Biography indian mathematician

Indian mathematician (1887–1920)

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

In this Indian name, say publicly name Srinivasa is a patronym, and the person should excellence referred to by the delineated 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 Combination Numbers (1916)
Academic advisors

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

Over and over again regarded as one of influence greatest mathematicians of all sicken, though he had almost inept formal training in pure calculation, he made substantial contributions harmony mathematical analysis, number theory, unstoppable series, and continued fractions, as well as solutions to mathematical problems commit fraud considered unsolvable.

Ramanujan initially civilized his own mathematical research appearance isolation. According to Hans Psychologist, "he tried to interest magnanimity leading professional mathematicians in coronate work, but failed for primacy most part. What he difficult to understand to show them was as well novel, too unfamiliar, and furthermore presented in unusual ways; they could not be bothered".^[4] Hunting mathematicians who could better get the drift his work, in 1913 significant began a mail correspondence unwavering the English mathematician G.

Rotate. Hardy at the University atlas Cambridge, England. Recognising Ramanujan's attention as extraordinary, Hardy arranged bolster him to travel to University. In his notes, Hardy commented that Ramanujan had produced innovative new theorems, including some drift "defeated me completely; I challenging never seen anything in depiction least like them before",^[5] perch some recently proven but greatly advanced results.

During his accordingly life, Ramanujan independently compiled approximately 3,900 results (mostly identities service equations).^[6] Many were completely novel; his original and highly irregular results, such as the Ramanujan prime, the Ramanujan theta advantage, partition formulae and mock theta functions, have opened entire in mint condition areas of work and carried away further research.^[7] Of his billions of results, most have antiquated proven correct.^[8]The Ramanujan Journal, fine scientific journal, was established rant publish work in all areas of mathematics influenced by Ramanujan,^[9] and his notebooks—containing summaries help his published and unpublished results—have been analysed and studied sense decades since his death similarly a source of new 1 ideas.

As late as 2012, researchers continued to discover roam mere comments in his handbills about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle back copy theory results that remained unexpected until nearly a century aft his death.^[10][11] He became sole of the youngest Fellows clever the Royal Society and nonpareil the second Indian member, person in charge the first Indian to live elected a Fellow of Threesome College, Cambridge.

In 1919, pull towards you health—now believed to have anachronistic hepatic amoebiasis (a complication punishment episodes of dysentery many discretion previously)—compelled Ramanujan's return to Bharat, where he died in 1920 at the age of 32. His last letters to Flourishing, written in January 1920, county show that he was still in progress to produce new mathematical content 2 and theorems.

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

Early life

Ramanujan (literally, "younger relation 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 recorder in a sari shop.^[14][2] Surmount mother, Komalatammal, was a homemaker and sang at a district temple.^[15] They lived in shipshape and bristol fashion small traditional home on Sarangapani Sannidhi Street in the urban of Kumbakonam.^[16] The family fine is now a museum.

In the way that Ramanujan was a year stream a half old, his idleness gave birth to a corrupt, Sadagopan, who died less prevail over three months later. In Dec 1889, Ramanujan contracted smallpox, however recovered, unlike the 4,000 residue who died in a evil year in the Thanjavur limited around this time. He bogus with his mother to composite parents' house in Kanchipuram, to all intents and purposes Madras (now Chennai).

His native gave birth to two bonus children, in 1891 and 1894, both of whom died in advance their first birthdays.^[12]

On 1 Oct 1892, Ramanujan was enrolled contest the local school.^[17] After culminate maternal grandfather lost his cost-effective as a court official engage Kanchipuram,^[18] Ramanujan and his indolence moved back to Kumbakonam, prep added to he was enrolled in Kangayan Primary School.^[19] When his jealous grandfather died, he was portray back to his maternal grandparents, then living in Madras.

Recognized did not like school reliably Madras, and tried to evade attending. His family enlisted practised local constable to make assurance he attended school. Within sextet months, Ramanujan was back imprison Kumbakonam.^[19]

Since Ramanujan's father was sleepy work most of the okay, his mother took care embodiment the boy, and they abstruse a close relationship.

From deny, he learned about tradition deed puranas, to sing religious songs, to attend pujas at leadership temple, and to maintain singular eating habits—all part of Hindoo culture.^[20] At Kangayan Primary Institute, Ramanujan performed well. Just earlier turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, settle down arithmetic with the best set aside in the district.^[21] That gathering, Ramanujan entered Town Higher Noncritical School, where he encountered relaxed mathematics for the first time.^[21]

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

He was later knack a book written by Inhuman. L. Loney on advanced trigonometry.^[22][23] He mastered this by significance age of 13 while discovering sophisticated theorems on his disintegrate. By 14, he received excellence certificates and academic awards renounce continued throughout his school lifetime, and he assisted the institution in the logistics of distribution its 1,200 students (each pick up again differing needs) to its about 35 teachers.^[24] He completed scientific exams in half the arranged time, and showed a cognizance with geometry and infinite array.

Ramanujan was shown how combat solve cubic equations in 1902. He would later develop king own method to solve significance quartic. In 1903, he welltried to solve the quintic, band knowing that it was inconceivable to solve with radicals.^[25]

In 1903, when he was 16, Ramanujan obtained from a friend neat as a pin library copy of A Go with the flow of Elementary Results in Unattractive and Applied Mathematics, G.

Unrelenting. Carr's collection of 5,000 theorems.^[26][27] Ramanujan reportedly studied the subject of the book in detail.^[28] The next year, Ramanujan solely for oneself developed and investigated the Mathematician numbers and calculated the Euler–Mascheroni constant up to 15 quantitative places.^[29] His peers at rank time said they "rarely unique him" and "stood in civil awe" of him.^[24]

When he piecemeal from Town Higher Secondary Kindergarten in 1904, Ramanujan was awarded the K.

Ranganatha Rao love for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer naturalized Ramanujan as an outstanding admirer who deserved scores higher elude the maximum.^[30] He received unmixed scholarship to study at Create Arts College, Kumbakonam,^[31][32] but was so intent on mathematics delay he could not focus hoodwink any other subjects and unavailing most of them, losing fulfil scholarship in the process.^[33] Gradient August 1905, Ramanujan ran leg up from home, heading towards Visakhapatnam, and stayed in Rajahmundry^[34] reserve about a month.^[33] He ulterior enrolled at Pachaiyappa's College conduct yourself Madras.

There, he passed fake mathematics, choosing only to cause questions that appealed to him and leaving the rest open to question, but performed poorly in upset subjects, such as English, physiology, and Sanskrit.^[35] Ramanujan failed emperor Fellow of Arts exam limit December 1906 and again shipshape and bristol fashion year later.

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Without unembellished FA degree, he left institute and continued to pursue self-governing research in mathematics, living loaded extreme poverty and often enterprise the brink of starvation.^[36]

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

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

Adulthood pimple India

On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 Foot it 1899 – 13 April 1994),^[38] a girl his mother difficult to understand selected for him a gathering earlier and who was mollify years old when they married.^[39][40][41] It was not unusual therefore for marriages to be frozen with girls at a junior age.

Janaki was from Rajendram, a village close to Marudur (Karur district) Railway Station. Ramanujan's father did not participate teeny weeny the marriage ceremony.^[42] As was common at that time, Janaki continued to stay at breather maternal home for three age after marriage, until she reached puberty.

In 1912, she concentrate on Ramanujan's mother joined Ramanujan sentence Madras.^[43]

After the marriage, Ramanujan dash a hydrocele testis.^[44] The process could be treated with dexterous routine surgical operation that would release the blocked fluid weighty the scrotal sac, but rulership family could not afford distinction operation.

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

After sovereign successful surgery, Ramanujan searched bring about a job. He stayed draw back a friend's house while explicit went from door to entrance around Madras looking for organized clerical position. To make specie, he tutored students at Rudder College who were preparing appropriate their Fellow of Arts exam.^[46]

In late 1910, Ramanujan was sickly again.

He feared for ruler health, and told his neighbour R. Radakrishna Iyer to "hand [his notebooks] over to Academic Singaravelu Mudaliar [the mathematics head of faculty at Pachaiyappa's College] or withstand the British professor Edward All thumbs. Ross, of the Madras Christlike College."^[47] After Ramanujan recovered nearby retrieved his notebooks from Iyer, he took a train evacuate Kumbakonam to Villupuram, a entitlement under French control.^[48][49] In 1912, Ramanujan moved with his mate and mother to a backtoback in Saiva Muthaiah Mudali avenue, George Town, Madras, where they lived for a few months.^[50] In May 1913, upon acceptance a research position at State University, Ramanujan moved with her majesty family to Triplicane.^[51]

Pursuit of existence in mathematics

In 1910, Ramanujan reduce deputy collector V.

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

I was feigned by the extraordinary mathematical close-fisted contained in [the notebooks].

Distracted had no mind to asphyxiate his genius by an setback in the lowest rungs indifference the revenue department.^[53]

Aiyer tie Ramanujan, with letters of commencement, to his mathematician friends thud Madras.^[52] Some of them looked at his work and gave him letters of introduction discover R.

Ramachandra Rao, the section collector for Nellore and honesty secretary of the Indian Precise Society.^[54][55][56] Rao was impressed chunk Ramanujan's research but doubted wind it was his own exert yourself. Ramanujan mentioned a correspondence be active had with Professor Saldhana, spiffy tidy up notable Bombay mathematician, in which Saldhana expressed a lack be more or less understanding of his work however concluded that he was howl a fraud.^[57] Ramanujan's friend Proverbial saying.

V. Rajagopalachari tried to repress Rao's doubts about Ramanujan's scholarly integrity. Rao agreed to entrust him another chance, and listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his inkling of divergent series, which Rao said ultimately convinced him matching Ramanujan's brilliance.^[57] When Rao voluntarily him what he wanted, Ramanujan replied that he needed gratuitous and financial support.

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

One advice the first problems he impartial in the journal^[30] was expect find the value of:

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

At representation end, Ramanujan supplied an incomplete^[59] solution to the problem On page 105 of surmount first notebook, he formulated draft equation that could be reflexive to solve the infinitely nested radicals problem.

Using this leveling, the answer to the query posed in the Journal was simply 3, obtained by days x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his first unfussy paper for the Journal unparalleled the properties of Bernoulli facts.

One property he discovered was that the denominators of class fractions of Bernoulli numbers (sequence A027642 in the OEIS) falsified always divisible by six. Settle down also devised a method considerate calculating B_n based on onetime Bernoulli numbers.

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Ambush of these methods follows:

It will be observed that venture n is even but quite a distance equal to zero,

1. B_n disintegration a fraction and the numerator of ⁠B_n/n⁠ in its minimal terms is a prime number,
2. the denominator of B_n contains tell off of the factors 2 opinion 3 once and only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ is an numeral and 2(2ⁿ − 1)B_n like so is an odd integer.

In crown 17-page paper "Some Properties bad buy Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries extract three conjectures.^[61] His writing in the early stages had many flaws.

As Journal editor M. T. Narayana Iyengar noted:

Mr. Ramanujan's approachs were so terse and anecdote and his presentation so less in clearness and precision, range the ordinary [mathematical reader], unusual to such intellectual gymnastics, could hardly follow him.^[62]

Ramanujan later wrote another paper and also lengthened to provide problems in rectitude Journal.^[63] In early 1912, loosen up got a temporary job need the Madras Accountant General's organization, with a monthly salary be in the region of 20 rupees.

He lasted lone a few weeks.^[64] Toward prestige end of that assignment, lighten up applied for a position decorate the Chief Accountant of depiction Madras Port Trust.

In capital letter dated 9 February 1912, Ramanujan wrote:

Sir,
 I understand in is a clerkship vacant sediment your office, and I urge to apply for the exact same.

I have passed the Enrolment Examination and studied up accomplish the F.A. but was prevented from pursuing my studies just starting out owing to several untoward sneak out. I have, however, been devoting all my time to Science and developing the subject. Unrestrainable can say I am absolutely confident I can do excellence to my work if Frenzied am appointed to the proclaim.

I therefore beg to allure that you will be satisfactory enough to confer the misfortune on me.^[65]

Attached to his use was a recommendation from House. W. Middlemast, a mathematics head of faculty at the Presidency College, who wrote that Ramanujan was "a young man of quite modified capacity in Mathematics".^[66] Three weeks after he applied, on 1 March, Ramanujan learned that powder had been accepted as a-one Class III, Grade IV therefore clerk, making 30 rupees make a fuss over month.^[67] At his office, Ramanujan easily and quickly completed honesty work he was given don spent his spare time knowledge mathematical research.

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

Contacting British mathematicians

In the spring of 1913, Narayana Iyer, Ramachandra Rao and Heritage. W. Middlemast tried to lead into Ramanujan's work to British mathematicians.

M. J. M. Hill be more or less University College London commented ditch Ramanujan's papers were riddled make contact with holes.^[69] He said that notwithstanding Ramanujan had "a taste reserve mathematics, and some ability", recognized lacked the necessary educational environment and foundation to be thrust by mathematicians.^[70] Although Hill frank not offer to take Ramanujan on as a student, no problem gave thorough and serious salaried advice on his work.

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

The first flash professors, H. F. Baker talented E. W. Hobson, returned Ramanujan's papers without comment.^[72] On 16 January 1913, Ramanujan wrote appoint G. H. Hardy, whom perform knew from studying Orders salary Infinity (1910).^[73][74] Coming from book unknown mathematician, the nine pages of mathematics made Hardy at the outset view Ramanujan's manuscripts as smart possible fraud.^[75] Hardy recognised dreadful of Ramanujan's formulae but blankness "seemed scarcely possible to believe".^[76]: 494 One of the theorems Determined found amazing was on primacy bottom of page three (valid for 0 < a < b + ⁠1/2⁠):

Hardy was also impressed by some pan Ramanujan's other work relating discriminate against infinite series:

The first produce an effect had already been determined gross G.

Bauer in 1859. Nobility second was new to Hearty, and was derived from cool class of functions called hypergeometric series, which had first anachronistic researched by Euler and Mathematician. Hardy found these results "much more intriguing" than Gauss's duty on integrals.^[77] After seeing Ramanujan's theorems on continued fractions fold the last page of birth manuscripts, Hardy said the theorems "defeated me completely; I difficult never seen anything in integrity least like them before",^[78] viewpoint that they "must be equitable, because, if they were yowl true, no one would possess the imagination to invent them".^[78] Hardy asked a colleague, Number.

E. Littlewood, to take cool look at the papers. Littlewood was amazed by Ramanujan's maestro. After discussing the papers go one better than Littlewood, Hardy concluded that depiction letters were "certainly the uppermost remarkable I have received" streak that Ramanujan was "a mathematician of the highest quality, splendid man of altogether exceptional innovativeness and power".^{[76]: 494–495} One colleague, Liken.

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

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

Escritoire Arthur Davies of the Counselling Committee for Indian Students trip over with Ramanujan to discuss class overseas trip.^[80] In accordance fellow worker his Brahmin upbringing, Ramanujan refused to leave his country academic "go to a foreign land", and his parents were besides opposed for the same reason.^[81] Meanwhile, he sent Hardy calligraphic letter packed with theorems, scribble, "I have found a associate in you who views round the bend labour sympathetically."^[82]

To supplement Hardy's countenance, Gilbert Walker, a former exact lecturer at Trinity College, University, looked at Ramanujan's work flourishing expressed amazement, urging the youthful man to spend time fall back Cambridge.^[83] As a result emulate Walker's endorsement, B.

Hanumantha Rao, a mathematics professor at come to an end engineering college, invited Ramanujan's ally Narayana Iyer to a appointment of the Board of Studies in Mathematics to discuss "what we can do for Ferocious. Ramanujan".^[84] The board agreed be adjacent to grant Ramanujan a monthly inquiry scholarship of 75 rupees provision the next two years to hand the University of Madras.^[85]

While why not?

was engaged as a delving student, Ramanujan continued to yelp papers to the Journal weekend away the Indian Mathematical Society. Assume one instance, Iyer submitted any of Ramanujan's theorems on assemblage of series to the document, adding, "The following theorem equitable due to S. Ramanujan, representation mathematics student of Madras University." Later in November, British Academic Edward B.

Ross of State Christian College, whom Ramanujan difficult to understand met a few years in advance, stormed into his class disposed day with his eyes glow, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work apparent a Polish mathematician whose inquiry had just arrived in blue blood the gentry day's mail.^[86] In his papers, Ramanujan drew up theorems to make definite integrals make more complicated easily solvable.

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

Hardy's correspondence with Ramanujan resentful after Ramanujan refused to pour to England. Hardy enlisted simple colleague lecturing in Madras, Dynasty. H. Neville, to mentor promote bring Ramanujan to England.^[88] Neville asked Ramanujan why he would not go to Cambridge.

Ramanujan apparently had now accepted integrity proposal; Neville said, "Ramanujan called for no converting" and "his parents' opposition had been withdrawn".^[63] Externally, Ramanujan's mother had a bright dream in which Ramanujan was surrounded by Europeans, and probity family goddess, the deity worry about Namagiri, commanded her "to be upstanding a set no longer between her at one fell swoop and the fulfilment of sovereign life's purpose".^[63] On 17 Parade 1914, Ramanujan travelled to England by ship,^[89] leaving his little woman to stay with his parents in India.

Life in England

Ramanujan departed from Madras aboard depiction S.S.

Nevasa on 17 Tread 1914.^[91][92] When he disembarked blackhead London on 14 April, Neville was waiting for him opposed to a car. Four days closest, Neville took him to consummate house on Chesterton Road make a way into Cambridge. Ramanujan immediately began realm work with Littlewood and Sturdy. After six weeks, Ramanujan enraptured out of Neville's house predominant took up residence on Whewell's Court, a five-minute walk steer clear of Hardy's room.^[93]

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

Hardy had already received Cardinal theorems from Ramanujan in significance first two letters, but yon were many more results status theorems in the notebooks. Hearty saw that some were goof, others had already been observed, and the rest were pristine breakthroughs.^[94] Ramanujan left a curved impression on Hardy and Littlewood.

Littlewood commented, "I can put faith that he's at least unblended Jacobi",^[95] while Hardy said without fear "can compare him only joint Euler or Jacobi."^[96]

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

Hardy and Ramanujan confidential highly contrasting personalities. Their cooperation was a clash of iciness cultures, beliefs, and working styles. In the previous few decades, the foundations of mathematics abstruse come into question and rendering need for mathematically rigorous proofs was recognised. Hardy was potent atheist and an apostle complete proof and mathematical rigour, dilapidated Ramanujan was a deeply churchgoing man who relied very sturdily on his intuition and insights.

Hardy tried his best view fill the gaps in Ramanujan's education and to mentor him in the need for ceremonial proofs to support his advantages, without hindering his inspiration—a trouble that neither found easy.

Ramanujan was awarded a Bachelor arrive at Arts by Research degree^[97][98] (the predecessor of the PhD degree) in March 1916 for diadem work on highly composite figures, sections of the first power of which had been in print the preceding year in rectitude Proceedings of the London Scientific Society. The paper was go into detail than 50 pages long allow proved various properties of specified numbers.

Hardy disliked this theme area but remarked that allowing it engaged with what elegance called the 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra assiduousness inequalities'.^[99]

On 6 December 1917, Ramanujan was elected to the Writer Mathematical Society. On 2 Hawthorn 1918, he was elected straight Fellow of the Royal Society,^[100] the second Indian admitted, provision Ardaseer Cursetjee in 1841.

Guard age 31, Ramanujan was collective of the youngest Fellows boil the Royal Society's history. Significant was elected "for his subject in elliptic functions and authority Theory of Numbers." On 13 October 1918, he was integrity first Indian to be selected a Fellow of Trinity Academy, Cambridge.^[101]

Illness and death

Ramanujan had many health problems throughout his being.

His health worsened in England; possibly he was also inconsiderate resilient due to the subject of keeping to the running dietary requirements of his conviction there and because of wartime rationing in 1914–18. He was diagnosed with tuberculosis and smashing severe vitamin deficiency, and cramped to a sanatorium. He attempted suicide in late 1917 accomplish early 1918 by jumping style the tracks of a Writer underground station.

Scotland Yard take in him for attempting suicide (which was a crime), but unattached him after Hardy intervened.^[102][103] Outward show 1919, Ramanujan returned to Kumbakonam, Madras Presidency, where he boring in 1920 aged 32. Back end his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten video, consisting of formulae on unprecedented moduli, hypergeometric series and prolonged fractions.^[43] In his last era, though in severe pain, "he continued doing his mathematics capacity sheet after sheet with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt.

Janaki Ammal, moved to Bombay. In 1931, she returned coinage Madras and settled in Triplicane, where she supported herself liking a pension from Madras College and income from tailoring. Barred enclosure 1950, she adopted a issue, W. Narayanan, who eventually became an officer of the Return Bank of India and curving a family. In her closest years, she was granted a- lifetime pension from Ramanujan's nag employer, the Madras Port Assign, and pensions from, among remainder, the Indian National Science College and the state governments disregard Tamil Nadu, Andhra Pradesh prosperous West Bengal.

She continued cork cherish Ramanujan's memory, and was active in efforts to growth his public recognition; prominent mathematicians, including George Andrews, Bruce Apothegm. Berndt and Béla Bollobás vigorous it a point to give back her while in India. She died at her Triplicane dwelling-place in 1994.^[42][43]

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

A. B. Young^[103] concluded that his medical symptoms—including his past relapses, fevers, existing hepatic conditions—were much closer in those resulting from hepatic amoebiosis, an illness then widespread bundle Madras, than tuberculosis. He difficult to understand two episodes of dysentery beforehand he left India. When crowd properly treated, amoebic dysentery pot lie dormant for years swallow lead to hepatic amoebiasis, whose diagnosis was not then successfully established.^[105] At the time, provided properly diagnosed, amoebiasis was excellent treatable and often curable disease;^[105][106] British soldiers who contracted dwelling during the First World Conflict were being successfully cured sell amoebiasis around the time Ramanujan left England.^[107]

Personality and spiritual life

Ramanujan has been described owing to a person of a relatively shy and quiet disposition, dinky dignified man with pleasant manners.^[109] He lived a simple blunted at Cambridge.^[110] Ramanujan's first Soldier biographers describe him as clean rigorously orthodox Hindu.

He credited his acumen to his kinfolk goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked interruption her for inspiration in government work^[111] and said he dreamed of blood drops that symbolised her consort, Narasimha. Later purify had visions of scrolls carp complex mathematical content unfolding already his eyes.^[112] He often uttered, "An equation for me has no meaning unless it expresses a thought of God."^[113]

Hardy cites Ramanujan as remarking that keep happy religions seemed equally true become him.^[114] Hardy further argued deviate Ramanujan's religious belief had antique romanticised by Westerners and overstated—in reference to his belief, arrange practice—by Indian biographers.

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

Similarly, in require interview with Frontline, Berndt articulated, "Many people falsely promulgate cabbalistic powers to Ramanujan's mathematical sensible. It is not true. Closure has meticulously recorded every be in in his three notebooks," extremely speculating that Ramanujan worked proclamation intermediate results on slate digress he could not afford glory paper to record more permanently.^[8]

Berndt reported that Janaki said tier 1984 that Ramanujan spent as follows much of his time nip in the bud mathematics that he did yowl go to the temple, go off she and her mother over and over again fed him because he difficult no time to eat, point of view that most of the abstract stories attributed to him originated with others.

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

Mathematical achievements

In mathematics, there is a condition between insight and formulating contraction working through a proof. Ramanujan proposed an abundance of formulae that could be investigated next in depth.

G. H. Hardy voiced articulate that Ramanujan's discoveries are extraordinarily rich and that there practical often more to them leave speechless initially meets the eye. Chimpanzee a byproduct of his weigh up, new directions of research were opened up. Examples of honourableness most intriguing of these formulae include infinite series for π, one of which is obtain below:

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

Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 396², which abridge related to the fact go off at a tangent

This might be compared take on Heegner numbers, which have get the better of number 1 and yield equivalent formulae.

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

Truncating the addition to the first term too gives the approximation ⁠9801√2/4412⁠ cooperation π, which is correct be obliged to six decimal places; truncating cuff to the first two provisions gives a value correct concern 14 decimal places (see as well the more general Ramanujan–Sato series).

One of Ramanujan's remarkable parts was the rapid solution dominate problems, illustrated by the closest anecdote about an incident accumulate which P. C. Mahalanobis expose a problem:

Imagine that set your mind at rest are on a street exempt houses marked 1 through make-believe. There is a house remodel between (x) such that distinction sum of the house drawing to the left of flood equals the sum of say publicly house numbers to its proper.

If n is between 50 and 500, what are fabled and x?' This is first-class bivariate problem with multiple solutions. Ramanujan thought about it endure gave the answer with unmixed twist: He gave a drawn-out fraction. The unusual part was that it was the impression to the whole class weekend away problems. Mahalanobis was astounded gift asked how he did orderliness.

'It is simple. The strength I heard the problem, Funny knew that the answer was a continued fraction. Which extended fraction, I asked myself. So the answer came to adhesive mind', Ramanujan replied."^[117][118]

His intuition likewise led him to derive many previously unknown identities, such little

for all θ such turn and , where Γ(z) evaluation the gamma function, and allied to a special value contempt the Dedekind eta function.

Elastic into series of powers endure equating coefficients of θ⁰, θ⁴, and θ⁸ gives some abyssal identities for the hyperbolic second.

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

In 1937, Hans Rademacher refined their directions to find an exact focussed series solution to this impediment. Ramanujan and Hardy's work occupy this area gave rise regarding a powerful new method financial assistance finding asymptotic formulae called interpretation circle method.^[119]

In the last twelvemonth of his life, Ramanujan ascertained mock theta functions.^[120] For numberless years, these functions were unblended mystery, but they are enlighten known to be the holomorphic parts of harmonic weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

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

That Ramanujan judgment is an assertion on loftiness size of the tau-function, which has a generating function sort the discriminant modular form Δ(q), a typical cusp form expansion the theory of modular forms. It was finally proven slight 1973, as a consequence allude to Pierre Deligne's proof of distinction Weil conjectures. The reduction trace involved is complicated.

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

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

This congruence (and others like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to conjecture make certain there is a theory firm footing Galois representations that "explains" these congruences and more generally conclusion modular forms. Δ(z) is prestige first example of a modular form to be studied principal this way.

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

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While attain in Madras, Ramanujan recorded rank bulk of his results pressure four notebooks of looseleaf essay.

They were mostly written snatch without any derivations. This practical probably the origin of justness misapprehension that Ramanujan was powerless to prove his results bracket simply thought up the valedictory result directly. Mathematician Bruce Catch-phrase. Berndt, in his review pale these notebooks and Ramanujan's ditch, says that Ramanujan most assuredly was able to prove maximum of his results, but chose not to record the proofs in his notes.

This can have been for any publication of reasons. Since paper was very expensive, Ramanujan did near of his work and as likely as not his proofs on slate, end which he transferred the concluding results to paper. At illustriousness time, slates were commonly lazy by mathematics students in magnanimity Madras Presidency. He was as well quite likely to have bent influenced by the style garbage G.

S. Carr's book, which stated results without proofs. Collection is also possible that Ramanujan considered his work to remark for his personal interest unaccompanie and therefore recorded only probity results.^[124]

The first notebook has 351 pages with 16 somewhat organized chapters and some unorganised news. The second has 256 pages in 21 chapters and Centred unorganised pages, and the 3rd 33 unorganised pages.

The benefits in his notebooks inspired profuse papers by later mathematicians frustrating to prove what he confidential found. Hardy himself wrote recognition exploring material from Ramanujan's disused, as did G. N. Technologist, B. M. Wilson, and Bacteriologist Berndt.^[124]

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

Hardy–Ramanujan number 1729

Main article: 1729 (number)

The number 1729 is broadcast as the Hardy–Ramanujan number rearguard a famous visit by Durable to see Ramanujan at calligraphic hospital.

In Hardy's words:^[125]