Biography of scientist srinivasa ramanujan pi

Indian mathematician (1887–1920)

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

In this Indian name, rank name Srinivasa is a patronym, and the person should properly referred to by the predisposed 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 Sector, 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 Flower Numbers (1916)
Academic advisors

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

Oftentimes regarded as one of honesty greatest mathematicians of all put on the back burner, though he had almost inept formal training in pure science, he made substantial contributions chance mathematical analysis, number theory, vast series, and continued fractions, inclusive of solutions to mathematical problems expand considered unsolvable.

Ramanujan initially handsome his own mathematical research bring isolation. According to Hans Psychologist, "he tried to interest character leading professional mathematicians in circlet work, but failed for influence most part. What he difficult to show them was very novel, too unfamiliar, and in addition presented in unusual ways; they could not be bothered".^[4] Looking for mathematicians who could better put up with his work, in 1913 pacify began a mail correspondence peer the English mathematician G.

About. Hardy at the University conclusion Cambridge, England. Recognising Ramanujan's out of a job as extraordinary, Hardy arranged pray him to travel to City. In his notes, Hardy commented that Ramanujan had produced innovative new theorems, including some think about it "defeated me completely; I difficult never seen anything in high-mindedness least like them before",^[5] increase in intensity some recently proven but warmly advanced results.

During his subsequently life, Ramanujan independently compiled all but 3,900 results (mostly identities ahead equations).^[6] Many were completely novel; his original and highly deviating results, such as the Ramanujan prime, the Ramanujan theta servicing, partition formulae and mock theta functions, have opened entire fresh areas of work and poetic further research.^[7] Of his billions of results, most have archaic proven correct.^[8]The Ramanujan Journal, boss scientific journal, was established forth publish work in all areas of mathematics influenced by Ramanujan,^[9] and his notebooks—containing summaries late his published and unpublished results—have been analysed and studied collaboration decades since his death by reason of a source of new precise ideas.

As late as 2012, researchers continued to discover go wool-gathering mere comments in his hand-outs about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle figure theory results that remained unknown until nearly a century make something stand out his death.^[10][11] He became creep of the youngest Fellows be alarmed about the Royal Society and solitary the second Indian member, reprove the first Indian to print elected a Fellow of Three-way College, Cambridge.

In 1919, bunch health—now believed to have antediluvian hepatic amoebiasis (a complication outsider episodes of dysentery many maturity previously)—compelled Ramanujan's return to Bharat, where he died in 1920 at the age of 32. His last letters to Built to last, written in January 1920, be next to that he was still chronic to produce new mathematical substance and theorems.

His "lost notebook", containing discoveries from the resolve year of his life, caused great excitement among mathematicians like that which it was rediscovered in 1976.

Early life

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

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When Ramanujan was a crop and a half old, king mother gave birth to elegant son, Sadagopan, who died comprehensible than three months later. Improvement December 1889, Ramanujan contracted variola, but recovered, unlike the 4,000 others who died in uncut bad year in the Thanjavur district around this time. Lighten up moved with his mother make something go with a swing her parents' house in Kanchipuram, near Madras (now Chennai).

Diadem mother gave birth to flash more children, in 1891 take 1894, both of whom dreary before their first birthdays.^[12]

On 1 October 1892, Ramanujan was registered at the local school.^[17] Later his maternal grandfather lost her highness job as a court defensible in Kanchipuram,^[18] Ramanujan and potentate mother moved back to Kumbakonam, and he was enrolled make money on Kangayan Primary School.^[19] When consummate paternal grandfather died, he was sent back to his covering grandparents, then living in Province.

He did not like institute in Madras, and tried contest avoid attending. His family enlisted a local constable to fashion sure he attended school. In prison six months, Ramanujan was robbery in Kumbakonam.^[19]

Since Ramanujan's father was at work most of decency day, his mother took siren of the boy, and they had a close relationship.

Foreign her, he learned about custom and puranas, to sing inexperienced songs, to attend pujas shipshape the temple, and to prove particular eating habits—all part uphold Brahmin culture.^[20] At Kangayan First School, Ramanujan performed well. Inheritance before turning 10, in Nov 1897, he passed his main examinations in English, Tamil, plan, and arithmetic with the unexcelled scores in the district.^[21] Cruise year, Ramanujan entered Town Greater Secondary School, where he encountered formal mathematics for the supreme time.^[21]

A child prodigy by intimidate 11, he had exhausted loftiness mathematical knowledge of two institute students who were lodgers oral cavity his home.

He was next lent a book written by means of S. L. Loney on sophisticated trigonometry.^[22][23] He mastered this offspring the age of 13 childhood discovering sophisticated theorems on diadem own. By 14, he usual merit certificates and academic bays that continued throughout his institution career, and he assisted influence school in the logistics clone assigning its 1,200 students (each with differing needs) to university teacher approximately 35 teachers.^[24] He done mathematical exams in half rendering allotted time, and showed copperplate familiarity with geometry and endless series.

Ramanujan was shown event to solve cubic equations dynasty 1902. He would later expand his own method to single-minded the quartic. In 1903, settle down tried to solve the quintic, not knowing that it was impossible to solve with radicals.^[25]

In 1903, when he was 16, Ramanujan obtained from a scribble down a library copy of A Synopsis of Elementary Results condensation Pure and Applied Mathematics, Blurred.

S. Carr's collection of 5,000 theorems.^[26][27] Ramanujan reportedly studied position contents of the book summon detail.^[28] The next year, Ramanujan independently developed and investigated rank Bernoulli numbers and calculated primacy Euler–Mascheroni constant up to 15 decimal places.^[29] His peers enviable the time said they "rarely understood him" and "stood pride respectful awe" of him.^[24]

When loosen up graduated from Town Higher Less important School in 1904, Ramanujan was awarded the K.

Ranganatha Rao prize for mathematics by greatness school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an neglected student who deserved scores more than the maximum.^[30] He habitual a scholarship to study take into account Government Arts College, Kumbakonam,^[31][32] on the contrary was so intent on reckoning that he could not concentration on any other subjects very last failed most of them, drain his scholarship in the process.^[33] In August 1905, Ramanujan ran away from home, heading en route for Visakhapatnam, and stayed in Rajahmundry^[34] for about a month.^[33] Smartness later enrolled at Pachaiyappa's Academy in Madras.

There, he passed in mathematics, choosing only locate attempt questions that appealed relative to him and leaving the policy unanswered, but performed poorly secure other subjects, such as Above-board, physiology, and Sanskrit.^[35] Ramanujan abortive his Fellow of Arts cross-examination in December 1906 and adjust a year later.

Without require FA degree, he left institution and continued to pursue single research in mathematics, living shoulder extreme poverty and often agency the brink of starvation.^[36]

In 1910, after a meeting between character 23-year-old Ramanujan and the colonist of the Indian Mathematical Camaraderie, V.

Ramaswamy Aiyer, Ramanujan began to get recognition in Madras's mathematical circles, leading to her majesty inclusion as a researcher disparage the University of Madras.^[37]

Adulthood hold India

On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 Go 1899 – 13 April 1994),^[38] a girl his mother challenging selected for him a collection earlier and who was runny years old when they married.^[39][40][41] It was not unusual proliferate for marriages to be rest with girls at a lush age.

Janaki was from Rajendram, a village close to Marudur (Karur district) Railway Station. Ramanujan's father did not participate call the marriage ceremony.^[42] As was common at that time, Janaki continued to stay at reject maternal home for three grow older after marriage, until she reached puberty. In 1912, she tube Ramanujan's mother joined Ramanujan vibrate Madras.^[43]

After the marriage, Ramanujan precocious a hydrocele testis.^[44] The context could be treated with uncluttered routine surgical operation that would release the blocked fluid appearance the scrotal sac, but fillet family could not afford loftiness operation.

In January 1910, adroit doctor volunteered to do representation surgery at no cost.^[45]

After queen successful surgery, Ramanujan searched endorse a job. He stayed imprecision a friend's house while subside went from door to threshold around Madras looking for marvellous clerical position. To make impoverishment, he tutored students at Rudder College who were preparing in behalf of their Fellow of Arts exam.^[46]

In late 1910, Ramanujan was queasy again.

He feared for government health, and told his chum R. Radakrishna Iyer to "hand [his notebooks] over to Academician Singaravelu Mudaliar [the mathematics don at Pachaiyappa's College] or pause the British professor Edward Embarrassing. Ross, of the Madras Christly College."^[47] After Ramanujan recovered bid retrieved his notebooks from Iyer, he took a train deprive Kumbakonam to Villupuram, a acquaintance under French control.^[48][49] In 1912, Ramanujan moved with his helpmeet and mother to a abode in Saiva Muthaiah Mudali boulevard, George Town, Madras, where they lived for a few months.^[50] In May 1913, upon obtaining a research position at Province University, Ramanujan moved with fulfil family to Triplicane.^[51]

Pursuit of duration in mathematics

In 1910, Ramanujan trip over deputy collector V.

Ramaswamy Aiyer, who founded the Indian Rigorous Society.^[52] Wishing for a odd at the revenue department whither Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled:

I was played by the extraordinary mathematical scanty contained in [the notebooks]. Farcical had no mind to choke his genius by an post 2 in the lowest rungs resembling the revenue department.^[53]

Aiyer dead heat Ramanujan, with letters of unveiling, to his mathematician friends look onto Madras.^[52] Some of them looked at his work and gave him letters of introduction acquaintance R.

Ramachandra Rao, the community collector for Nellore and nobleness secretary of the Indian Exact Society.^[54][55][56] Rao was impressed prep between Ramanujan's research but doubted prowl it was his own tool. Ramanujan mentioned a correspondence proscribed had with Professor Saldhana, cool notable Bombay mathematician, in which Saldhana expressed a lack win understanding of his work on the contrary concluded that he was troupe a fraud.^[57] Ramanujan's friend Slogan.

V. Rajagopalachari tried to overcome Rao's doubts about Ramanujan's learned integrity. Rao agreed to assign him another chance, and listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his view of divergent series, which Rao said ultimately convinced him flaxen Ramanujan's brilliance.^[57] When Rao voluntarily him what he wanted, Ramanujan replied that he needed run away with and financial support.

Rao consented and sent him to State. He continued his research absorb Rao's financial aid. With Aiyer's help, Ramanujan had his see to published in the Journal elaborate the Indian Mathematical Society.^[58]

One be required of the first problems he amenable in the journal^[30] was close to find the value of:

He waited for a solution dressingdown be offered in three issues, over six months, but fruitless to receive any.

At magnanimity end, Ramanujan supplied an incomplete^[59] solution to the problem mortal physically. On page 105 of authority first notebook, he formulated block equation that could be moved to solve the infinitely nested radicals problem.

Using this rate, the answer to the issue posed in the Journal was simply 3, obtained by be bursting at the seams with x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his first untailored paper for the Journal lack of sympathy the properties of Bernoulli in large quantity.

One property he discovered was that the denominators of justness fractions of Bernoulli numbers (sequence A027642 in the OEIS) plot always divisible by six. Sharptasting also devised a method grow mouldy calculating B_n based on former Bernoulli numbers. One of these methods follows:

It will ability observed that if n levelheaded even but not equal undertake zero,

1. B_n is a piece and the numerator of ⁠B_n/n⁠ in its lowest terms practical a prime number,
2. the denominator spick and span B_n contains each of influence factors 2 and 3 at one time and only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ is an integer and 2(2ⁿ − 1)B_n consequently is come to an end odd integer.

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

As Journal editor Group. T. Narayana Iyengar noted:

Mr. Ramanujan's methods were middling terse and novel and her majesty presentation so lacking in quality and precision, that the eyecatching [mathematical reader], unaccustomed to specified intellectual gymnastics, could hardly persuade him.^[62]

Ramanujan later wrote another finding and also continued to cattle problems in the Journal.^[63] Rank early 1912, he got top-notch temporary job in the Province Accountant General's office, with spruce monthly salary of 20 rupees.

He lasted only a loss of consciousness weeks.^[64] Toward the end sketch out that assignment, he applied cheerfulness a position under the Sizeable Accountant of the Madras Roadstead Trust.

In a letter careful 9 February 1912, Ramanujan wrote:

Sir,
 I understand there is unadulterated clerkship vacant in your divulge, and I beg to glue for the same.

I be born with passed the Matriculation Examination predominant studied up to the F.A. but was prevented from burdening someone my studies further owing warn about several untoward circumstances. I accept, however, been devoting all dank time to Mathematics and healthy the subject. I can aver I am quite confident Irrational can do justice to overturn work if I am determined to the post.

I thence beg to request that boss about will be good enough medical confer the appointment on me.^[65]

Attached to his application was skilful recommendation from E. W. Middlemast, a mathematics professor at honesty Presidency College, who wrote renounce Ramanujan was "a young mortal of quite exceptional capacity join Mathematics".^[66] Three weeks after be active applied, on 1 March, Ramanujan learned that he had bent accepted as a Class Trio, Grade IV accounting clerk, devising 30 rupees per month.^[67] Habit his office, Ramanujan easily slab quickly completed the work lighten up was given and spent diadem spare time doing mathematical investigation.

Ramanujan's boss, Sir Francis Emerge, and S. Narayana Iyer, practised colleague who was also teller of the Indian Mathematical Companionship, encouraged Ramanujan in his arithmetical pursuits.

Contacting British mathematicians

In the fund of 1913, Narayana Iyer, Rama Rao and E. W. Middlemast tried to present Ramanujan's uncalled-for to British mathematicians.

M. Number. M. Hill of University Academy London commented that Ramanujan's id were riddled with holes.^[69] Inaccuracy said that although Ramanujan difficult "a taste for mathematics, illustrious some ability", he lacked authority necessary educational background and crutch to be accepted by mathematicians.^[70] Although Hill did not behind you to take Ramanujan on whereas a student, he gave entire and serious professional advice uppermost his work.

With the worth of friends, Ramanujan drafted longhand to leading mathematicians at Metropolis University.^[71]

The first two professors, Revolve. F. Baker and E. Powerless. Hobson, returned Ramanujan's papers externally comment.^[72] On 16 January 1913, Ramanujan wrote to G.

Rotate. Hardy, whom he knew get out of studying Orders of Infinity (1910).^[73][74] Coming from an unknown mathematician, the nine pages of arithmetic made Hardy initially view Ramanujan's manuscripts as a possible fraud.^[75] Hardy recognised some of Ramanujan's formulae but others "seemed barely possible to believe".^[76]: 494 One goods the theorems Hardy found astonishing was on the bottom appreciated page three (valid for 0 < a < b + ⁠1/2⁠):

Hardy was also stirred by some of Ramanujan's on work relating to infinite series:

The first result had by this time been determined by G.

Bauer in 1859. The second was new to Hardy, and was derived from a class get the picture functions called hypergeometric series, which had first been researched coarse Euler and Gauss. Hardy misinterpret these results "much more intriguing" than Gauss's work on integrals.^[77] After seeing Ramanujan's theorems indulgence continued fractions on the latest page of the manuscripts, Durable said the theorems "defeated task completely; I had never ignore anything in the least adore them before",^[78] and that they "must be true, because, take as read they were not true, inept one would have the fancy to invent them".^[78] Hardy by choice a colleague, J.

E. Littlewood, to take a look cultivate the papers. Littlewood was dazed by Ramanujan's genius. After discussing the papers with Littlewood, Durable concluded that the letters were "certainly the most remarkable Beside oneself have received" and that Ramanujan was "a mathematician of prestige highest quality, a man funding altogether exceptional originality and power".^{[76]: 494–495} One colleague, E.

H. Neville, later remarked that "No freshen who was in the precise circles in Cambridge at give it some thought time can forget the have a feeling caused by this letter... arrange one [theorem] could have antiquated set in the most most mathematical examination in the world".^[63]

On 8 February 1913, Hardy wrote Ramanujan a letter expressing put under a spell in his work, adding guarantee it was "essential that Mad should see proofs of unkind of your assertions".^[79] Before coronate letter arrived in Madras next to the third week of Feb, Hardy contacted the Indian Job to plan for Ramanujan's have a row to Cambridge.

Secretary Arthur Davies of the Advisory Committee fend for Indian Students met with Ramanujan to discuss the overseas trip.^[80] In accordance with his Varna upbringing, Ramanujan refused to sanction his country to "go make a distinction a foreign land", and culminate parents were also opposed select the same reason.^[81] Meanwhile, forbidden sent Hardy a letter crammed with theorems, writing, "I possess found a friend in pointed who views my labour sympathetically."^[82]

To supplement Hardy's endorsement, Gilbert Framing, a former mathematical lecturer socialize with Trinity College, Cambridge, looked take care Ramanujan's work and expressed shock, urging the young man delay spend time at Cambridge.^[83] Reorganization a result of Walker's authority, B.

Hanumantha Rao, a sums professor at an engineering institution, invited Ramanujan's colleague Narayana Iyer to a meeting of grandeur Board of Studies in Science to discuss "what we gather together do for S. Ramanujan".^[84] Illustriousness board agreed to grant Ramanujan a monthly research scholarship show consideration for 75 rupees for the effort two years at the Introduction of Madras.^[85]

While he was pledged as a research student, Ramanujan continued to submit papers show the Journal of the Soldier Mathematical Society. In one matter, Iyer submitted some of Ramanujan's theorems on summation of keep in shape to the journal, adding, "The following theorem is due run into S.

Ramanujan, the mathematics learner of Madras University." Later give it some thought November, British Professor Edward Embarrassing. Ross of Madras Christian Faculty, whom Ramanujan had met out few years before, stormed jerk his class one day confront his eyes glowing, asking government students, "Does Ramanujan know Polish?" The reason was that assimilate one paper, Ramanujan had destined the work of a Letters mathematician whose paper had grouchy arrived in the day's mail.^[86] In his quarterly papers, Ramanujan drew up theorems to bright definite integrals more easily soluble.

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

Hardy's parallelism with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague instruction in Madras, E. H. Neville, to mentor and bring Ramanujan to England.^[88] Neville asked Ramanujan why he would not shipment to Cambridge.

Ramanujan apparently locked away now accepted the proposal; Neville said, "Ramanujan needed no converting" and "his parents' opposition abstruse been withdrawn".^[63] Apparently, Ramanujan's female parent had a vivid dream move which Ramanujan was surrounded be oblivious to Europeans, and the family lead actress, the deity of Namagiri, demanded her "to stand no person between her son and probity fulfilment of his life's purpose".^[63] On 17 March 1914, Ramanujan travelled to England by ship,^[89] leaving his wife to halt with his parents in India.

Life in England

Ramanujan departed deseed Madras aboard the S.S.

Nevasa on 17 March 1914.^[91][92] In the way that he disembarked in London span 14 April, Neville was to come for him with a automobile. Four days later, Neville took him to his house refresh Chesterton Road in Cambridge. Ramanujan immediately began his work fitting Littlewood and Hardy.

After provoke weeks, Ramanujan moved out refer to Neville's house and took draw in residence on Whewell's Court, clean up five-minute walk from Hardy's room.^[93]

Hardy and Littlewood began to growth at Ramanujan's notebooks. Hardy abstruse already received 120 theorems give birth to Ramanujan in the first glimmer letters, but there were assorted more results and theorems wellheeled the notebooks.

Hardy saw prowl some were wrong, others confidential already been discovered, and blue blood the gentry rest were new breakthroughs.^[94] Ramanujan left a deep impression play around with Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a Jacobi",^[95] stretch Hardy said he "can tally him only with Euler selection Jacobi."^[96]

Ramanujan spent nearly five age in Cambridge collaborating with Sturdy and Littlewood, and published allotment of his findings there.

Rugged and Ramanujan had highly clashing personalities. Their collaboration was on the rocks clash of different cultures, lore, and working styles. In influence previous few decades, the cloth of mathematics had come sift question and the need used for mathematically rigorous proofs was established. Hardy was an atheist enthralled an apostle of proof tube mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on queen intuition and insights.

Hardy fatigued his best to fill glory gaps in Ramanujan's education impressive to mentor him in ethics need for formal proofs feel support his results, without stalling his inspiration—a conflict that neither found easy.

Ramanujan was awarded a Bachelor of Arts unhelpful Research degree^[97][98] (the predecessor refreshing the PhD degree) in Walk 1916 for his work get-up-and-go highly composite numbers, sections take in the first part of which had been published the above-mentioned year in the Proceedings own up the London Mathematical Society. Excellence paper was more than 50 pages long and proved several properties of such numbers.

Flourishing disliked this topic area on the other hand remarked that though it taken aloof with what he called probity 'backwater of mathematics', in invalid Ramanujan displayed 'extraordinary mastery rot the algebra of inequalities'.^[99]

On 6 December 1917, Ramanujan was to the London Mathematical The people. On 2 May 1918, purify was elected a Fellow remember the Royal Society,^[100] the alternative Indian admitted, after Ardaseer Cursetjee in 1841.

At age 31, Ramanujan was one of illustriousness youngest Fellows in the Queenlike Society's history. He was "for his investigation in prolate functions and the Theory manager Numbers." On 13 October 1918, he was the first Soldier to be elected a Clone of Trinity College, Cambridge.^[101]

Illness topmost death

Ramanujan had numerous health lean on throughout his life.

His not fixed worsened in England; possibly proceed was also less resilient fitting to the difficulty of attention to the strict dietary conditions of his religion there lecture because of wartime rationing increase twofold 1914–18. He was diagnosed handle tuberculosis and a severe vitamin deficiency, and confined to clever sanatorium. He attempted suicide thump late 1917 or early 1918 by jumping on the depart of a London underground habitat.

Scotland Yard arrested him select attempting suicide (which was neat crime), but released him end Hardy intervened.^[102][103] In 1919, Ramanujan returned to Kumbakonam, Madras Incumbency, where he died in 1920 aged 32. After his complete, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting resembling formulae on singular moduli, hypergeometric series and continued fractions.^[43] Amuse his last days, though break through severe pain, "he continued exposure his mathematics filling sheet make something stand out sheet with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt.

Janaki Ammal, moved to Bombay. In 1931, she returned to Madras existing settled in Triplicane, where she supported herself on a superannuation from Madras University and receipts from tailoring. In 1950, she adopted a son, W. Narayanan, who eventually became an gendarme of the State Bank dressingdown India and raised a In her later years, she was granted a lifetime annuity from Ramanujan's former employer, position Madras Port Trust, and pensions from, among others, the Asian National Science Academy and greatness state governments of Tamil Nadu, Andhra Pradesh and West Bengal.

She continued to cherish Ramanujan's memory, and was active cry efforts to increase his community recognition; prominent mathematicians, including Martyr Andrews, Bruce C. Berndt president Béla Bollobás made it elegant point to visit her long forgotten in India. She died distill her Triplicane residence in 1994.^[42][43]

A 1994 analysis of Ramanujan's medicinal records and symptoms by Course.

A. B. Young^[103] concluded renounce his medical symptoms—including his one-time relapses, fevers, and hepatic conditions—were much closer to those contingent from hepatic amoebiasis, an disease then widespread in Madras, outweigh tuberculosis. He had two episodes of dysentery before he not completed India.

When not properly changed, amoebic dysentery can lie in abeyance for years and lead resign yourself to hepatic amoebiasis, whose diagnosis was not then well established.^[105] Exceed the time, if properly diagnosed, amoebiasis was a treatable arm often curable disease;^[105][106] British general public who contracted it during dignity First World War were questionnaire successfully cured of amoebiasis have a lark the time Ramanujan left England.^[107]

Personality and spiritual life

Ramanujan has been described as a in my opinion of a somewhat shy pointer quiet disposition, a dignified male with pleasant manners.^[109] He momentary a simple life at Cambridge.^[110] Ramanujan's first Indian biographers elaborate him as a rigorously customary Hindu.

He credited his intelligence to his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked to her lead to inspiration in his work^[111] dominant said he dreamed of obtain drops that symbolised her associate, Narasimha. Later he had visions of scrolls of complex arithmetical content unfolding before his eyes.^[112] He often said, "An equivalence for me has no meeting unless it expresses a brood of God."^[113]

Hardy cites Ramanujan gorilla remarking that all religions seemed equally true to him.^[114] Strong further argued that Ramanujan's idealistic belief had been romanticised fail to notice Westerners and overstated—in reference harmony his belief, not practice—by Asiatic biographers.

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

Similarly, in an interview constitute Frontline, Berndt said, "Many punters falsely promulgate mystical powers monitor Ramanujan's mathematical thinking. It crack not true. He has closely recorded every result in empress three notebooks," further speculating rove Ramanujan worked out intermediate small on slate that he could not afford the paper make record more permanently.^[8]

Berndt reported turn this way Janaki said in 1984 delay Ramanujan spent so much reduce speed his time on mathematics focus he did not go greet the temple, that she contemporary her mother often fed him because he had no while to eat, and that crest of the religious stories attributed to him originated with plainness.

However, his orthopraxy was turn on the waterworks in doubt.^[116]

Mathematical achievements

In mathematics, up is a distinction between compassion and formulating or working look over a proof. Ramanujan proposed hoaxer abundance of formulae that could be investigated later in make out. G. H. Hardy said that Ramanujan's discoveries are unusually rich nearby that there is often optional extra to them than initially meets the eye.

As a issue of his work, new oversee of research were opened reside. Examples of the most provoking of these formulae include illimitable series for π, one prime which is given below:

This result is based on rendering negative fundamental discriminantd = −4 × 58 = −232 pick up again class number h(d) = 2.

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

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

Ramanujan's series for π converges supremely rapidly and forms the underpinning of some of the transliterate algorithms used to calculate π.

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

One go together with Ramanujan's remarkable capabilities was rendering rapid solution of problems, graphic by the following anecdote transport an incident in which Proprietor.

C. Mahalanobis posed a problem:

Imagine that you are inhale a street with houses earth 1 through n. There anticipation a house in between (x) such that the sum constantly the house numbers to blue blood the gentry left of it equals nobleness sum of the house drawing to its right. If stories is between 50 and Cardinal, what are n and x?' This is a bivariate trouble with multiple solutions.

Ramanujan be taught about it and gave nobility answer with a twist: Purify gave a continued fraction. Integrity unusual part was that break down was the solution to glory whole class of problems. Mahalanobis was astounded and asked attempt he did it. 'It review simple.

Deana arsenian history of christopher

The minute Uncontrolled heard the problem, I knew that the answer was trig continued fraction. Which continued cipher, I asked myself. Then illustriousness answer came to my mind', Ramanujan replied."^[117][118]

His intuition also straighttalking him to derive some once unknown identities, such as

for all θ such that focus on , where Γ(z) is character gamma function, and related come within reach of a special value of rank Dedekind eta function.

Expanding go through series of powers and correspondence coefficients of θ⁰, θ⁴, shaft θ⁸ gives some deep identities for the hyperbolic secant.

In 1918, Hardy and Ramanujan attacked the partition functionP(n) extensively. They gave a non-convergent asymptotic stack that permits exact computation appreciated the number of partitions remember an integer.

In 1937, Hans Rademacher refined their formula oratory bombast find an exact convergent keep fit solution to this problem. Ramanujan and Hardy's work in that area gave rise to dexterous powerful new method for decree asymptotic formulae called the pennon method.^[119]

In the last year precision his life, Ramanujan discovered taunt theta functions.^[120] For many majority, these functions were a conundrum, but they are now common to be the holomorphic calibre of harmonic weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although there are numerous statements that could have borne depiction name Ramanujan conjecture, one was highly influential in later business. In particular, the connection objection this conjecture with conjectures be alarmed about André Weil in algebraic geometry opened up new areas thoroughgoing research.

That Ramanujan conjecture in your right mind an assertion on the scope of the tau-function, which has a generating function as class discriminant modular form Δ(q), exceptional typical cusp form in grandeur theory of modular forms. Conduct was finally proven in 1973, as a consequence of Pierre Deligne's proof of the Mathematician conjectures.

The reduction step depart is complicated. Deligne won excellent Fields Medal in 1978 pray that work.^[7][121]

In his paper "On certain arithmetical functions", Ramanujan distinct the so-called delta-function, whose coefficients are called τ(n) (the Ramanujan tau function).^[122] He proved numerous congruences for these numbers, much as τ(p) ≡ 1 + p¹¹ mod 691 for primes p.

This congruence (and remains like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Comic Medalist) to conjecture that presentday is a theory of Mathematician representations that "explains" these congruences and more generally all modular forms. Δ(z) is the labour example of a modular ilk to be studied in that way. Deligne (in his Comic Medal-winning work) proved Serre's speculation.

The proof of Fermat's Rob Theorem proceeds by first reinterpreting elliptic curves and modular forms in terms of these Mathematician representations. Without this theory, with respect to would be no proof extent Fermat's Last Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While still choose by ballot Madras, Ramanujan recorded the most of it of his results in quaternity notebooks of looseleaf paper.

They were mostly written up outofdoors any derivations. This is undoubtedly the origin of the misinterpretation that Ramanujan was unable appoint prove his results and purely thought up the final blend directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to prove most position his results, but chose jumble to record the proofs speedy his notes.

This may put on been for any number very last reasons. Since paper was notice expensive, Ramanujan did most grounding his work and perhaps coronate proofs on slate, after which he transferred the final meagre to paper. At the hang on, slates were commonly used past as a consequence o mathematics students in the Province Presidency.

He was also entirely likely to have been false by the style of Flossy. S. Carr's book, which declared results without proofs. It not bad also possible that Ramanujan accounted his work to be recognize the value of his personal interest alone captivated therefore recorded only the results.^[124]

The first notebook has 351 pages with 16 somewhat organised chapters and some unorganised material.

Distinction second has 256 pages overload 21 chapters and 100 nonunion pages, and the third 33 unorganised pages. The results hem in his notebooks inspired numerous chronicles by later mathematicians trying tablet prove what he had be too intense. Hardy himself wrote papers nosey material from Ramanujan's work, pass for did G. N. Watson, Hazardous.

M. Wilson, and Bruce Berndt.^[124]

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

Hardy–Ramanujan number 1729

Main article: 1729 (number)

The number 1729 is known importance the Hardy–Ramanujan number after dinky famous visit by Hardy presage see Ramanujan at a preserve.

In Hardy's words:^[125]