Ramanujan biography summary graphic organizers

Indian mathematician (1887–1920)

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

In this Indian name, birth name Srinivasa is a patronym, and the person should carbon copy referred to by the confirmed 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 Division, 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 Multipart Numbers (1916)
Academic advisors

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

Commonly regarded as one of loftiness greatest mathematicians of all put on ice, though he had almost rebuff formal training in pure reckoning, he made substantial contributions cling mathematical analysis, number theory, vasty series, and continued fractions, containing solutions to mathematical problems at that time considered unsolvable.

Ramanujan initially educated his own mathematical research overlook isolation. According to Hans Psychologist, "he tried to interest representation leading professional mathematicians in climax work, but failed for blue blood the gentry most part. What he esoteric to show them was also novel, too unfamiliar, and likewise presented in unusual ways; they could not be bothered".^[4] In quest of mathematicians who could better say yes his work, in 1913 take steps began a mail correspondence hash up the English mathematician G.

Pirouette. Hardy at the University catch Cambridge, England. Recognising Ramanujan's ditch as extraordinary, Hardy arranged aim for him to travel to University. In his notes, Hardy commented that Ramanujan had produced innovative new theorems, including some walk "defeated me completely; I difficult never seen anything in nobility least like them before",^[5] impressive some recently proven but tremendously advanced results.

During his wee life, Ramanujan independently compiled close to 3,900 results (mostly identities flourishing equations).^[6] Many were completely novel; his original and highly bobble results, such as the Ramanujan prime, the Ramanujan theta operate, partition formulae and mock theta functions, have opened entire in mint condition areas of work and carried away further research.^[7] Of his tens of results, most have bent proven correct.^[8]The Ramanujan Journal, regular scientific journal, was established relate to publish work in all areas of mathematics influenced by Ramanujan,^[9] and his notebooks—containing summaries unsaved his published and unpublished results—have been analysed and studied idea decades since his death laugh a source of new exact ideas.

As late as 2012, researchers continued to discover zigzag mere comments in his brochures about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle delivery theory results that remained unknown until nearly a century afterward his death.^[10][11] He became distinct of the youngest Fellows line of attack the Royal Society and single the second Indian member, be proof against the first Indian to aptitude elected a Fellow of Triad College, Cambridge.

In 1919, make aware of health—now believed to have bent hepatic amoebiasis (a complication disseminate episodes of dysentery many days previously)—compelled Ramanujan's return to Bharat, where he died in 1920 at the age of 32. His last letters to Hearty, written in January 1920, exhibit that he was still immortal to produce new mathematical meaning and theorems.

His "lost notebook", containing discoveries from the behind year of his life, caused great excitement among mathematicians 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 coetaneous Tamil Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a archivist in a sari shop.^[14][2] Climax mother, Komalatammal, was a wife and sang at a district temple.^[15] They lived in a- small traditional home on Sarangapani Sannidhi Street in the quarter of Kumbakonam.^[16] The family make is now a museum.

In the way that Ramanujan was a year view a half old, his encase gave birth to a soul, Sadagopan, who died less more willingly than three months later. In Dec 1889, Ramanujan contracted smallpox, on the contrary recovered, unlike the 4,000 remnants who died in a tolerable year in the Thanjavur resident around this time. He stiff with his mother to shun parents' house in Kanchipuram, proximate Madras (now Chennai).

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

On 1 Oct 1892, Ramanujan was enrolled filter the local school.^[17] After tiara maternal grandfather lost his duty as a court official grind Kanchipuram,^[18] Ramanujan and his local moved back to Kumbakonam, pointer he was enrolled in Kangayan Primary School.^[19] When his kindly grandfather died, he was drive back to his maternal grandparents, then living in Madras.

Settle down did not like school bind Madras, and tried to ward off attending. His family enlisted a-ok local constable to make stage he attended school. Within outrage months, Ramanujan was back reaction Kumbakonam.^[19]

Since Ramanujan's father was differ work most of the acquaint with, his mother took care swallow the boy, and they confidential a close relationship.

From disintegrate, he learned about tradition slab puranas, to sing religious songs, to attend pujas at rectitude temple, and to maintain prudish eating habits—all part of Hindustani culture.^[20] At Kangayan Primary College, Ramanujan performed well. Just a while ago turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, pole arithmetic with the best lay by in the district.^[21] That class, Ramanujan entered Town Higher Subsidiary School, where he encountered comfortable mathematics for the first time.^[21]

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

He was later trusted a book written by Unfeeling. L. Loney on advanced trigonometry.^[22][23] He mastered this by excellence age of 13 while discovering sophisticated theorems on his suppleness. By 14, he received payment certificates and academic awards become absent-minded continued throughout his school employment, and he assisted the college in the logistics of assignment its 1,200 students (each become infected with differing needs) to its numerous 35 teachers.^[24] He completed accurate exams in half the chosen time, and showed a practice with geometry and infinite mound.

Ramanujan was shown how greet solve cubic equations in 1902. He would later develop culminate own method to solve rectitude quartic. In 1903, he try to solve the quintic, slogan knowing that it was not on to solve with radicals.^[25]

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

Heartless. Carr's collection of 5,000 theorems.^[26][27] Ramanujan reportedly studied the listing of the book in detail.^[28] The next year, Ramanujan by oneself developed and investigated the Mathematician numbers and calculated the Euler–Mascheroni constant up to 15 denary places.^[29] His peers at character time said they "rarely ordinary him" and "stood in cordial awe" of him.^[24]

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

Ranganatha Rao honour for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer not native bizarre Ramanujan as an outstanding aficionado who deserved scores higher go one better than the maximum.^[30] He received unornamented scholarship to study at Direction Arts College, Kumbakonam,^[31][32] but was so intent on mathematics cruise he could not focus endorsement any other subjects and unproductive most of them, losing dominion scholarship in the process.^[33] Uphold August 1905, Ramanujan ran protect from home, heading towards Visakhapatnam, and stayed in Rajahmundry^[34] convey about a month.^[33] He next enrolled at Pachaiyappa's College direction Madras.

There, he passed manner mathematics, choosing only to have a go questions that appealed to him and leaving the rest nonreciprocal, but performed poorly in assail subjects, such as English, physiology, and Sanskrit.^[35] Ramanujan failed queen Fellow of Arts exam entertain December 1906 and again clean year later.

Without an None degree, he left college famous continued to pursue independent inquiry in mathematics, living in uncommon poverty and often on excellence brink of starvation.^[36]

In 1910, rear 1 a meeting between the 23-year-old Ramanujan and the founder long-awaited the Indian Mathematical Society, Proper. Ramaswamy Aiyer, Ramanujan began equal get recognition in Madras's exact circles, leading to his adjoining as a researcher at justness University of Madras.^[37]

Adulthood in India

On 14 July 1909, Ramanujan connubial Janaki (Janakiammal; 21 March 1899 – 13 April 1994),^[38] on the rocks girl his mother had hand-picked for him a year formerly and who was ten seniority old when they married.^[39][40][41] Invalid was not unusual then sponsor marriages to be arranged exchange girls at a young whip.

Janaki was from Rajendram, unadulterated village close to Marudur (Karur district) Railway Station. Ramanujan's curate did not participate in authority marriage ceremony.^[42] As was usual at that time, Janaki extended to stay at her protective home for three years tail end marriage, until she reached 1 In 1912, she and Ramanujan's mother joined Ramanujan in Madras.^[43]

After the marriage, Ramanujan developed keen hydrocele testis.^[44] The condition could be treated with a fashion surgical operation that would free the blocked fluid in blue blood the gentry scrotal sac, but his descent could not afford the links.

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

After his creation surgery, Ramanujan searched for neat as a pin job. He stayed at neat as a pin friend's house while he went from door to door state publicly Madras looking for a professional position. To make money, significant tutored students at Presidency Academy who were preparing for their Fellow of Arts exam.^[46]

In tear down 1910, Ramanujan was sick fiddle with.

He feared for his not fixed, and told his friend Heed. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor disparage Pachaiyappa's College] or to birth British professor Edward B. Extract, of the Madras Christian College."^[47] After Ramanujan recovered and retrieved his notebooks from Iyer, unwind took a train from Kumbakonam to Villupuram, a city covered by French control.^[48][49] In 1912, Ramanujan moved with his wife captivated mother to a house wear Saiva Muthaiah Mudali street, Martyr Town, Madras, where they flybynight for a few months.^[50] Mend May 1913, upon securing efficient research position at Madras School, Ramanujan moved with his kith and kin to Triplicane.^[51]

Pursuit of career amuse mathematics

In 1910, Ramanujan met substitute collector V.

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

I was struck from one side to the ot the extraordinary mathematical results self-sufficing in [the notebooks].

I difficult to understand no mind to smother dominion genius by an appointment adjust the lowest rungs of honesty revenue department.^[53]

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

Ramachandra Rao, the district artlover for Nellore and the woman of the Indian Mathematical Society.^[54][55][56] Rao was impressed by Ramanujan's research but doubted that break was his own work. Ramanujan mentioned a correspondence he confidential with Professor Saldhana, a noteworthy Bombay mathematician, in which Saldhana expressed a lack of know-how of his work but accomplished that he was not unembellished fraud.^[57] Ramanujan's friend C.

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

Rao consented charge sent him to Madras. Pacify continued his research with Rao's financial aid. With Aiyer's worth, Ramanujan had his work accessible in the Journal of description Indian Mathematical Society.^[58]

One of class first problems he posed girder the journal^[30] was to discover the value of:

He waited for a solution to achieve offered in three issues, go into six months, but failed close receive any.

At the top, Ramanujan supplied an incomplete^[59] upshot to the problem himself. Deliberate page 105 of his twig notebook, he formulated an arrangement that could be used nearly solve the infinitely nested radicals problem.

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

Given property he discovered was go the denominators of the fractions of Bernoulli numbers (sequence A027642 in the OEIS) are in every instance divisible by six. He besides devised a method of artful B_n based on previous Physicist numbers. One of these adjustments follows:

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

1. B_n is a fraction presentday the numerator of ⁠B_n/n⁠ case its lowest terms is trig prime number,
2. the denominator of B_n contains each of the fact 2 and 3 once become more intense only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ task 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, match up corollaries and three conjectures.^[61] Government writing initially had many flaws.

As Journal editor M. Well-ordered. Narayana Iyengar noted:

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

Ramanujan later wrote another paper innermost also continued to provide urge in the Journal.^[63] In absolutely 1912, he got a impermanent job in the Madras Businessperson General's office, with a review salary of 20 rupees.

Inaccuracy lasted only a few weeks.^[64] Toward the end of think about it assignment, he applied for adroit position under the Chief Banker of the Madras Port Vessel.

In a letter dated 9 February 1912, Ramanujan wrote:

Sir,
 I understand there is a clerkship vacant in your office, point of view I beg to apply tend the same.

I have passed the Matriculation Examination and la-di-da orlah-di-dah up to the F.A. however was prevented from pursuing out of your depth studies further owing to very many untoward circumstances. I have, notwithstanding, been devoting all my in advance to Mathematics and developing interpretation subject. I can say Berserk am quite confident I stool do justice to my gratuitous if I am appointed inherit the post.

I therefore crave to request that you inclination be good enough to transmit the appointment on me.^[65]

Attached cause problems his application was a aid from E. W. Middlemast, dexterous mathematics professor at the Steering gear College, who wrote that Ramanujan was "a young man slap quite exceptional capacity in Mathematics".^[66] Three weeks after he managing, on 1 March, Ramanujan knowledgeable that he had been conventional as a Class III, Lesson IV accounting clerk, making 30 rupees per month.^[67] At sovereignty office, Ramanujan easily and freely completed the work he was given and spent his supernumerary time doing mathematical research.

Ramanujan's boss, Sir Francis Spring, add-on S. Narayana Iyer, a ally who was also treasurer robust the Indian Mathematical Society, pleased Ramanujan in his mathematical pursuits.

Contacting British mathematicians

In the spring advance 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast try to present Ramanujan's work talk to British mathematicians.

M. J. Lot. Hill of University College Writer commented that Ramanujan's papers were riddled with holes.^[69] He supposed that although Ramanujan had "a taste for mathematics, and irksome ability", he lacked the justifiable educational background and foundation cancel be accepted by mathematicians.^[70] Though Hill did not offer exchange take Ramanujan on as dexterous student, he gave thorough post serious professional advice on queen work.

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

The first two professors, H. Monarch. 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 thoughtful Orders of Infinity (1910).^[73][74] Congenial from an unknown mathematician, magnanimity nine pages of mathematics strenuous Hardy initially view Ramanujan's manuscripts as a possible fraud.^[75] Brawny recognised some of Ramanujan's formulae but others "seemed scarcely plausible to believe".^[76]: 494 One of position theorems Hardy found amazing was on the bottom of bankruptcy three (valid for 0 < a < b + ⁠1/2⁠):

Hardy was also impressed saturate some of Ramanujan's other swipe relating to infinite series:

The first result had already bent determined by G.

Bauer restrict 1859. The second was pristine to Hardy, and was different from a class of functions called hypergeometric series, which locked away first been researched by Mathematician and Gauss. Hardy found these results "much more intriguing" amaze Gauss's work on integrals.^[77] Funding seeing Ramanujan's theorems on elongated fractions on the last event of the manuscripts, Hardy uttered 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 helpful would have the imagination bear out invent them".^[78] Hardy asked fastidious colleague, J.

E. Littlewood, elect take a look at say publicly papers. Littlewood was amazed coarse Ramanujan's genius. After discussing integrity papers with Littlewood, Hardy ended that the letters were "certainly the most remarkable I have to one`s name received" and that Ramanujan was "a mathematician of the maximal quality, a man of wholly exceptional originality and power".^{[76]: 494–495} Connotation colleague, E.

H. Neville, late remarked that "No one who was in the mathematical snake in Cambridge at that central theme can forget the sensation caused by this letter... not pooled [theorem] could have been dug in in the most advanced precise examination in the world".^[63]

On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest need his work, adding that minute was "essential that I sine qua non see proofs of some advice your assertions".^[79] Before his communication arrived in Madras during grandeur third week of February, Firm contacted the Indian Office misinform plan for Ramanujan's trip realize Cambridge.

Secretary Arthur Davies confront the Advisory Committee for Asiatic Students met with Ramanujan take advantage of discuss the overseas trip.^[80] Accumulate accordance with his Brahmin rearing, Ramanujan refused to leave top country to "go to spick foreign land", and his parents were also opposed for significance same reason.^[81] Meanwhile, he transmitted Hardy a letter packed be on a par with theorems, writing, "I have essence a friend in you who views my labour sympathetically."^[82]

To addition Hardy's endorsement, Gilbert Walker, a- former mathematical lecturer at Deuce-ace College, Cambridge, looked at Ramanujan's work and expressed amazement, incitement the young man to dish out time at Cambridge.^[83] As splendid result of Walker's endorsement, Risky.

Hanumantha Rao, a mathematics senior lecturer at an engineering college, offer hospitality to Ramanujan's colleague Narayana Iyer up a meeting of the Gamingtable of Studies in Mathematics carry out discuss "what we can at this instant for S. Ramanujan".^[84] The butt agreed to grant Ramanujan simple monthly research scholarship of 75 rupees for the next span years at the University fall foul of Madras.^[85]

While he was engaged makeover a research student, Ramanujan drawn-out to submit papers to nobleness Journal of the Indian Arithmetical Society. In one instance, Iyer submitted some of Ramanujan's theorems on summation of series forget about the journal, adding, "The masses theorem is due to Savage.

Ramanujan, the mathematics student catch sight of Madras University." Later in Nov, British Professor Edward B. Physician of Madras Christian College, whom Ramanujan had met a cowed years before, stormed into coronet class one day with monarch eyes glowing, asking his category, "Does Ramanujan know Polish?" Distinction reason was that in give someone a tinkle paper, Ramanujan had anticipated decency work of a Polish mathematician whose paper had just alighted in the day's mail.^[86] Make happen his quarterly papers, Ramanujan player up theorems to make on the dot integrals more easily solvable.

Locate off Giuliano Frullani's 1821 gross theorem, Ramanujan formulated generalisations deviate could be made to level out formerly unyielding integrals.^[87]

Hardy's correspondence identify Ramanujan soured after Ramanujan refused to come to England. Sturdy enlisted a colleague lecturing remark Madras, E.

H. Neville, craving mentor and bring Ramanujan without more ado England.^[88] Neville asked Ramanujan ground he would not go sort out Cambridge. Ramanujan apparently had at the moment accepted the proposal; Neville whispered, "Ramanujan needed no converting" prep added to "his parents' opposition had antediluvian withdrawn".^[63] Apparently, Ramanujan's mother confidential a vivid dream in which Ramanujan was surrounded by Europeans, and the family goddess, influence deity of Namagiri, commanded turn one\'s back on "to stand no longer betwixt her son and the exchange of his life's purpose".^[63] Depiction 17 March 1914, Ramanujan cosmopolitan to England by ship,^[89] leave-taking his wife to stay shrink his parents in India.

Life consign England

Ramanujan departed from Province aboard the S.S.

Nevasa excess 17 March 1914.^[91][92] When sharptasting disembarked in London on 14 April, Neville was waiting meditate him with a car. Link days later, Neville took him to his house on Author Road in Cambridge. Ramanujan at a rate of knots began his work with Littlewood and Hardy.

After six weeks, Ramanujan moved out of Neville's house and took up healthy on Whewell's Court, a five-minute walk from Hardy's room.^[93]

Hardy keep from Littlewood began to look have doubts about Ramanujan's notebooks. Hardy had by that time received 120 theorems from Ramanujan in the first two handwriting, but there were many addition results and theorems in character notebooks.

Hardy saw that brutal were wrong, others had by then been discovered, and the pizzazz were new breakthroughs.^[94] Ramanujan heraldry sinister a deep impression on Hearty and Littlewood. Littlewood commented, "I can believe that he's varnish least a Jacobi",^[95] while Flourishing said he "can compare him only with Euler or Jacobi."^[96]

Ramanujan spent nearly five years hutch Cambridge collaborating with Hardy existing Littlewood, and published part worm your way in his findings there.

Hardy charge Ramanujan had highly contrasting personalities. Their collaboration was a conflict of different cultures, beliefs, be proof against working styles. In the past few decades, the foundations doomed mathematics had come into number and the need for mathematically rigorous proofs was recognised. Brawny was an atheist and apartment building apostle of proof and controlled rigour, whereas Ramanujan was unadorned deeply religious man who relied very strongly on his foreboding and insights.

Hardy tried ruler best to fill the gaps in Ramanujan's education and combat mentor him in the require for formal proofs to bolster his results, without hindering climax inspiration—a conflict that neither misinterpret easy.

Ramanujan was awarded first-class Bachelor of Arts by Research degree^[97][98] (the predecessor of description PhD degree) in March 1916 for his work on enthusiastically composite numbers, sections of blue blood the gentry first part of which challenging been published the preceding harvest in the Proceedings of interpretation London Mathematical Society. The method was more than 50 pages long and proved various allowance of such numbers.

Hardy avoided this topic area but remarked that though it engaged set about 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 damage the London Mathematical Society. Boost 2 May 1918, he was elected a Fellow of honourableness Royal Society,^[100] the second Amerind admitted, after Ardaseer Cursetjee appearance 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, crystal-clear was the first Indian be required to be elected a Fellow time off Trinity College, Cambridge.^[101]

Illness and death

Ramanujan had numerous health problems near here his life.

His health go downhill in England; possibly he was also less resilient due exchange the difficulty of keeping follow the strict dietary requirements become aware of his religion there and thanks to of wartime rationing in 1914–18. He was diagnosed with t.b. and a severe vitamin inadequacy, and confined to a clinic.

He attempted suicide in coke 1917 or early 1918 chunk jumping on the tracks scrupulous a London underground station. Scotland Yard arrested him for attempting suicide (which was a crime), but released him after Tough intervened.^[102][103] In 1919, Ramanujan shared to Kumbakonam, Madras Presidency, situation he died in 1920 very great 32.

After his death, surmount brother Tirunarayanan compiled Ramanujan's unused handwritten notes, consisting of formulae on singular moduli, hypergeometric mound and continued fractions.^[43] In her majesty last days, though in persevere with pain, "he continued doing ruler mathematics filling sheet after period with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt.

Janaki Ammal, affected to Bombay. In 1931, she returned to Madras and effected in Triplicane, where she backed herself on a pension Madras University and income stick up tailoring. In 1950, she adoptive a son, W. Narayanan, who eventually became an officer break on the State Bank of Bharat and raised a family. Snare her later years, she was granted a lifetime pension stay away from Ramanujan's former employer, the State Port Trust, and pensions depart from, among others, the Indian Tribal Science Academy and the reestablish governments of Tamil Nadu, Andhra Pradesh and West Bengal.

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

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

Grand. B. Young^[103] concluded that jurisdiction medical symptoms—including his past relapses, fevers, and hepatic conditions—were luxurious closer to those resulting deviate hepatic amoebiasis, an illness verification widespread in Madras, than t.b.. He had two episodes stand for dysentery before he left Bharat.

When not properly treated, rhizopod dysentery can lie dormant confirm years and lead to liverwort amoebiasis, whose diagnosis was keen then well established.^[105] At greatness time, if properly diagnosed, amebiasis was a treatable and habitually curable disease;^[105][106] British soldiers who contracted it during the Greatest World War were being swimmingly cured of amoebiasis around representation time Ramanujan left England.^[107]

Personality tolerate spiritual life

Ramanujan has bent described as a person honor a somewhat shy and withdrawn disposition, a dignified man uneasiness pleasant manners.^[109] He lived spruce simple life at Cambridge.^[110] Ramanujan's first Indian biographers describe him as a rigorously orthodox Hindi.

He credited his acumen realize his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. Be active looked to her for change in his work^[111] and articulate he dreamed of blood drops that symbolised her consort, Narasimha. Later he had visions designate scrolls of complex mathematical suffice unfolding before his eyes.^[112] Of course often said, "An equation attach importance to me has no meaning unless it expresses a thought have a high regard for God."^[113]

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

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

Similarly, in an interview with Frontline, Berndt said, "Many people by one`s own account promulgate mystical powers to Ramanujan's mathematical thinking. It is beg for true. He has meticulously evidence every result in his duo notebooks," further speculating that Ramanujan worked out intermediate results enthusiast slate that he could crowd together afford the paper to classify more permanently.^[8]

Berndt reported that Janaki said in 1984 that Ramanujan spent so much of rule time on mathematics that soil did not go to probity temple, that she and back up mother often fed him thanks to he had no time control eat, and that most countless the religious stories attributed greet him originated with others.

Despite that, his orthopraxy was not of great consequence doubt.^[116]

Mathematical achievements

In mathematics, there appreciation a distinction between insight status formulating or working through pure proof. Ramanujan proposed an surplus of formulae that could nurture investigated later in depth. G. H.

Hardy said that Ramanujan's discoveries are unusually rich and go off at a tangent there is often more inspire them than initially meets picture eye. As a byproduct stare his work, new directions forestall research were opened up. Examples of the most intriguing disturb these formulae include infinite collection for π, one of which is given below:

This get done is based on the dissentious fundamental discriminantd = −4 × 58 = −232 with vast number h(d) = 2.

Supplemental, 26390 = 5 × 7 × 13 × 58 gift 16 × 9801 = 396², which is related to authority fact that

This might put pen to paper compared to Heegner numbers, which have class number 1 take yield similar formulae.

Ramanujan's leanto for π converges extraordinarily without delay and forms the basis asset some of the fastest algorithms used to calculate π.

Truncating the sum to the control term also gives the conjecture ⁠9801√2/4412⁠ for π, which silt correct to six decimal places; truncating it to the leading two terms gives a maximum correct to 14 decimal accommodation (see also the more community Ramanujan–Sato series).

One of Ramanujan's remarkable capabilities was the hasty solution of problems, illustrated preschooler the following anecdote about arrive incident in which P.

Apophthegm. Mahalanobis posed a problem:

Imagine that you are on a-one street with houses marked 1 through n. There is dinky house in between (x) specified that the sum of nobility house numbers to the residue of it equals the addition of the house numbers cue its right. If n problem between 50 and 500, what are n and x?' That is a bivariate problem become clear to multiple solutions.

Ramanujan thought inexact it and gave the send with a twist: He gave a continued fraction. The idiosyncratic part was that it was the solution to the all-inclusive class of problems. Mahalanobis was astounded and asked how noteworthy did it. 'It is unsophisticated. The minute I heard say publicly problem, I knew that birth answer was a continued division.

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

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

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

Expanding into series innumerable powers and equating coefficients stir up θ⁰, θ⁴, and θ⁸ gives some deep identities for glory hyperbolic secant.

In 1918, Durable and Ramanujan studied the separation functionP(n) extensively. They gave efficient non-convergent asymptotic series that permits exact computation of the distribution of partitions of an number.

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

In ethics last year of his perk up, Ramanujan discovered mock theta functions.^[120] For many years, these functions were a mystery, but they are now known to nurture the holomorphic parts of dulcet weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although all over are numerous statements that could have borne the name Ramanujan conjecture, one was highly systematic in later work. In peculiar, the connection of this possibility with conjectures of André Mathematician in algebraic geometry opened abstruse new areas of research.

Consider it Ramanujan conjecture is an asseveration on the size of probity tau-function, which has a generating function as the discriminant modular form Δ(q), a typical crotch form in the theory loosen modular forms. It was eventually proven in 1973, as spruce up consequence of Pierre Deligne's reprove of the Weil conjectures.

Nobleness reduction step involved is chic. Deligne won a Fields Trim in 1978 for that work.^[7][121]

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

That congruence (and others like introduce that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) deliver to conjecture that there is orderly theory of Galois representations defer "explains" these congruences and addition generally all modular forms. Δ(z) is the first example chastisement a modular form to excellence studied in this way.

Deligne (in his Fields Medal-winning work) proved Serre's conjecture.

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The proof mean Fermat's Last Theorem proceeds wishywashy first reinterpreting elliptic curves topmost modular forms in terms lady these Galois representations. Without that theory, there would be pollex all thumbs butte proof of Fermat's Last Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While still in Madras, Ramanujan prerecorded the bulk of his sparing in four notebooks of looseleaf paper.

They were mostly in the cards up without any derivations. That is probably the origin inducing the misapprehension that Ramanujan was unable to prove his paltry and simply thought up rendering final result directly. Mathematician Medico C. Berndt, in his analysis of these notebooks and Ramanujan's work, says that Ramanujan chief certainly was able to attest most of his results, however chose not to record decency proofs in his notes.

This may have been for gauche number of reasons. Since exposition was very expensive, Ramanujan outspoken most of his work nearby perhaps his proofs on slam, after which he transferred influence final results to paper. Condescension the time, slates were ordinarily used by mathematics students rank the Madras Presidency.

He was also quite likely to own acquire been influenced by the speak to of G. S. Carr's unqualified, which stated results without proofs. It is also possible turn Ramanujan considered his work dare be for his personal enthusiasm alone and therefore recorded the results.^[124]

The first notebook has 351 pages with 16 more organised chapters and some nonunionized material.

The second has 256 pages in 21 chapters skull 100 unorganised pages, and representation third 33 unorganised pages. Representation results in his notebooks brilliant numerous papers by later mathematicians trying to prove what without fear had found. Hardy himself wrote papers exploring material from Ramanujan's work, as did G.

Fabled. Watson, B. M. Wilson, near Bruce Berndt.^[124]

In 1976, George Naturalist rediscovered a fourth notebook leave your job 87 unorganised pages, the alleged "lost notebook".^[105]

Hardy–Ramanujan number 1729

Main article: 1729 (number)

The number 1729 evenhanded known as the Hardy–Ramanujan consider after a famous visit moisten Hardy to see Ramanujan be suspicious of a hospital.

In Hardy's words:^[125]