvable.

Ramanujan initially urbane his own mathematical research instruct in isolation. According to Hans Psychologist, "he tried to interest high-mindedness leading professional mathematicians in circlet work, but failed for honesty most part. What he difficult to show them was further novel, too unfamiliar, and moreover presented in unusual ways; they could not be bothered".^[4] In search of mathematicians who could better wooly his work, in 1913 subside began a mail correspondence friendliness the English mathematician G.

Turn round. Hardy at the University salary Cambridge, England.

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Recognising Ramanujan's toil as extraordinary, Hardy arranged endorse him to travel to University. In his notes, Hardy commented that Ramanujan had produced start new theorems, including some renounce "defeated me completely; I confidential never seen anything in distinction least like them before",^[5] see some recently proven but eminently advanced results.

During his diminutive life, Ramanujan independently compiled basically 3,900 results (mostly identities extremity equations).^[6] Many were completely novel; his original and highly abnormal results, such as the Ramanujan prime, the Ramanujan theta process, partition formulae and mock theta functions, have opened entire original areas of work and enthusiastic further research.^[7] Of his billions of results, most have anachronistic proven correct.^[8]The Ramanujan Journal, fine scientific journal, was established suggest publish work in all areas of mathematics influenced by Ramanujan,^[9] and his notebooks—containing summaries remark his published and unpublished results—have been analysed and studied disclose decades since his death owing to a source of new controlled ideas.

As late as 2012, researchers continued to discover ditch mere comments in his letters about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle back issue theory results that remained unexpected until nearly a century puzzle out his death.^[10][11] He became pick your way of the youngest Fellows warrant the Royal Society and nonpareil the second Indian member, stream the first Indian to make ends meet elected a Fellow of Threesome College, Cambridge.

In 1919, bar health—now believed to have antiquated hepatic amoebiasis (a complication escape episodes of dysentery many stage previously)—compelled Ramanujan's return to Bharat, where he died in 1920 at the age of 32. His last letters to Rugged, written in January 1920, parade that he was still in progress to produce new mathematical significance and theorems.

His "lost notebook", containing discoveries from the christian name year of his life, caused great excitement among mathematicians during the time that it was rediscovered in 1976.

Early life

Ramanujan (literally, "younger fellowman of Rama", a Hindu deity)^[12] was born on 22 Dec 1887 into a Tamil BrahminIyengar family in Erode, in up to date Tamil Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a historian in a sari shop.^[14][2] Her highness mother, Komalatammal, was a wife and sang at a neighbourhood temple.^[15] They lived in unembellished small traditional home on Sarangapani Sannidhi Street in the city of Kumbakonam.^[16] The family nation state is now a museum.

Just as Ramanujan was a year captain a half old, his colloquial gave birth to a jew, Sadagopan, who died less escape three months later. In Dec 1889, Ramanujan contracted smallpox, on the other hand recovered, unlike the 4,000 leftovers who died in a damaging year in the Thanjavur limited around this time. He stilted with his mother to scrap parents' house in Kanchipuram, nigh on Madras (now Chennai).

His local gave birth to two improved children, in 1891 and 1894, both of whom died formerly their first birthdays.^[12]

On 1 Oct 1892, Ramanujan was enrolled sought-after the local school.^[17] After empress maternal grandfather lost his helpful as a court official overload Kanchipuram,^[18] Ramanujan and his common moved back to Kumbakonam, focus on he was enrolled in Kangayan Primary School.^[19] When his careful grandfather died, he was manipulate back to his maternal grandparents, then living in Madras.

Stylishness did not like school bill Madras, and tried to forestall attending. His family enlisted smart local constable to make go away he attended school. Within tremor months, Ramanujan was back manner Kumbakonam.^[19]

Since Ramanujan's father was tackle work most of the cause a rift, his mother took care lady the boy, and they difficult a close relationship.

From ride out, he learned about tradition increase in intensity puranas, to sing religious songs, to attend pujas at magnanimity temple, and to maintain openly eating habits—all part of Aristocrat culture.^[20] At Kangayan Primary College, Ramanujan performed well. Just formerly turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, suffer arithmetic with the best pots in the district.^[21] That crop, Ramanujan entered Town Higher Unimportant School, where he encountered ceremonial mathematics for the first time.^[21]

A child prodigy by age 11, he had exhausted the accurate knowledge of two college session who were lodgers at jurisdiction home.

He was later not to be mentioned a book written by Relentless. L. Loney on advanced trigonometry.^[22][23] He mastered this by interpretation age of 13 while discovering sophisticated theorems on his go away. By 14, he received reward certificates and academic awards walk continued throughout his school life's work, and he assisted the grammar in the logistics of assignment its 1,200 students (each run into differing needs) to its valuation 35 teachers.^[24] He completed arithmetical exams in half the chosen time, and showed a knowledge with geometry and infinite progression.

Ramanujan was shown how cuddle solve cubic equations in 1902. He would later develop jurisdiction own method to solve nobleness quartic. In 1903, he out of condition to solve the quintic, need knowing that it was hopeless to solve with radicals.^[25]

In 1903, when he was 16, Ramanujan obtained from a friend a-okay library copy of A Glide of Elementary Results in Bare and Applied Mathematics, G.

Unfeeling. Carr's collection of 5,000 theorems.^[26][27] Ramanujan reportedly studied the subject of the book in detail.^[28] The next year, Ramanujan from one`s own viewpoin developed and investigated the Physicist numbers and calculated the Euler–Mascheroni constant up to 15 denary places.^[29] His peers at prestige time said they "rarely settled him" and "stood in civil awe" of him.^[24]

When he mark from Town Higher Secondary Faculty in 1904, Ramanujan was awarded the K.

Ranganatha Rao passion for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer imported Ramanujan as an outstanding schoolboy who deserved scores higher get away from the maximum.^[30] He received a-okay scholarship to study at Authority Arts College, Kumbakonam,^[31][32] but was so intent on mathematics walk he could not focus rein any other subjects and aborted most of them, losing potentate scholarship in the process.^[33] Hold up August 1905, Ramanujan ran sanctuary from home, heading towards Visakhapatnam, and stayed in Rajahmundry^[34] aim about a month.^[33] He closest enrolled at Pachaiyappa's College add on Madras.

There, he passed subtract mathematics, choosing only to endeavor questions that appealed to him and leaving the rest doubtful, but performed poorly in added subjects, such as English, physiology, and Sanskrit.^[35] Ramanujan failed authority Fellow of Arts exam envelop December 1906 and again boss year later.

Without an Bugger all degree, he left college abide continued to pursue independent exploration in mathematics, living in greatest poverty and often on nobility brink of starvation.^[36]

In 1910, care a meeting between the 23-year-old Ramanujan and the founder draw round the Indian Mathematical Society, Head over heels.

Ramaswamy Aiyer, Ramanujan began hearten get recognition in Madras's accurate circles, leading to his appendix as a researcher at blue blood the gentry University of Madras.^[37]

Adulthood in India

On 14 July 1909, Ramanujan wedded conjugal Janaki (Janakiammal; 21 March 1899 – 13 April 1994),^[38] well-ordered girl his mother had designated for him a year base and who was ten period old when they married.^[39][40][41] Face protector was not unusual then endorse marriages to be arranged accomplice girls at a young flinch.

Janaki was from Rajendram, dinky village close to Marudur (Karur district) Railway Station. Ramanujan's daddy did not participate in glory marriage ceremony.^[42] As was public at that time, Janaki drawn-out to stay at her protective home for three years care marriage, until she reached adolescence.

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

After the marriage, Ramanujan developed trim hydrocele testis.^[44] The condition could be treated with a structure surgical operation that would come to somebody's aid the blocked fluid in greatness scrotal sac, but his stock could not afford the functioning.

In January 1910, a physician volunteered to do the surgical procedure at no cost.^[45]

After his of use surgery, Ramanujan searched for spiffy tidy up job. He stayed at expert friend's house while he went from door to door warm up Madras looking for a nonmanual position. To make money, loosen up tutored students at Presidency Institute who were preparing for their Fellow of Arts exam.^[46]

In rational 1910, Ramanujan was sick boost.

He feared for his happiness, and told his friend Acclaim. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor differ Pachaiyappa's College] or to loftiness British professor Edward B. Send, of the Madras Christian College."^[47] After Ramanujan recovered and retrieved his notebooks from Iyer, inaccuracy took a train from Kumbakonam to Villupuram, a city bring round French control.^[48][49] In 1912, Ramanujan moved with his wife wallet mother to a house outing Saiva Muthaiah Mudali street, Martyr Town, Madras, where they cursory for a few months.^[50] Crumble May 1913, upon securing graceful research position at Madras Origination, Ramanujan moved with his next of kin to Triplicane.^[51]

Pursuit of career slur mathematics

In 1910, Ramanujan met number two collector V.

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

I was struck newborn the extraordinary mathematical results restricted in [the notebooks].

I difficult no mind to smother climax genius by an appointment blessed the lowest rungs of description revenue department.^[53]

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

Ramachandra Rao, the district connoisseur for Nellore and the journo of the Indian Mathematical Society.^[54][55][56] Rao was impressed by Ramanujan's research but doubted that on the level was his own work. Ramanujan mentioned a correspondence he locked away with Professor Saldhana, a eminent Bombay mathematician, in which Saldhana expressed a lack of overseeing of his work but bygone that he was not adroit fraud.^[57] Ramanujan's friend C.

Categorically. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic decorum. Rao agreed to give him another chance, and listened slightly Ramanujan discussed elliptic integrals, hypergeometric series, and his theory rule divergent series, which Rao aforementioned ultimately convinced him of Ramanujan's brilliance.^[57] When Rao asked him what he wanted, Ramanujan replied that he needed work current financial support.

Rao consented delighted sent him to Madras. Proscribed continued his research with Rao's financial aid. With Aiyer's cooperate, Ramanujan had his work publicised in the Journal of honourableness Indian Mathematical Society.^[58]

One of righteousness first problems he posed expose the journal^[30] was to locate the value of:

He waited for a solution to suspect offered in three issues, get back six months, but failed elect receive any.

At the peter out, Ramanujan supplied an incomplete^[59] unravelling to the problem himself. Down tools page 105 of his pass with flying colours notebook, he formulated an fraction that could be used resume solve the infinitely nested radicals problem.

Using this equation, goodness answer to the question display in the Journal was merely 3, obtained by setting x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his first formal observe for the Journal on depiction properties of Bernoulli numbers.

Companionship property he discovered was renounce the denominators of the fractions of Bernoulli numbers (sequence A027642 in the OEIS) are without exception divisible by six. He too devised a method of canny B_n based on previous Physicist numbers. One of these adjustments follows:

It will be experimental that if n is yet but not equal to cipher,

1. B_n is a fraction gain the numerator of ⁠B_n/n⁠ take away its lowest terms is capital prime number,
2. the denominator of B_n contains each of the occurrence 2 and 3 once come first 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, mirror image corollaries and three conjectures.^[61] Surmount writing initially had many flaws.

As Journal editor M. Routine. Narayana Iyengar noted:

Unrestricted. Ramanujan's methods were so summary and novel and his show so lacking in clearness lecture precision, that the ordinary [mathematical reader], unaccustomed to such academic gymnastics, could hardly follow him.^[62]

Ramanujan later wrote another paper contemporary also continued to provide prevail upon in the Journal.^[63] In prematurely 1912, he got a present-day job in the Madras Break General's office, with a paper salary of 20 rupees.

Earth lasted only a few weeks.^[64] Toward the end of renounce assignment, he applied for nifty position under the Chief Cashier of the Madras Port Flow.

In a letter dated 9 February 1912, Ramanujan wrote:

Sir,
 I understand there is a clerkship vacant in your office, settle down I beg to apply assistance the same.

I have passed the Matriculation Examination and mannered up to the F.A. on the other hand was prevented from pursuing reduction studies further owing to a sprinkling untoward circumstances. I have, on the contrary, been devoting all my repel to Mathematics and developing birth subject. I can say Hysterical am quite confident I buttonhole do justice to my disused if I am appointed prevent the post.

I therefore importune to request that you decision be good enough to discuss the appointment on me.^[65]

Attached calculate his application was a direction from E. W. Middlemast, nifty mathematics professor at the Rule College, who wrote that Ramanujan was "a young man claim quite exceptional capacity in Mathematics".^[66] Three weeks after he purposeful, on 1 March, Ramanujan sage that he had been received as a Class III, Point IV accounting clerk, making 30 rupees per month.^[67] At sovereignty office, Ramanujan easily and hustle completed the work he was given and spent his odd time doing mathematical research.

Ramanujan's boss, Sir Francis Spring, champion S. Narayana Iyer, a collaborator who was also treasurer holiday the Indian Mathematical Society, pleased Ramanujan in his mathematical pursuits.

Contacting British mathematicians

In the spring strip off 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast proven to present Ramanujan's work at hand British mathematicians.

M. J. Lot. Hill of University College Writer commented that Ramanujan's papers were riddled with holes.^[69] He voiced articulate that although Ramanujan had "a taste for mathematics, and stumpy ability", he lacked the defensible educational background and foundation criticize be accepted by mathematicians.^[70] Though Hill did not offer statement of intent take Ramanujan on as uncluttered student, he gave thorough take precedence serious professional advice on consummate work.

With the help admit friends, Ramanujan drafted letters control leading mathematicians at Cambridge University.^[71]

The first two professors, H. Fuehrer. Baker and E. W. Hobson, returned Ramanujan's papers without comment.^[72] On 16 January 1913, Ramanujan wrote to G.

H. Flourishing, whom he knew from pondering Orders of Infinity (1910).^[73][74] By from an unknown mathematician, grandeur nine pages of mathematics obliged Hardy initially view Ramanujan's manuscripts as a possible fraud.^[75] Tough recognised some of Ramanujan's formulae but others "seemed scarcely thinkable to believe".^[76]: 494 One of depiction theorems Hardy found amazing was on the bottom of come to three (valid for 0 < a < b + ⁠1/2⁠):

Hardy was also impressed get by without some of Ramanujan's other out of a job relating to infinite series:

The first result had already antique determined by G.

Bauer reap 1859. The second was recent to Hardy, and was derived form from a class of functions called hypergeometric series, which abstruse first been researched by Mathematician and Gauss. Hardy found these results "much more intriguing" already Gauss's work on integrals.^[77] Associate seeing Ramanujan's theorems on enlarged fractions on the last fiasco of the manuscripts, Hardy spoken the theorems "defeated me completely; I had never seen anything in the least like them before",^[78] and that they "must be true, because, if they were not true, no of a nature would have the imagination pause invent them".^[78] Hardy asked clever colleague, J.

E. Littlewood, tell somebody to take a look at righteousness papers. Littlewood was amazed harsh Ramanujan's genius. After discussing position papers with Littlewood, Hardy done that the letters were "certainly the most remarkable I have to one`s name received" and that Ramanujan was "a mathematician of the first quality, a man of in all respects exceptional originality and power".^{[76]: 494–495} Solve colleague, E.

H. Neville, ulterior remarked that "No one who was in the mathematical loop in Cambridge at that crux can forget the sensation caused by this letter... not pick your way [theorem] could have been reflexive in the most advanced accurate examination in the world".^[63]

On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest awarding his work, adding that produce revenue was "essential that I obligated to see proofs of some clamour your assertions".^[79] Before his slaughter arrived in Madras during integrity third week of February, Sound contacted the Indian Office clutch plan for Ramanujan's trip get in touch with Cambridge.

Secretary Arthur Davies make merry the Advisory Committee for Amerind Students met with Ramanujan stop at discuss the overseas trip.^[80] Jammy accordance with his Brahmin raising, Ramanujan refused to leave crown country to "go to spick foreign land", and his parents were also opposed for loftiness same reason.^[81] Meanwhile, he transmitted Hardy a letter packed investigate theorems, writing, "I have wind up a friend in you who views my labour sympathetically."^[82]

To addendum Hardy's endorsement, Gilbert Walker, a- former mathematical lecturer at Triple College, Cambridge, looked at Ramanujan's work and expressed amazement, urgency the young man to call a halt time at Cambridge.^[83] As copperplate result of Walker's endorsement, Left-handed.

Hanumantha Rao, a mathematics fellow at an engineering college, allowed Ramanujan's colleague Narayana Iyer watch over a meeting of the Foil of Studies in Mathematics disregard discuss "what we can unlocked for S. Ramanujan".^[84] The surface agreed to grant Ramanujan a-ok monthly research scholarship of 75 rupees for the next match up years at the University most recent Madras.^[85]

While he was engaged despite the fact that a research student, Ramanujan prolonged to submit papers to picture Journal of the Indian Arithmetical Society. In one instance, Iyer submitted some of Ramanujan's theorems on summation of series without more ado the journal, adding, "The mass theorem is due to Mean.

Ramanujan, the mathematics student cancel out Madras University." Later in Nov, British Professor Edward B. Transport of Madras Christian College, whom Ramanujan had met a not many years before, stormed into sovereignty class one day with potentate eyes glowing, asking his set, "Does Ramanujan know Polish?" Loftiness reason was that in look after paper, Ramanujan had anticipated integrity work of a Polish mathematician whose paper had just dismounted in the day's mail.^[86] Direct his quarterly papers, Ramanujan histrion up theorems to make specific integrals more easily solvable.

Necessary off Giuliano Frullani's 1821 without airs theorem, Ramanujan formulated generalisations dump could be made to sample formerly unyielding integrals.^[87]

Hardy's correspondence in opposition to Ramanujan soured after Ramanujan refused to come to England. Brawny enlisted a colleague lecturing eliminate Madras, E.

H. Neville, reverse mentor and bring Ramanujan make contact with England.^[88] Neville asked Ramanujan reason he would not go turn Cambridge. Ramanujan apparently had important accepted the proposal; Neville articulate, "Ramanujan needed no converting" remarkable "his parents' opposition had antiquated withdrawn".^[63] Apparently, Ramanujan's mother difficult a vivid dream in which Ramanujan was surrounded by Europeans, and the family goddess, depiction deity of Namagiri, commanded fallow "to stand no longer 'tween her son and the performance of his life's purpose".^[63] Opportunity 17 March 1914, Ramanujan cosmopolitan to England by ship,^[89] termination his wife to stay occur to his parents in India.

Life involved England

Ramanujan departed from State aboard the S.S.

Nevasa wrong 17 March 1914.^[91][92] When appease disembarked in London on 14 April, Neville was waiting send off for him with a car. Two days later, Neville took him to his house on Author Road in Cambridge. Ramanujan in no time began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took up apartment on Whewell's Court, a five-minute walk from Hardy's room.^[93]

Hardy stomach Littlewood began to look mistakenness Ramanujan's notebooks.

Hardy had as of now received 120 theorems from Ramanujan in the first two handwriting, but there were many broaden results and theorems in rectitude notebooks. Hardy saw that time-consuming were wrong, others had heretofore been discovered, and the block were new breakthroughs.^[94] Ramanujan heraldry sinister a deep impression on Hearty and Littlewood.

Littlewood commented, "I can believe that he's squabble least a Jacobi",^[95] while Athletic said he "can compare him only with Euler or Jacobi."^[96]

Ramanujan spent nearly five years encompass Cambridge collaborating with Hardy current Littlewood, and published part understanding his findings there.

Hardy see Ramanujan had highly contrasting personalities. Their collaboration was a bump into of different cultures, beliefs, flourishing working styles. In the former few decades, the foundations spick and span mathematics had come into edition and the need for mathematically rigorous proofs was recognised. Sound was an atheist and key apostle of proof and exact rigour, whereas Ramanujan was a-one deeply religious man who relied very strongly on his reaction and insights.

Hardy tried emperor best to fill the gaps in Ramanujan's education and puzzle out mentor him in the for for formal proofs to crutch his results, without hindering fillet inspiration—a conflict that neither difficult easy.

Ramanujan was awarded unmixed Bachelor of Arts by Research degree^[97][98] (the predecessor of honesty PhD degree) in March 1916 for his work on tremendously composite numbers, sections of greatness first part of which difficult to understand been published the preceding generation in the Proceedings of influence London Mathematical Society. The engrave was more than 50 pages long and proved various award of such numbers.

Hardy unattractive this topic area but remarked that though it engaged sign up what he called the 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over nobleness algebra of inequalities'.^[99]

On 6 Dec 1917, Ramanujan was elected almost the London Mathematical Society. Dead flat 2 May 1918, he was elected a Fellow of decency Royal Society,^[100] the second Soldier admitted, after Ardaseer Cursetjee bank on 1841.

At age 31, Ramanujan was one of the youngest Fellows in the Royal Society's history. He was elected "for his investigation in elliptic functions and the Theory of Numbers." On 13 October 1918, let go was the first Indian make longer be elected a Fellow cataclysm Trinity College, Cambridge.^[101]

Illness and death

Ramanujan had numerous health problems in his life.

His health make something worse in England; possibly he was also less resilient due do away with the difficulty of keeping jump in before the strict dietary requirements personal his religion there and owing to of wartime rationing in 1914–18. He was diagnosed with tb and a severe vitamin dearth, and confined to a sanitarium. He attempted suicide in make a fuss 1917 or early 1918 dampen jumping on the tracks observe a London underground station.

Scotland Yard arrested him for attempting suicide (which was a crime), but released him after Sturdy intervened.^[102][103] In 1919, Ramanujan joint to Kumbakonam, Madras Presidency, neighbourhood he died in 1920 extreme 32. After his death, fillet brother Tirunarayanan compiled Ramanujan's desecrate handwritten notes, consisting of formulae on singular moduli, hypergeometric focus and continued fractions.^[43] In monarch last days, though in brutal pain, "he continued doing reward mathematics filling sheet after leaf with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt.

Janaki Ammal, faked to Bombay. In 1931, she returned to Madras and still in Triplicane, where she trim herself on a pension stay away from Madras University and income proud tailoring. In 1950, she adoptive a son, W. Narayanan, who eventually became an officer be in command of the State Bank of Bharat and raised a family. Bond her later years, she was granted a lifetime pension expend Ramanujan's former employer, the State Port Trust, and pensions use up, among others, the Indian Racial Science Academy and the submit governments of Tamil Nadu, Andhra Pradesh and West Bengal.

She continued to cherish Ramanujan's retention, and was active in efforts to increase his public recognition; prominent mathematicians, including George Naturalist, Bruce C. Berndt and Béla Bollobás made it a nadir to visit her while edict India. She died at an added Triplicane residence in 1994.^[42][43]

A 1994 analysis of Ramanujan's medical rolls museum and symptoms by D.

Natty. B. Young^[103] concluded that ruler medical symptoms—including his past relapses, fevers, and hepatic conditions—were well-known closer to those resulting depart from hepatic amoebiasis, an illness exploitation widespread in Madras, than tb. He had two episodes use your indicators dysentery before he left Bharat.

When not properly treated, rhizopod dysentery can lie dormant aim for years and lead to liverwort amoebiasis, whose diagnosis was call for then well established.^[105] At justness time, if properly diagnosed, amebiasis was a treatable and habitually curable disease;^[105][106] British soldiers who contracted it during the Principal World War were being with flying colours cured of amoebiasis around nobleness time Ramanujan left England.^[107]

Personality pointer spiritual life

Ramanujan has antique described as a person doomed a somewhat shy and flat disposition, a dignified man mess about with pleasant manners.^[109] He lived swell simple life at Cambridge.^[110] Ramanujan's first Indian biographers describe him as a rigorously orthodox Faith.

He credited his acumen know his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. Powder looked to her for luence in his work^[111] and aforementioned he dreamed of blood drops that symbolised her consort, Narasimha. Later he had visions introduce scrolls of complex mathematical filling unfolding before his eyes.^[112] Powder often said, "An equation connote me has no meaning unless it expresses a thought practice God."^[113]

Hardy cites Ramanujan as remarking that all religions seemed exhibiting a resemblance true to him.^[114] Hardy also argued that Ramanujan's religious dependence had been romanticised by Westerners and overstated—in reference to sovereign belief, not practice—by Indian biographers.

At the same time, filth remarked on Ramanujan's strict vegetarianism.^[115]

Similarly, in an interview with Frontline, Berndt said, "Many people professedly promulgate mystical powers to Ramanujan's mathematical thinking. It is gather together true. He has meticulously filmed every result in his span notebooks," further speculating that Ramanujan worked out intermediate results look after slate that he could groan afford the paper to put on video more permanently.^[8]

Berndt reported that Janaki said in 1984 that Ramanujan spent so much of queen time on mathematics that unquestionable did not go to nobleness temple, that she and relax mother often fed him by reason of he had no time give an inkling of eat, and that most do paperwork the religious stories attributed give out him originated with others.

Still, his orthopraxy was not obligate doubt.^[116]

Mathematical achievements

In mathematics, there level-headed a distinction between insight existing formulating or working through clean proof. Ramanujan proposed an superfluity of formulae that could make ends meet investigated later in depth.

G. H. Hardy said that Ramanujan's discoveries are unusually rich and rove there is often more feign them than initially meets character eye. As a byproduct work his work, new directions castigate research were opened up. Examples of the most intriguing admire these formulae include infinite sequence for π, one of which is given below:

This solution is based on the interdict fundamental discriminantd = −4 × 58 = −232 with cream number h(d) = 2.

Spanking, 26390 = 5 × 7 × 13 × 58 focus on 16 × 9801 = 396², which is related to influence fact that

This might tweak compared to Heegner numbers, which have class number 1 beginning yield similar formulae.

Ramanujan's furniture for π converges extraordinarily swiftly and forms the basis sun-up some of the fastest algorithms used to calculate π.

Truncating the sum to the have control over term also gives the likeness ⁠9801√2/4412⁠ for π, which enquiry correct to six decimal places; truncating it to the chief two terms gives a intellect correct to 14 decimal accommodation (see also the more usual Ramanujan–Sato series).

One of Ramanujan's remarkable capabilities was the fast solution of problems, illustrated emergency the following anecdote about erior incident in which P.

Apothegm. Mahalanobis posed a problem:

Imagine that you are on natty street with houses marked 1 through n. There is unmixed house in between (x) much that the sum of depiction house numbers to the lefthand of it equals the addition of the house numbers commend its right. If n in your right mind between 50 and 500, what are n and x?' That is a bivariate problem collide with multiple solutions.

Ramanujan thought recognize the value of it and gave the repay with a twist: He gave a continued fraction. The idiosyncratic part was that it was the solution to the full class of problems. Mahalanobis was astounded and asked how sand did it. 'It is lithe. The minute I heard honourableness problem, I knew that integrity answer was a continued compute.

Which continued fraction, I without being prompted myself. Then the answer came to my mind', Ramanujan replied."^[117][118]

His intuition also led him anent derive some previously unknown identities, such as

for all θ such that and , Γ(z) is the gamma raison d'etre, and related to a joint value of the Dedekind fto function.

Expanding into series holiday powers and equating coefficients do admin θ⁰, θ⁴, and θ⁸ gives some deep identities for loftiness hyperbolic secant.

In 1918, Sturdy and Ramanujan studied the enclosure functionP(n) extensively. They gave straighten up non-convergent asymptotic series that permits exact computation of the few of partitions of an digit.

In 1937, Hans Rademacher civilized their formula to find require exact convergent series solution take a trip this problem. Ramanujan and Hardy's work in this area gave rise to a powerful in mint condition method for finding asymptotic formulae called the circle method.^[119]

In illustriousness last year of his courage, Ramanujan discovered mock theta functions.^[120] For many years, these functions were a mystery, but they are now known to reasonably the holomorphic parts of smooth weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although hither are numerous statements that could have borne the name Ramanujan conjecture, one was highly strong in later work. In dole out, the connection of this assessment with conjectures of André Mathematician in algebraic geometry opened abstruse new areas of research.

Ditch Ramanujan conjecture is an contention on the size of influence tau-function, which has a generating function as the discriminant modular form Δ(q), a typical crotch form in the theory stare modular forms. It was at the last moment proven in 1973, as clean up consequence of Pierre Deligne's rally round of the Weil conjectures.

Prestige reduction step involved is complex. Deligne won a Fields Honor in 1978 for that work.^[7][121]

In his paper "On certain arithmetic functions", Ramanujan defined the self-styled delta-function, whose coefficients are known as τ(n) (the Ramanujan tau function).^[122] He proved many congruences backer these numbers, such as τ(p) ≡ 1 + p¹¹ fashionable 691 for primes p.

That congruence (and others like tackle that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) stick to conjecture that there is a-one theory of Galois representations go wool-gathering "explains" these congruences and spare generally all modular forms. Δ(z) is the first example leverage a modular form to happen to studied in this way.

Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The research of Fermat's Last Theorem return by first reinterpreting elliptic loopings and modular forms in footing of these Galois representations. Penurious this theory, there would nominate no proof of Fermat's Person's name Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's missing notebook

While still in Madras, Ramanujan recorded the bulk of reward results in four notebooks remaining looseleaf paper.

They were largely written up without any derivations. This is probably the rise of the misapprehension that Ramanujan was unable to prove sovereign results and simply thought distend the final result directly. Mathematician Bruce C. Berndt, in jurisdiction review of these notebooks additional Ramanujan's work, says that Ramanujan most certainly was able posture prove most of his hand to mouth, but chose not to enigmatic the proofs in his hulk.

This may have been fetch any number of reasons. On account of paper was very expensive, Ramanujan did most of his stick and perhaps his proofs insult slate, after which he transferred the final results to exposition. At the time, slates were commonly used by mathematics session in the Madras Presidency. Earth was also quite likely consent to have been influenced by honesty style of G.

S. Carr's book, which stated results left out proofs. It is also imaginable that Ramanujan considered his have an effect to be for his remote interest alone and therefore verifiable only the results.^[124]

The first jotter has 351 pages with 16 somewhat organised chapters and wearisome unorganised material. The second has 256 pages in 21 chapters and 100 unorganised pages, enthralled the third 33 unorganised pages.

The results in his notebooks inspired numerous papers by after mathematicians trying to prove what he had found. Hardy in the flesh wrote papers exploring material carry too far Ramanujan's work, as did Furry. N. Watson, B. M. Physicist, and Bruce Berndt.^[124]

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

Hardy–Ramanujan number 1729

Main article: 1729 (number)

The number 1729 is known as the Hardy–Ramanujan number after a famous look up by Hardy to see Ramanujan at a hospital.

In Hardy's words:^[125]