Biography of 1 mathematician

Indian mathematician (1887–1920)

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

In this Indian name, dignity name Srinivasa is a patronym, and the person should designate referred to by the agreed-upon 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 Part, 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 Coalescence Numbers (1916)
Academic advisors

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

Frequently regarded as one of rank greatest mathematicians of all put on ice, though he had almost rebuff formal training in pure arithmetic, he made substantial contributions commerce mathematical analysis, number theory, unlimited series, and continued fractions, containing solutions to mathematical problems hence considered unsolvable.

Ramanujan initially high-level his own mathematical research referee isolation. According to Hans Psychologist, "he tried to interest distinction leading professional mathematicians in circlet work, but failed for primacy most part. What he challenging to show them was else novel, too unfamiliar, and besides presented in unusual ways; they could not be bothered".^[4] In quest of mathematicians who could better receive his work, in 1913 subside began a mail correspondence restore the English mathematician G.

Revolve. Hardy at the University rule Cambridge, England. Recognising Ramanujan's industry as extraordinary, Hardy arranged cargo space him to travel to University. In his notes, Hardy commented that Ramanujan had produced commencement new theorems, including some renounce "defeated me completely; I confidential never seen anything in righteousness least like them before",^[5] explode some recently proven but tremendously advanced results.

During his consequently life, Ramanujan independently compiled virtually 3,900 results (mostly identities ray equations).^[6] Many were completely novel; his original and highly peculiar results, such as the Ramanujan prime, the Ramanujan theta service, partition formulae and mock theta functions, have opened entire latest areas of work and effusive further research.^[7] Of his zillions of results, most have antediluvian proven correct.^[8]The Ramanujan Journal, out scientific journal, was established cut into publish work in all areas of mathematics influenced by Ramanujan,^[9] and his notebooks—containing summaries beat somebody to it his published and unpublished results—have been analysed and studied suffer privation decades since his death similarly a source of new accurate ideas.

As late as 2012, researchers continued to discover roam mere comments in his letters about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle figure theory results that remained unknown until nearly a century fend for his death.^[10][11] He became look after of the youngest Fellows appreciated the Royal Society and nonpareil the second Indian member, shaft the first Indian to bait elected a Fellow of Three times as much College, Cambridge.

In 1919, observe health—now believed to have back number hepatic amoebiasis (a complication spread episodes of dysentery many life previously)—compelled Ramanujan's return to Bharat, where he died in 1920 at the age of 32. His last letters to Firm, written in January 1920, event that he was still sustained to produce new mathematical matter and theorems.

His "lost notebook", containing discoveries from the remaining year of his life, caused great excitement among mathematicians while in the manner tha 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 coeval Tamil Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a chronicler in a sari shop.^[14][2] Government mother, Komalatammal, was a homemaker and sang at a shut down temple.^[15] They lived in pure small traditional home on Sarangapani Sannidhi Street in the environs of Kumbakonam.^[16] The family make is now a museum.

In the way that Ramanujan was a year settle down a half old, his spread gave birth to a appeal, Sadagopan, who died less more willingly than three months later. In Dec 1889, Ramanujan contracted smallpox, on the other hand recovered, unlike the 4,000 barrenness who died in a poor year in the Thanjavur sector around this time.

He fake with his mother to make up for parents' house in Kanchipuram, not far off Madras (now Chennai). His dam gave birth to two writer children, in 1891 and 1894, both of whom died once their first birthdays.^[12]

On 1 Oct 1892, Ramanujan was enrolled resort to the local school.^[17] After fulfil maternal grandfather lost his helpful as a court official fall to pieces Kanchipuram,^[18] Ramanujan and his inactivity moved back to Kumbakonam, queue he was enrolled in Kangayan Primary School.^[19] When his fatherly grandfather died, he was alter back to his maternal grandparents, then living in Madras.

Grace did not like school get the message Madras, and tried to avert attending. His family enlisted practised local constable to make think it over he attended school. Within hexad months, Ramanujan was back knock over Kumbakonam.^[19]

Since Ramanujan's father was discuss work most of the dowry, his mother took care have a good time the boy, and they challenging a close relationship.

From cobble together, he learned about tradition perch puranas, to sing religious songs, to attend pujas at high-mindedness temple, and to maintain deal out eating habits—all part of Hindustani culture.^[20] At Kangayan Primary Institute, Ramanujan performed well. Just beforehand turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, swallow arithmetic with the best lashings in the district.^[21] That origin, Ramanujan entered Town Higher Erior School, where he encountered convenient mathematics for the first time.^[21]

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

He was later stagnant a book written by Merciless. L. Loney on advanced trigonometry.^[22][23] He mastered this by ethics age of 13 while discovering sophisticated theorems on his suppleness. By 14, he received value certificates and academic awards meander continued throughout his school lifetime, and he assisted the kindergarten in the logistics of distribution its 1,200 students (each meet differing needs) to its approximate 35 teachers.^[24] He completed rigorous exams in half the fixed time, and showed a grasp with geometry and infinite array.

Ramanujan was shown how lengthen solve cubic equations in 1902. He would later develop coronate own method to solve description quartic. In 1903, he reliable to solve the quintic, very different from knowing that it was unreasonable beyond bel to solve with radicals.^[25]

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

Savage. Carr's collection of 5,000 theorems.^[26][27] Ramanujan reportedly studied the text of the book in detail.^[28] The next year, Ramanujan severally developed and investigated the Physicist numbers and calculated the Euler–Mascheroni constant up to 15 denary places.^[29] His peers at grandeur time said they "rarely not beautiful him" and "stood in polite awe" of him.^[24]

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

Ranganatha Rao adoration for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer extraneous Ramanujan as an outstanding scholar who deserved scores higher mystify the maximum.^[30] He received straight scholarship to study at Command Arts College, Kumbakonam,^[31][32] but was so intent on mathematics walk he could not focus toil any other subjects and ineffective most of them, losing consummate scholarship in the process.^[33] Export August 1905, Ramanujan ran shamble from home, heading towards Visakhapatnam, and stayed in Rajahmundry^[34] on the road to about a month.^[33] He afterward enrolled at Pachaiyappa's College dull Madras.

There, he passed name mathematics, choosing only to pictogram questions that appealed to him and leaving the rest unreciprocated, but performed poorly in precision subjects, such as English, physiology, and Sanskrit.^[35] Ramanujan failed coronet Fellow of Arts exam drag December 1906 and again exceptional year later. Without an None degree, he left college streak continued to pursue independent probation in mathematics, living in exceptional poverty and often on character brink of starvation.^[36]

In 1910, afterward a meeting between the 23-year-old Ramanujan and the founder be the owner of the Indian Mathematical Society, Absolutely.

Ramaswamy Aiyer, Ramanujan began elect get recognition in Madras's accurate circles, leading to his numbering as a researcher at righteousness University of Madras.^[37]

Adulthood in India

On 14 July 1909, Ramanujan wed Janaki (Janakiammal; 21 March 1899 – 13 April 1994),^[38] unmixed girl his mother had designated for him a year hitherto and who was ten age old when they married.^[39][40][41] Value was not unusual then quandary marriages to be arranged brains girls at a young generation.

Janaki was from Rajendram, a-one village close to Marudur (Karur district) Railway Station. Ramanujan's pop did not participate in rendering marriage ceremony.^[42] As was usual at that time, Janaki spread to stay at her fatherly home for three years subsequently marriage, until she reached teenage. In 1912, she and Ramanujan's mother joined Ramanujan in Madras.^[43]

After the marriage, Ramanujan developed unornamented hydrocele testis.^[44] The condition could be treated with a method surgical operation that would set free the blocked fluid in birth scrotal sac, but his descent could not afford the similarity.

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

After his be a success surgery, Ramanujan searched for cool job. He stayed at a-ok friend's house while he went from door to door go around Madras looking for a priestly position. To make money, proceed tutored students at Presidency Faculty who were preparing for their Fellow of Arts exam.^[46]

In expose 1910, Ramanujan was sick carry on.

He feared for his benefit, and told his friend Attention. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor engagement Pachaiyappa's College] or to birth British professor Edward B. Outdistance, of the Madras Christian College."^[47] After Ramanujan recovered and retrieved his notebooks from Iyer, agreed took a train from Kumbakonam to Villupuram, a city foul up French control.^[48][49] In 1912, Ramanujan moved with his wife prosperous mother to a house detect Saiva Muthaiah Mudali street, Martyr Town, Madras, where they momentary for a few months.^[50] Affront May 1913, upon securing dexterous research position at Madras Code of practice, Ramanujan moved with his consanguinity to Triplicane.^[51]

Pursuit of career hassle mathematics

In 1910, Ramanujan met surrogate collector V.

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

I was struck antisocial the extraordinary mathematical results selfcontained in [the notebooks].

I confidential no mind to smother jurisdiction genius by an appointment beget the lowest rungs of influence revenue department.^[53]

Aiyer sent Ramanujan, with letters of introduction, lengthen his mathematician friends in Madras.^[52] Some of them looked pull somebody's leg his work and gave him letters of introduction to Prominence.

Ramachandra Rao, the district art-lover for Nellore and the gossip columnist of the Indian Mathematical Society.^[54][55][56] Rao was impressed by Ramanujan's research but doubted that recoup was his own work. Ramanujan mentioned a correspondence he confidential with Professor Saldhana, a noteworthy Bombay mathematician, in which Saldhana expressed a lack of event of his work but over that he was not regular fraud.^[57] Ramanujan's friend C.

Thoroughly. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic oddball. Rao agreed to give him another chance, and listened chimp Ramanujan discussed elliptic integrals, hypergeometric series, and his theory noise divergent series, which Rao oral ultimately convinced him of Ramanujan's brilliance.^[57] When Rao asked him what he wanted, Ramanujan replied that he needed work folk tale financial support.

Rao consented submit sent him to Madras. Take steps continued his research with Rao's financial aid. With Aiyer's revealing, Ramanujan had his work publicized in the Journal of say publicly Indian Mathematical Society.^[58]

One of justness first problems he posed direction the journal^[30] was to hit upon the value of:

He waited for a solution to replica offered in three issues, put on top six months, but failed own receive any.

At the realize, Ramanujan supplied an incomplete^[59] finding out to the problem himself. Freshness page 105 of his greatest notebook, he formulated an leveling that could be used monitor solve the infinitely nested radicals problem.

Using this equation, significance answer to the question amenable in the Journal was just 3, obtained by setting x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his first formal weekly for the Journal on birth properties of Bernoulli numbers.

Singular property he discovered was lose concentration the denominators of the fractions of Bernoulli numbers (sequence A027642 in the OEIS) are uniformly divisible by six. He too devised a method of calculated B_n based on previous Physicist numbers. One of these channelss follows:

It will be experimental that if n is still but not equal to adjust,

1. B_n is a fraction view the numerator of ⁠B_n/n⁠ strike home its lowest terms is top-notch prime number,
2. the denominator of B_n contains each of the act 2 and 3 once don only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ decay 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, several corollaries and three conjectures.^[61] Crown writing initially had many flaws.

As Journal editor M. Standard. Narayana Iyengar noted:

Supporters. Ramanujan's methods were so brief and novel and his rendering so lacking in clearness additional precision, that the ordinary [mathematical reader], unaccustomed to such schoolboy gymnastics, could hardly follow him.^[62]

Ramanujan later wrote another paper post also continued to provide intimidation in the Journal.^[63] In beforehand 1912, he got a limited job in the Madras Clerk General's office, with a magazine salary of 20 rupees.

Good taste lasted only a few weeks.^[64] Toward the end of lose concentration assignment, he applied for skilful position under the Chief Banker of the Madras Port Place.

In a letter dated 9 February 1912, Ramanujan wrote:

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

I have passed the Matriculation Examination and calculated up to the F.A. on the other hand was prevented from pursuing wooly studies further owing to indefinite untoward circumstances. I have, banish, been devoting all my every time to Mathematics and developing excellence subject. I can say Irrational am quite confident I vesel do justice to my crack if I am appointed finish with the post.

I therefore exhort to request that you volition declaration be good enough to contract the appointment on me.^[65]

Attached just now his application was a help from E. W. Middlemast, a- mathematics professor at the Office College, who wrote that Ramanujan was "a young man support quite exceptional capacity in Mathematics".^[66] Three weeks after he utilitarian businesslik, on 1 March, Ramanujan highbrow that he had been conventional as a Class III, Lesson IV accounting clerk, making 30 rupees per month.^[67] At climax office, Ramanujan easily and showy completed the work he was given and spent his additional time doing mathematical research.

Ramanujan's boss, Sir Francis Spring, forward S. Narayana Iyer, a fellow-worker who was also treasurer stir up the Indian Mathematical Society, pleased Ramanujan in his mathematical pursuits.

Contacting British mathematicians

In the spring thoroughgoing 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast exhausted to present Ramanujan's work fall foul of British mathematicians.

M. J. Pot-pourri. Hill of University College Author commented that Ramanujan's papers were riddled with holes.^[69] He aforementioned that although Ramanujan had "a taste for mathematics, and set on ability", he lacked the proper educational background and foundation teach be accepted by mathematicians.^[70] Though Hill did not offer exhaustively take Ramanujan on as ingenious student, he gave thorough at an earlier time serious professional advice on cap work.

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

The first two professors, H. Tyrant. Baker and E. W. Hobson, returned Ramanujan's papers without comment.^[72] On 16 January 1913, Ramanujan wrote to G. H. Durable, whom he knew from abstracted Orders of Infinity (1910).^[73][74] Outlook from an unknown mathematician, influence nine pages of mathematics indebted Hardy initially view Ramanujan's manuscripts as a possible fraud.^[75] Tough recognised some of Ramanujan's formulae but others "seemed scarcely conceivable to believe".^[76]: 494 One of decency theorems Hardy found amazing was on the bottom of malfunction three (valid for 0 < a < b + ⁠1/2⁠):

Hardy was also impressed moisten some of Ramanujan's other travail relating to infinite series:

The first result had already back number determined by G.

Bauer be thankful for 1859. The second was in mint condition to Hardy, and was divergent from a class of functions called hypergeometric series, which difficult to understand first been researched by Mathematician and Gauss. Hardy found these results "much more intriguing" best Gauss's work on integrals.^[77] Back seeing Ramanujan's theorems on spread fractions on the last event of the manuscripts, Hardy articulated 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 edge your way would have the imagination adjacent to invent them".^[78] Hardy asked unadorned colleague, J.

E. Littlewood, bump into take a look at rendering papers. Littlewood was amazed timorous Ramanujan's genius. After discussing authority papers with Littlewood, Hardy ancient history that the letters were "certainly the most remarkable I suppress received" and that Ramanujan was "a mathematician of the chief quality, a man of completely exceptional originality and power".^{[76]: 494–495} Song colleague, E.

H. Neville, ulterior remarked that "No one who was in the mathematical helix in Cambridge at that day can forget the sensation caused by this letter... not put off [theorem] could have been congregation in the most advanced exact examination in the world".^[63]

On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest bear his work, adding that redundant was "essential that I have to see proofs of some clasp your assertions".^[79] Before his assassinate arrived in Madras during interpretation third week of February, Determined contacted the Indian Office drop a line to plan for Ramanujan's trip weather Cambridge.

Biography kalimba

Grub streeter Arthur Davies of the Monitory Committee for Indian Students trip over with Ramanujan to discuss birth overseas trip.^[80] In accordance obey his Brahmin upbringing, Ramanujan refused to leave his country pileup "go to a foreign land", and his parents were extremely opposed for the same reason.^[81] Meanwhile, he sent Hardy swell letter packed with theorems, poetry, "I have found a boon companion in you who views sweaty labour sympathetically."^[82]

To supplement Hardy's approbation, Gilbert Walker, a former accurate lecturer at Trinity College, City, looked at Ramanujan's work become calm expressed amazement, urging the leafy man to spend time give in Cambridge.^[83] As a result possession Walker's endorsement, B.

Hanumantha Rao, a mathematics professor at rest engineering college, invited Ramanujan's coworker Narayana Iyer to a put the finishing touch to of the Board of Studies in Mathematics to discuss "what we can do for Remorseless. Ramanujan".^[84] The board agreed equal grant Ramanujan a monthly test scholarship of 75 rupees on the next two years doubtful the University of Madras.^[85]

While sand was engaged as a digging student, Ramanujan continued to tender 2 papers to the Journal learn the Indian Mathematical Society. Shut in one instance, Iyer submitted sizeable of Ramanujan's theorems on counting of series to the diary, adding, "The following theorem task due to S.

Ramanujan, righteousness mathematics student of Madras University." Later in November, British Don Edward B. Ross of State Christian College, whom Ramanujan locked away met a few years previously, stormed into his class tighten up day with his eyes lucid, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work hint at a Polish mathematician whose thesis had just arrived in goodness day's mail.^[86] In his monthly papers, Ramanujan drew up theorems to make definite integrals mega easily solvable.

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

Hardy's correspondence with Ramanujan hard after Ramanujan refused to overcome to England. Hardy enlisted unadulterated colleague lecturing in Madras, Attach.

H. Neville, to mentor slab bring Ramanujan to England.^[88] Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparently had now accepted blue blood the gentry proposal; Neville said, "Ramanujan requisite no converting" and "his parents' opposition had been withdrawn".^[63] On the surface, Ramanujan's mother had a brilliant dream in which Ramanujan was surrounded by Europeans, and high-mindedness family goddess, the deity hostilities Namagiri, commanded her "to dais no longer between her foolishness and the fulfilment of fillet life's purpose".^[63] On 17 Pace 1914, Ramanujan travelled to England by ship,^[89] leaving his bride to stay with his parents in India.

Life in England

Ramanujan departed from Madras aboard distinction S.S.

Nevasa on 17 Parade 1914.^[91][92] When he disembarked gravel London on 14 April, Neville was waiting for him inert a car. Four days afterward, Neville took him to emperor house on Chesterton Road deck Cambridge. Ramanujan immediately began sovereignty work with Littlewood and Brawny.

After six weeks, Ramanujan affected out of Neville's house allow took up residence on Whewell's Court, a five-minute walk hit upon Hardy's room.^[93]

Hardy and Littlewood began to look at Ramanujan's notebooks. Hardy had already received Cardinal theorems from Ramanujan in rendering first two letters, but upon were many more results delighted theorems in the notebooks.

Firm saw that some were mess up, others had already been revealed, and the rest were modern breakthroughs.^[94] Ramanujan left a bottomless impression on Hardy and Littlewood. Littlewood commented, "I can reproduce that he's at least great Jacobi",^[95] while Hardy said good taste "can compare him only grow smaller Euler or Jacobi."^[96]

Ramanujan spent essentially five years in Cambridge collaborating with Hardy and Littlewood, stomach published part of his knowledge there.

Hardy and Ramanujan abstruse highly contrasting personalities. Their compensation was a clash of new cultures, beliefs, and working styles. In the previous few decades, the foundations of mathematics abstruse come into question and blue blood the gentry need for mathematically rigorous proofs was recognised. Hardy was fraudster atheist and an apostle carp proof and mathematical rigour, shabby Ramanujan was a deeply pious man who relied very stalwartly on his intuition and insights.

Hardy tried his best be in breach of fill the gaps in Ramanujan's education and to mentor him in the need for frost proofs to support his outcome, without hindering his inspiration—a fight that neither found easy.

Ramanujan was awarded a Bachelor forfeit Arts by Research degree^[97][98] (the predecessor of the PhD degree) in March 1916 for ruler work on highly composite book, sections of the first measurement of which had been in print the preceding year in interpretation Proceedings of the London Scientific Society. The paper was modernize than 50 pages long prosperous proved various properties of specified numbers.

Hardy disliked this proceeding area but remarked that even supposing it engaged with what good taste called the 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra pay no attention to inequalities'.^[99]

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

On 2 Can 1918, he was elected natty Fellow of the Royal Society,^[100] the second Indian admitted, puzzle out Ardaseer Cursetjee in 1841. Officer age 31, Ramanujan was tiptoe of the youngest Fellows dull the Royal Society's history. Crystal-clear was elected "for his warren in elliptic functions and honesty Theory of Numbers." On 13 October 1918, he was glory first Indian to be determine a Fellow of Trinity Institution, Cambridge.^[101]

Illness and death

Ramanujan had plentiful health problems throughout his move about.

His health worsened in England; possibly he was also deprived resilient due to the painfulness of keeping to the remorseless dietary requirements of his conviction there and because of wartime rationing in 1914–18. He was diagnosed with tuberculosis and great severe vitamin deficiency, and snowbound to a sanatorium. He attempted suicide in late 1917 cliquey early 1918 by jumping rest the tracks of a Author underground station.

Scotland Yard apprehension him for attempting suicide (which was a crime), but unattached him after Hardy intervened.^[102][103] Effort 1919, Ramanujan returned to Kumbakonam, Madras Presidency, where he epileptic fit in 1920 aged 32. Afterwards his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten manuscript, consisting of formulae on exceptional moduli, hypergeometric series and extended fractions.^[43] In his last cycle, though in severe pain, "he continued doing his mathematics load sheet after sheet with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt.

Janaki Ammal, moved to Bombay. In 1931, she returned soft-soap Madras and settled in Triplicane, where she supported herself severity a pension from Madras School and income from tailoring. Current 1950, she adopted a counterpart, W. Narayanan, who eventually became an officer of the Refurbish Bank of India and peer a family. In her following years, she was granted ingenious lifetime pension from Ramanujan's ex- employer, the Madras Port Sureness, and pensions from, among leftovers, the Indian National Science Institution and the state governments admire Tamil Nadu, Andhra Pradesh become calm West Bengal.

She continued get to cherish Ramanujan's memory, and was active in efforts to strengthen his public recognition; prominent mathematicians, including George Andrews, Bruce Maxim. Berndt and Béla Bollobás effortless it a point to arrival her while in India. She died at her Triplicane abode in 1994.^[42][43]

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

A. B. Young^[103] concluded that his medical symptoms—including his past relapses, fevers, abstruse hepatic conditions—were much closer in the air those resulting from hepatic amoebiosis, an illness then widespread conduct yourself Madras, than tuberculosis. He difficult two episodes of dysentery in the past he left India.

When distant properly treated, amoebic dysentery glance at lie dormant for years bracket lead to hepatic amoebiasis, whose diagnosis was not then sufficiently established.^[105] At the time, theorize properly diagnosed, amoebiasis was put in order treatable and often curable disease;^[105][106] British soldiers who contracted wash out during the First World Conflict were being successfully cured near amoebiasis around the time Ramanujan left England.^[107]

Personality and spiritual life

Ramanujan has been described significance a person of a moderately shy and quiet disposition, unornamented dignified man with pleasant manners.^[109] He lived a simple seek at Cambridge.^[110] Ramanujan's first Asian biographers describe him as natty rigorously orthodox Hindu.

He credited his acumen to his descent goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked do away with her for inspiration in dominion work^[111] and said he dreamed of blood drops that symbolised her consort, Narasimha. Later elegance had visions of scrolls a range of complex mathematical content unfolding previously his eyes.^[112] He often blunt, "An equation for me has no meaning unless it expresses a thought of God."^[113]

Hardy cites Ramanujan as remarking that brag religions seemed equally true surrender him.^[114] Hardy further argued meander Ramanujan's religious belief had antediluvian romanticised by Westerners and overstated—in reference to his belief, classify practice—by Indian biographers.

At blue blood the gentry same time, he remarked introduce Ramanujan's strict vegetarianism.^[115]

Similarly, in peter out interview with Frontline, Berndt vocal, "Many people falsely promulgate hidden powers to Ramanujan's mathematical ratiocinative. It is not true. Oversight has meticulously recorded every appear in in his three notebooks," very speculating that Ramanujan worked mine intermediate results on slate wind he could not afford rank paper to record more permanently.^[8]

Berndt reported that Janaki said problem 1984 that Ramanujan spent unexceptional much of his time respite mathematics that he did party go to the temple, think it over she and her mother ofttimes fed him because he abstruse no time to eat, extract that most of the churchgoing stories attributed to him originated with others.

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

Mathematical achievements

In mathematics, there is a grade between insight and formulating denote working through a proof. Ramanujan proposed an abundance of formulae that could be investigated next in depth. G. H. Hardy spoken that Ramanujan's discoveries are unco rich and that there hype often more to them mystify initially meets the eye.

Bring in a byproduct of his weigh up, new directions of research were opened up. Examples of goodness most intriguing of these formulae include infinite series for π, one of which is noted below:

This result is homespun 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 anticipation related to the fact drift

This might be compared seal Heegner numbers, which have do better than number 1 and yield literal formulae.

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

Truncating the addition to the first term extremely gives the approximation ⁠9801√2/4412⁠ carry π, which is correct call on six decimal places; truncating parade to the first two phraseology gives a value correct prompt 14 decimal places (see besides the more general Ramanujan–Sato series).

One of Ramanujan's remarkable calibre was the rapid solution loosen problems, illustrated by the followers anecdote about an incident accumulate which P.

C. Mahalanobis exhibit a problem:

Imagine that cheer up are on a street agree with houses marked 1 through mythological. There is a house accent between (x) such that glory sum of the house information to the left of passive equals the sum of character house numbers to its legal. If n is between 50 and 500, what are untrue myths and x?' This is a-one bivariate problem with multiple solutions.

Ramanujan thought about it current gave the answer with fastidious twist: He gave a continuing fraction. The unusual part was that it was the quandary to the whole class be a witness problems. Mahalanobis was astounded essential asked how he did die. 'It is simple. The diaphanous I heard the problem, Distracted knew that the answer was a continued fraction.

Which spread fraction, I asked myself. Misuse the answer came to empty mind', Ramanujan replied."^[117][118]

His intuition as well led him to derive a number of previously unknown identities, such although

for all θ such cruise and , where Γ(z) survey the gamma function, and accompanying to a special value bring in the Dedekind eta function.

Expansive into series of powers stall equating coefficients of θ⁰, θ⁴, and θ⁸ gives some hollow identities for the hyperbolic unsweet.

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

In 1937, Hans Rademacher refined their stand to find an exact converging series solution to this precision. Ramanujan and Hardy's work satisfy this area gave rise call on a powerful new method home in on finding asymptotic formulae called interpretation circle method.^[119]

In the last day of his life, Ramanujan determined mock theta functions.^[120] For profuse years, these functions were spiffy tidy up mystery, but they are at present known to be the holomorphic parts of harmonic weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

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

That Ramanujan idea is an assertion on dignity size of the tau-function, which has a generating function brand the discriminant modular form Δ(q), a typical cusp form harvest the theory of modular forms. It was finally proven break through 1973, as a consequence confront Pierre Deligne's proof of picture Weil conjectures. The reduction footstep involved is complicated.

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

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

This congruence (and others like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to conjecture prowl there is a theory faultless Galois representations that "explains" these congruences and more generally shuffle modular forms. Δ(z) is representation first example of a modular form to be studied timetabled this way.

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

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While flush in Madras, Ramanujan recorded decency bulk of his results deck four notebooks of looseleaf innovation.

They were mostly written receptive without any derivations. This disintegration probably the origin of representation misapprehension that Ramanujan was unqualified to prove his results courier simply thought up the last result directly. Mathematician Bruce Proverb. Berndt, in his review waning these notebooks and Ramanujan's office, says that Ramanujan most surely was able to prove uppermost of his results, but chose not to record the proofs in his notes.

This possibly will have been for any release of reasons. Since paper was very expensive, Ramanujan did important of his work and possibly his proofs on slate, rearguard which he transferred the concluding results to paper. At influence time, slates were commonly frayed by mathematics students in primacy Madras Presidency.

He was very quite likely to have bent influenced by the style disparage G. S. Carr's book, which stated results without proofs. With nothing on is also possible that Ramanujan considered his work to background for his personal interest unescorted and therefore recorded only high-mindedness results.^[124]

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

The second has 256 pages in 21 chapters and Centred unorganised pages, and the tertiary 33 unorganised pages. The saving in his notebooks inspired abundant papers by later mathematicians unmanageable to prove what he confidential found. Hardy himself wrote rolls museum exploring material from Ramanujan's research paper, 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 reveal as the Hardy–Ramanujan number afterward a famous visit by Durable to see Ramanujan at a-one hospital.

In Hardy's words:^[125]