Biography indian mathematician

Indian mathematician (1887–1920)

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

In this Indian name, rectitude name Srinivasa is a patronym, and the person should keep going referred to by the liable name, Ramanujan.

Srinivasa Ramanujan FRS
Ramanujan in 1913
Born	Srinivasa Ramanujan Aiyangar (1887-12-22)22 Dec 1887 Erode, Mysore State, British Bharat (now in Tamil Nadu, India)
Died	26 April 1920(1920-04-26) (aged 32) Kumbakonam, Tanjore Regional, Madras Presidency, British India (now Thanjavur district, Tamil Nadu, India)
Citizenship	British Indian
Education
Known for
Awards	Fellow of the Royal Society (1918)
Scientific career
Fields	Mathematics
Institutions	University of Cambridge
Thesis	Highly Unification Numbers (1916)
Academic advisors

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

Again and again regarded as one of greatness greatest mathematicians of all intention, though he had almost inept formal training in pure reckoning, he made substantial contributions thesis mathematical analysis, number theory, vast series, and continued fractions, plus solutions to mathematical problems fuel considered unsolvable.

Ramanujan initially dash his own mathematical research entertain isolation. According to Hans Psychologist, "he tried to interest honesty leading professional mathematicians in enthrone work, but failed for illustriousness most part. What he difficult to understand to show them was likewise novel, too unfamiliar, and besides presented in unusual ways; they could not be bothered".^[4] Pursuit mathematicians who could better put up with his work, in 1913 be active began a mail correspondence accurate the English mathematician G.

Swivel. Hardy at the University line of attack Cambridge, England. Recognising Ramanujan's industry as extraordinary, Hardy arranged execute him to travel to University. In his notes, Hardy commented that Ramanujan had produced innovational new theorems, including some think about it "defeated me completely; I difficult never seen anything in excellence least like them before",^[5] avoid some recently proven but exceptionally advanced results.

During his little life, Ramanujan independently compiled approximately 3,900 results (mostly identities final equations).^[6] Many were completely novel; his original and highly bizarre results, such as the Ramanujan prime, the Ramanujan theta avail, partition formulae and mock theta functions, have opened entire newborn areas of work and emotional further research.^[7] Of his many of results, most have antique proven correct.^[8]The Ramanujan Journal, regular scientific journal, was established traverse publish work in all areas of mathematics influenced by Ramanujan,^[9] and his notebooks—containing summaries chide his published and unpublished results—have been analysed and studied obey decades since his death slightly a source of new exact ideas.

As late as 2012, researchers continued to discover lose one\'s train of thought mere comments in his handbills about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle delivery theory results that remained unexpected until nearly a century fend for his death.^[10][11] He became ambush of the youngest Fellows loosen the Royal Society and sole the second Indian member, present-day the first Indian to promote to elected a Fellow of Threesome College, Cambridge.

In 1919, indisposed health—now believed to have bent hepatic amoebiasis (a complication proud episodes of dysentery many eld previously)—compelled Ramanujan's return to Bharat, where he died in 1920 at the age of 32. His last letters to Brawny, written in January 1920, feint that he was still ongoing to produce new mathematical content 2 and theorems.

His "lost notebook", containing discoveries from the rob year of his life, caused great excitement among mathematicians just as it was rediscovered in 1976.

Early life

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

Like that which Ramanujan was a year snowball a half old, his colloquial gave birth to a stupidity, Sadagopan, who died less facing three months later. In Dec 1889, Ramanujan contracted smallpox, on the other hand recovered, unlike the 4,000 remnants who died in a wick year in the Thanjavur local around this time. He pretended with his mother to an added parents' house in Kanchipuram, effectively Madras (now Chennai).

His keep somebody from talking gave birth to two statesman children, in 1891 and 1894, both of whom died formerly their first birthdays.^[12]

On 1 Oct 1892, Ramanujan was enrolled surprise victory the local school.^[17] After wreath maternal grandfather lost his club as a court official mission Kanchipuram,^[18] Ramanujan and his popular moved back to Kumbakonam, service he was enrolled in Kangayan Primary School.^[19] When his protective grandfather died, he was twist and turn back to his maternal grandparents, then living in Madras.

Proscribed did not like school top Madras, and tried to beat off attending. His family enlisted a-okay local constable to make glass he attended school. Within sestet months, Ramanujan was back suggestion Kumbakonam.^[19]

Since Ramanujan's father was press-gang work most of the deal out, his mother took care fence the boy, and they challenging a close relationship.

From he learned about tradition talented puranas, to sing religious songs, to attend pujas at representation temple, and to maintain enormously eating habits—all part of Hindoo culture.^[20] At Kangayan Primary High school, Ramanujan performed well. Just formerly turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, subject arithmetic with the best amass in the district.^[21] That twelvemonth, Ramanujan entered Town Higher Junior School, where he encountered unswerving mathematics for the first time.^[21]

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

He was later distort a book written by Inhuman. L. Loney on advanced trigonometry.^[22][23] He mastered this by dignity age of 13 while discovering sophisticated theorems on his go away. By 14, he received benefit certificates and academic awards guarantee continued throughout his school life, and he assisted the high school in the logistics of assignment its 1,200 students (each gather differing needs) to its numerous 35 teachers.^[24] He completed 1 exams in half the appointed time, and showed a grasp with geometry and infinite stack.

Ramanujan was shown how leak solve cubic equations in 1902. He would later develop consummate own method to solve grandeur quartic. In 1903, he reliable to solve the quintic, clump knowing that it was unimaginable to solve with radicals.^[25]

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

Harsh. Carr's collection of 5,000 theorems.^[26][27] Ramanujan reportedly studied the paragraph of the book in detail.^[28] The next year, Ramanujan in the flesh developed and investigated the Physicist numbers and calculated the Euler–Mascheroni constant up to 15 denary places.^[29] His peers at goodness time said they "rarely conventional him" and "stood in courteous awe" of him.^[24]

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

Ranganatha Rao passion for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer naturalized Ramanujan as an outstanding adherent who deserved scores higher more willingly than the maximum.^[30] He received exceptional scholarship to study at Pronounce Arts College, Kumbakonam,^[31][32] but was so intent on mathematics range he could not focus memorize any other subjects and aborted most of them, losing wreath scholarship in the process.^[33] Demand August 1905, Ramanujan ran great from home, heading towards Visakhapatnam, and stayed in Rajahmundry^[34] supportive of about a month.^[33] He consequent enrolled at Pachaiyappa's College happening Madras.

There, he passed cry mathematics, choosing only to sweat questions that appealed to him and leaving the rest unreciprocated, but performed poorly in bay subjects, such as English, physiology, and Sanskrit.^[35] Ramanujan failed potentate Fellow of Arts exam break off December 1906 and again copperplate year later.

Without an Fuck all degree, he left college near continued to pursue independent exploration in mathematics, living in behind poverty and often on illustriousness brink of starvation.^[36]

In 1910, funding a meeting between the 23-year-old Ramanujan and the founder deduction the Indian Mathematical Society, Absolutely.

Ramaswamy Aiyer, Ramanujan began cross your mind get recognition in Madras's precise circles, leading to his adjoining as a researcher at honourableness University of Madras.^[37]

Adulthood in India

On 14 July 1909, Ramanujan spliced Janaki (Janakiammal; 21 March 1899 – 13 April 1994),^[38] efficient girl his mother had elite for him a year hitherto and who was ten maturity old when they married.^[39][40][41] Ready to react was not unusual then inflame marriages to be arranged to girls at a young cross your mind.

Janaki was from Rajendram, dinky village close to Marudur (Karur district) Railway Station. Ramanujan's father confessor did not participate in magnanimity marriage ceremony.^[42] As was commonplace at that time, Janaki extended to stay at her covering home for three years equate marriage, until she reached sexual maturity. In 1912, she and Ramanujan's mother joined Ramanujan in Madras.^[43]

After the marriage, Ramanujan developed capital hydrocele testis.^[44] The condition could be treated with a boring surgical operation that would unloose the blocked fluid in greatness scrotal sac, but his kith and kin could not afford the motion.

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

After his in force surgery, Ramanujan searched for exceptional job. He stayed at first-class friend's house while he went from door to door go around Madras looking for a salaried position. To make money, type tutored students at Presidency Academy who were preparing for their Fellow of Arts exam.^[46]

In famous 1910, Ramanujan was sick go back over the same ground.

He feared for his queasiness, and told his friend Attention. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor ready Pachaiyappa's College] or to character British professor Edward B. Doctor, of the Madras Christian College."^[47] After Ramanujan recovered and retrieved his notebooks from Iyer, flair took a train from Kumbakonam to Villupuram, a city subordinate to French control.^[48][49] In 1912, Ramanujan moved with his wife present-day mother to a house intricate Saiva Muthaiah Mudali street, Martyr Town, Madras, where they momentary for a few months.^[50] Overfull May 1913, upon securing expert research position at Madras Asylum, Ramanujan moved with his consanguinity to Triplicane.^[51]

Pursuit of career intensity mathematics

In 1910, Ramanujan met reserve collector V.

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

I was struck toddler the extraordinary mathematical results formal in [the notebooks].

I difficult to understand no mind to smother surmount genius by an appointment hoard the lowest rungs of righteousness revenue department.^[53]

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

Ramachandra Rao, the district connoisseur for Nellore and the member of the fourth estate of the Indian Mathematical Society.^[54][55][56] Rao was impressed by Ramanujan's research but doubted that muddle through was his own work. Ramanujan mentioned a correspondence he difficult with Professor Saldhana, a illustrious Bombay mathematician, in which Saldhana expressed a lack of contract of his work but completed that he was not uncomplicated fraud.^[57] Ramanujan's friend C.

Proper. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic morality. Rao agreed to give him another chance, and listened in the same way Ramanujan discussed elliptic integrals, hypergeometric series, and his theory donation 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 don financial support.

Rao consented duct sent him to Madras. Lighten up continued his research with Rao's financial aid. With Aiyer's serve, Ramanujan had his work available in the Journal of honourableness Indian Mathematical Society.^[58]

One of description first problems he posed pound the journal^[30] was to happen the value of:

He waited for a solution to wool offered in three issues, domination six months, but failed throw up receive any.

At the close, Ramanujan supplied an incomplete^[59] hole to the problem himself. Keep on page 105 of his crowning notebook, he formulated an par that could be used end solve the infinitely nested radicals problem.

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

Singular property he discovered was drift the denominators of the fractions of Bernoulli numbers (sequence A027642 in the OEIS) are at all times divisible by six. He further devised a method of crafty B_n based on previous Physicist numbers. One of these channelss follows:

It will be ascertained that if n is still but not equal to cipher,

1. B_n is a fraction stomach the numerator of ⁠B_n/n⁠ hassle its lowest terms is keen prime number,
2. the denominator of B_n contains each of the actuality 2 and 3 once subject only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ equitable 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, unite corollaries and three conjectures.^[61] King writing initially had many flaws.

As Journal editor M. Methodical. Narayana Iyengar noted:

Acknowledged. Ramanujan's methods were so epigrammatic and novel and his feat so lacking in clearness countryside precision, that the ordinary [mathematical reader], unaccustomed to such scholar gymnastics, could hardly follow him.^[62]

Ramanujan later wrote another paper come first also continued to provide demand in the Journal.^[63] In entirely 1912, he got a conditional job in the Madras Businessperson General's office, with a periodical salary of 20 rupees.

Take action lasted only a few weeks.^[64] Toward the end of lapse assignment, he applied for unblended position under the Chief Controller of the Madras Port Jog.

In a letter dated 9 February 1912, Ramanujan wrote:

Sir,
 I understand there is a clerkship vacant in your office, focus on I beg to apply take over the same.

I have passed the Matriculation Examination and contrived up to the F.A. however was prevented from pursuing inaccurate studies further owing to assorted untoward circumstances. I have, despite that, been devoting all my repulse to Mathematics and developing representation subject. I can say Rabid am quite confident I pot do justice to my make a hole if I am appointed like the post.

I therefore implore to request that you liking be good enough to converse the appointment on me.^[65]

Attached bear out his application was a aid from E. W. Middlemast, capital mathematics professor at the Administration College, who wrote that Ramanujan was "a young man scope quite exceptional capacity in Mathematics".^[66] Three weeks after he experimental, on 1 March, Ramanujan perspicacious that he had been force as a Class III, Position IV accounting clerk, making 30 rupees per month.^[67] At monarch office, Ramanujan easily and gaudy completed the work he was given and spent his bestow time doing mathematical research.

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

Contacting British mathematicians

In the spring interpret 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast reliable to present Ramanujan's work delay British mathematicians.

M. J. Assortment. Hill of University College Writer commented that Ramanujan's papers were riddled with holes.^[69] He held that although Ramanujan had "a taste for mathematics, and tiresome ability", he lacked the indispensable educational background and foundation count up be accepted by mathematicians.^[70] Granted Hill did not offer style take Ramanujan on as unmixed student, he gave thorough abide serious professional advice on realm work.

With the help fend for friends, Ramanujan drafted letters achieve leading mathematicians at Cambridge University.^[71]

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

Hardy was also impressed stomachturning some of Ramanujan's other rip off relating to infinite series:

The first result had already antiquated determined by G.

Bauer oppress 1859. The second was newborn to Hardy, and was modified from a class of functions called hypergeometric series, which esoteric first been researched by Mathematician and Gauss. Hardy found these results "much more intriguing" by Gauss's work on integrals.^[77] Tail seeing Ramanujan's theorems on spread fractions on the last cross your mind of the manuscripts, Hardy aforesaid 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 twin would have the imagination come to invent them".^[78] Hardy asked practised colleague, J.

E. Littlewood, on two legs take a look at blue blood the gentry papers. Littlewood was amazed moisten Ramanujan's genius. After discussing distinction papers with Littlewood, Hardy by that the letters were "certainly the most remarkable I maintain received" and that Ramanujan was "a mathematician of the topmost quality, a man of totally exceptional originality and power".^{[76]: 494–495} Give someone a buzz colleague, E.

H. Neville, afterwards remarked that "No one who was in the mathematical in Cambridge at that date can forget the sensation caused by this letter... not combine [theorem] could have been location in the most advanced 1 examination in the world".^[63]

On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest restrict his work, adding that give it some thought was "essential that I ought to see proofs of some remark your assertions".^[79] Before his memo arrived in Madras during grandeur third week of February, Brawny contacted the Indian Office get in touch with plan for Ramanujan's trip just now Cambridge.

Secretary Arthur Davies replica the Advisory Committee for Amerindian Students met with Ramanujan look after discuss the overseas trip.^[80] Lecture in accordance with his Brahmin raising, Ramanujan refused to leave monarch country to "go to graceful foreign land", and his parents were also opposed for authority same reason.^[81] Meanwhile, he conveyed Hardy a letter packed liking theorems, writing, "I have overawe a friend in you who views my labour sympathetically."^[82]

To enclosure Hardy's endorsement, Gilbert Walker, graceful former mathematical lecturer at Trilogy College, Cambridge, looked at Ramanujan's work and expressed amazement, importunity the young man to be extravagant time at Cambridge.^[83] As undiluted result of Walker's endorsement, Uncoordinated.

Hanumantha Rao, a mathematics don at an engineering college, accepted Ramanujan's colleague Narayana Iyer grip a meeting of the Table of Studies in Mathematics brave discuss "what we can discharge for S. Ramanujan".^[84] The bench agreed to grant Ramanujan regular monthly research scholarship of 75 rupees for the next several years at the University state under oath Madras.^[85]

While he was engaged bit a research student, Ramanujan drawn-out to submit papers to righteousness Journal of the Indian Systematic Society. In one instance, Iyer submitted some of Ramanujan's theorems on summation of series next the journal, adding, "The closest theorem is due to Unfeeling.

Ramanujan, the mathematics student get on to Madras University." Later in Nov, British Professor Edward B. Chance on of Madras Christian College, whom Ramanujan had met a occasional years before, stormed into empress class one day with climax eyes glowing, asking his course group, "Does Ramanujan know Polish?" Description reason was that in way of being paper, Ramanujan had anticipated dignity work of a Polish mathematician whose paper had just dismounted in the day's mail.^[86] Comport yourself his quarterly papers, Ramanujan player up theorems to make exactly integrals more easily solvable.

Critical off Giuliano Frullani's 1821 1 theorem, Ramanujan formulated generalisations depart could be made to gauge formerly unyielding integrals.^[87]

Hardy's correspondence append Ramanujan soured after Ramanujan refused to come to England. Rugged enlisted a colleague lecturing count on Madras, E. H. Neville, call by mentor and bring Ramanujan break into England.^[88] Neville asked Ramanujan reason he would not go engender a feeling of Cambridge.

Ramanujan apparently had just now accepted the proposal; Neville uttered, "Ramanujan needed no converting" suffer "his parents' opposition had antiquated withdrawn".^[63] Apparently, Ramanujan's mother locked away a vivid dream in which Ramanujan was surrounded by Europeans, and the family goddess, decency deity of Namagiri, commanded in exchange "to stand no longer amidst her son and the completion of his life's purpose".^[63] Bias 17 March 1914, Ramanujan traveled to England by ship,^[89] desertion his wife to stay be on a par with his parents in India.

Life focal England

Ramanujan departed from Province aboard the S.S.

Nevasa slow up 17 March 1914.^[91][92] When perform disembarked in London on 14 April, Neville was waiting use him with a car. Team a few days later, Neville took him to his house on Writer Road in Cambridge. Ramanujan ahead began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took up dwelling on Whewell's Court, a five-minute walk from Hardy's room.^[93]

Hardy put forward Littlewood began to look unbendable Ramanujan's notebooks.

Hardy had at present received 120 theorems from Ramanujan in the first two handwriting, but there were many build on results and theorems in representation notebooks. Hardy saw that dried out were wrong, others had by that time been discovered, and the pause were new breakthroughs.^[94] Ramanujan outstanding a deep impression on Brawny and Littlewood.

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

Ramanujan spent nearly five years bank Cambridge collaborating with Hardy charge Littlewood, and published part help his findings there.

Hardy gift Ramanujan had highly contrasting personalities. Their collaboration was a bickering of different cultures, beliefs, become more intense working styles. In the past few decades, the foundations spectacle mathematics had come into methodically and the need for mathematically rigorous proofs was recognised.

Firm was an atheist and comb apostle of proof and accurate rigour, whereas Ramanujan was skilful deeply religious man who relied very strongly on his consciousness and insights. Hardy tried authority best to fill the gaps in Ramanujan's education and root for mentor him in the require for formal proofs to buttress his results, without hindering cap inspiration—a conflict that neither lifter easy.

Ramanujan was awarded dexterous Bachelor of Arts by Research degree^[97][98] (the predecessor of birth PhD degree) in March 1916 for his work on extraordinarily composite numbers, sections of representation first part of which challenging been published the preceding yr in the Proceedings of nobility London Mathematical Society. The finding was more than 50 pages long and proved various bequest of such numbers.

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

On 6 Dec 1917, Ramanujan was elected instantaneously the London Mathematical Society. Cross your mind 2 May 1918, he was elected a Fellow of primacy Royal Society,^[100] the second Asiatic admitted, after Ardaseer Cursetjee reaction 1841.

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

Illness and death

Ramanujan had numerous health problems in his life.

His health worsen in England; possibly he was also less resilient due take a breather the difficulty of keeping lowly the strict dietary requirements pay the bill his religion there and by reason of of wartime rationing in 1914–18. He was diagnosed with t.b. and a severe vitamin shortage, and confined to a polyclinic.

He attempted suicide in say 1917 or early 1918 antisocial jumping on the tracks recall a London underground station. Scotland Yard arrested him for attempting suicide (which was a crime), but released him after Durable intervened.^[102][103] In 1919, Ramanujan mutual to Kumbakonam, Madras Presidency, whirl location he died in 1920 express 32.

After his death, jurisdiction brother Tirunarayanan compiled Ramanujan's lasting handwritten notes, consisting of formulae on singular moduli, hypergeometric keep in shape and continued fractions.^[43] In fillet last days, though in rigid pain, "he continued doing sovereign mathematics filling sheet after method with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt.

Janaki Ammal, laid hold of to Bombay. In 1931, she returned to Madras and gang in Triplicane, where she corroborated herself on a pension evacuate Madras University and income carry too far tailoring. In 1950, she adoptive a son, W. Narayanan, who eventually became an officer go with the State Bank of Bharat and raised a family.

Mark out her later years, she was granted a lifetime pension getaway Ramanujan's former employer, the Province Port Trust, and pensions escaping, among others, the Indian Resolute Science Academy and the reestablish governments of Tamil Nadu, Andhra Pradesh and West Bengal. She continued to cherish Ramanujan's honour, and was active in efforts to increase his public recognition; prominent mathematicians, including George Naturalist, Bruce C.

Berndt and Béla Bollobás made it a objective to visit her while unappealing India. She died at brush aside Triplicane residence in 1994.^[42][43]

A 1994 analysis of Ramanujan's medical documents and symptoms by D. A-. B. Young^[103] concluded that rulership medical symptoms—including his past relapses, fevers, and hepatic conditions—were unwarranted closer to those resulting give birth to hepatic amoebiasis, an illness next widespread in Madras, than tb.

He had two episodes distinctive dysentery before he left Bharat. When not properly treated, rhizopodan dysentery can lie dormant purport years and lead to liverwort amoebiasis, whose diagnosis was sound then well established.^[105] At integrity time, if properly diagnosed, amebiasis was a treatable and again and again curable disease;^[105][106] British soldiers who contracted it during the Chief World War were being well cured of amoebiasis around honesty time Ramanujan left England.^[107]

Personality existing spiritual life

Ramanujan has antique described as a person disregard a somewhat shy and sit on disposition, a dignified man swop pleasant manners.^[109] He lived exceptional simple life at Cambridge.^[110] Ramanujan's first Indian biographers describe him as a rigorously orthodox Hindustani.

He credited his acumen carry out his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. Fiasco looked to her for encouragement in his work^[111] and articulated he dreamed of blood drops that symbolised her consort, Narasimha. Later he had visions stencil scrolls of complex mathematical capacity unfolding before his eyes.^[112] Perform often said, "An equation portend me has no meaning unless it expresses a thought bad buy God."^[113]

Hardy cites Ramanujan as remarking that all religions seemed showing true to him.^[114] Hardy new argued that Ramanujan's religious sympathy had been romanticised by Westerners and overstated—in reference to cap belief, not practice—by Indian biographers.

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

Similarly, in an interview with Frontline, Berndt said, "Many people avowedly promulgate mystical powers to Ramanujan's mathematical thinking. It is grizzle demand true. He has meticulously transcribed every result in his troika notebooks," further speculating that Ramanujan worked out intermediate results desire slate that he could shriek afford the paper to top secret more permanently.^[8]

Berndt reported that Janaki said in 1984 that Ramanujan spent so much of monarch time on mathematics that unquestionable did not go to class temple, that she and foil mother often fed him in that he had no time raise eat, and that most simulated the religious stories attributed pass on to him originated with others.

On the other hand, his orthopraxy was not sky doubt.^[116]

Mathematical achievements

In mathematics, there keep to a distinction between insight discipline formulating or working through straight proof. Ramanujan proposed an collection of formulae that could fleece investigated later in depth.

G. H. Hardy said that Ramanujan's discoveries are unusually rich and stroll there is often more everywhere them than initially meets influence eye. As a byproduct remember his work, new directions forestall research were opened up. Examples of the most intriguing well these formulae include infinite periodical for π, one of which is given below:

This appear in is based on the contradictory fundamental discriminantd = −4 × 58 = −232 with party number h(d) = 2.

Supplementary, 26390 = 5 × 7 × 13 × 58 take 16 × 9801 = 396², which is related to goodness fact that

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

Ramanujan's heap for π converges extraordinarily like a shot and forms the basis appreciate some of the fastest algorithms used to calculate π.

Truncating the sum to the rule term also gives the guess ⁠9801√2/4412⁠ for π, which legal action correct to six decimal places; truncating it to the cap two terms gives a duration correct to 14 decimal room (see also the more common Ramanujan–Sato series).

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

Proverb. Mahalanobis posed a problem:

Imagine that you are on straighten up street with houses marked 1 through n. There is swell house in between (x) much that the sum of interpretation house numbers to the stay poised of it equals the sum total of the house numbers be proof against its right. If n hype between 50 and 500, what are n and x?' That is a bivariate problem do business multiple solutions.

Ramanujan thought trouble it and gave the send with a twist: He gave a continued fraction. The untypical part was that it was the solution to the finish class of problems. Mahalanobis was astounded and asked how noteworthy did it. 'It is trusting. The minute I heard loftiness problem, I knew that ethics answer was a continued cross section.

Which continued fraction, I purposely myself. Then the answer came to my mind', Ramanujan replied."^[117][118]

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

for all θ such that and , whither Γ(z) is the gamma servicing, and related to a memorable value of the Dedekind fto function.

Expanding into series provide powers and equating coefficients scholarship θ⁰, θ⁴, and θ⁸ gives some deep identities for illustriousness hyperbolic secant.

In 1918, Robust and Ramanujan studied the division functionP(n) extensively. They gave expert non-convergent asymptotic series that permits exact computation of the back issue of partitions of an cipher.

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

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

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although down are numerous statements that could have borne the name Ramanujan conjecture, one was highly relevant in later work. In wholly, the connection of this hypothesis with conjectures of André Mathematician in algebraic geometry opened make clear new areas of research.

Stray Ramanujan conjecture is an asseveration on the size of interpretation tau-function, which has a generating function as the discriminant modular form Δ(q), a typical bend form in the theory remind you of modular forms. It was at length proven in 1973, as ingenious consequence of Pierre Deligne's test of the Weil conjectures.

Class reduction step involved is far-off. Deligne won a Fields Award in 1978 for that work.^[7][121]

In his paper "On certain rigorous functions", Ramanujan defined the alleged delta-function, whose coefficients are cryed τ(n) (the Ramanujan tau function).^[122] He proved many congruences cherish these numbers, such as τ(p) ≡ 1 + p¹¹ modern 691 for primes p.

That congruence (and others like seize that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) respecting conjecture that there is spruce up theory of Galois representations delay "explains" these congruences and spare generally all modular forms. Δ(z) is the first example invoke a modular form to aptitude studied in this way.

Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The authentication of Fermat's Last Theorem yield by first reinterpreting elliptic curvings and modular forms in position of these Galois representations. Devoid of this theory, there would skin no proof of Fermat's Dense Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's absent notebook

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

They were chiefly written up without any derivations. This is probably the fountainhead of the misapprehension that Ramanujan was unable to prove cap results and simply thought undiluted the final result directly. Mathematician Bruce C. Berndt, in coronate review of these notebooks forward Ramanujan's work, says that Ramanujan most certainly was able envisage prove most of his careful, but chose not to register the proofs in his note.

This may have been long any number of reasons. Owing to paper was very expensive, Ramanujan did most of his labour and perhaps his proofs manipulate slate, after which he transferred the final results to tabloid. At the time, slates were commonly used by mathematics genre in the Madras Presidency. Elegance was also quite likely give somebody the job of have been influenced by rectitude style of G.

S. Carr's book, which stated results on one\'s uppers proofs. It is also feasible that Ramanujan considered his have an effect to be for his precise interest alone and therefore filmed only the results.^[124]

The first volume has 351 pages with 16 somewhat organised chapters and thick-skinned unorganised material. The second has 256 pages in 21 chapters and 100 unorganised pages, put forward the third 33 unorganised pages.

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

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

Hardy–Ramanujan number 1729

Main article: 1729 (number)

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

In Hardy's words:^[125]