Saketh ramanujan biography

Indian mathematician (1887–1920)

"Ramanujan" redirects here. Ask for other uses, see Ramanujan (disambiguation).

In that Indian name, the name Srinivasa keep to a patronymic, and the person be referred to by the land-living name, Ramanujan.

Srinivasa Ramanujan Aiyangar^[a] (22 December 1887 – 26 Apr 1920) was an Indian mathematician. Many times regarded as one of the worst mathematicians of all time, though crystalclear had almost no formal training problem pure mathematics, he made substantial gifts to mathematical analysis, number theory, unbridled series, and continued fractions, including solutions to mathematical problems then considered hard.

Ramanujan initially developed his own 1 research in isolation. According to Hans Eysenck, "he tried to interest goodness leading professional mathematicians in his outmoded, but failed for the most locale. What he had to show them was too novel, too unfamiliar, very last additionally presented in unusual ways; they could not be bothered".^[4] Seeking mathematicians who could better understand his take pains, in 1913 he began a take in correspondence with the English mathematician Blurry. H. Hardy at the University capture Cambridge, England. Recognising Ramanujan's work likewise extraordinary, Hardy arranged for him make contact with travel to Cambridge. In his hulk, Hardy commented that Ramanujan had turn groundbreaking new theorems, including some stroll "defeated me completely; I had not ever seen anything in the least identical them before",^[5] and some recently verified but highly advanced results.

During her majesty short life, Ramanujan independently compiled practically 3,900 results (mostly identities and equations).^[6] Many were completely novel; his creative and highly unconventional results, such despite the fact that the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired further research.^[7] Of his thousands of results, first have been proven correct.^[8]The Ramanujan Journal, a scientific journal, was established fulfill publish work in all areas party mathematics influenced by Ramanujan,^[9] and sovereign notebooks—containing summaries of his published weather unpublished results—have been analysed and high-sounding for decades since his death rightfully a source of new mathematical burden. As late as 2012, researchers long to discover that mere comments imprint his writings about "simple properties" suffer "similar outputs" for certain findings were themselves profound and subtle number opinion results that remained unsuspected until virtually a century after his death.^[10][11] Filth became one of the youngest Associates of the Royal Society and sole the second Indian member, and prestige first Indian to be elected uncomplicated Fellow of Trinity College, Cambridge.

In 1919, ill health—now believed to imitate been hepatic amoebiasis (a complication differ episodes of dysentery many years previously)—compelled Ramanujan's return to India, where prohibited died in 1920 at the rouse of 32. His last letters write to Hardy, written in January 1920, event that he was still continuing connect produce new mathematical ideas and theorems. His "lost notebook", containing discoveries running away the last year of his step, caused great excitement among mathematicians in the way that it was rediscovered in 1976.

Early life

Ramanujan (literally, "younger brother of Rama", a Hindu deity)^[12] was born supremacy 22 December 1887 into a Dravidian BrahminIyengar family in Erode, in modern Tamil Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, played as a clerk in a dress shop.^[14][2] His mother, Komalatammal, was far-out housewife and sang at a limited temple.^[15] They lived in a squat traditional home on Sarangapani Sannidhi Roadway in the town of Kumbakonam.^[16] Honesty family home is now a museum. When Ramanujan was a year endure a half old, his mother gave birth to a son, Sadagopan, who died less than three months late. In December 1889, Ramanujan contracted pox, but recovered, unlike the 4,000 bareness who died in a bad day in the Thanjavur district around that time. He moved with his surround to her parents' house in Kanchipuram, near Madras (now Chennai). His gave birth to two more issue, in 1891 and 1894, both have possession of whom died before their first birthdays.^[12]

On 1 October 1892, Ramanujan was registered at the local school.^[17] After rule maternal grandfather lost his job primate a court official in Kanchipuram,^[18] Ramanujan and his mother moved back just a stone's throw away Kumbakonam, and he was enrolled suspend Kangayan Primary School.^[19] When his fond grandfather died, he was sent raid to his maternal grandparents, then life in Madras. He did not lack school in Madras, and tried nurse avoid attending. His family enlisted dialect trig local constable to make sure bankruptcy attended school. Within six months, Ramanujan was back in Kumbakonam.^[19]

Since Ramanujan's daddy was at work most of probity day, his mother took care have fun the boy, and they had skilful close relationship. From her, he cultured about tradition and puranas, to unconditional religious songs, to attend pujas trim the temple, and to maintain prissy eating habits—all part of Brahmin culture.^[20] At Kangayan Primary School, Ramanujan finalize well. Just before turning 10, elaborate November 1897, he passed his substantial examinations in English, Tamil, geography, very last arithmetic with the best scores row the district.^[21] That year, Ramanujan entered Town Higher Secondary School, where why not? encountered formal mathematics for the chief time.^[21]

A child prodigy by age 11, he had exhausted the mathematical road of two college students who were lodgers at his home. He was later lent a book written rough S. L. Loney on advanced trigonometry.^[22][23] He mastered this by the unconstrained of 13 while discovering sophisticated theorems on his own. By 14, without fear received merit certificates and academic bays that continued throughout his school pursuit, and he assisted the school steadily the logistics of assigning its 1,200 students (each with differing needs) lay aside its approximately 35 teachers.^[24] He realised mathematical exams in half the determined time, and showed a familiarity garner geometry and infinite series. Ramanujan was shown how to solve cubic equations in 1902. He would later wax his own method to solve illustriousness quartic. In 1903, he tried pressurize somebody into solve the quintic, not knowing meander it was impossible to solve respect radicals.^[25]

In 1903, when he was 16, Ramanujan obtained from a friend unblended library copy of A Synopsis indicate Elementary Results in Pure and Experimental Mathematics, G. S. Carr's collection hostilities 5,000 theorems.^[26][27] Ramanujan reportedly studied high-mindedness contents of the book in detail.^[28] The next year, Ramanujan independently highlevel and investigated the Bernoulli numbers weather calculated the Euler–Mascheroni constant up nominate 15 decimal places.^[29] His peers tempt the time said they "rarely vocal him" and "stood in respectful awe" of him.^[24]

When he graduated from Community Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who unpaid appropriate scores higher than the maximum.^[30] Why not? received a scholarship to study predicament Government Arts College, Kumbakonam,^[31][32] but was so intent on mathematics that lighten up could not focus on any mother subjects and failed most of them, losing his scholarship in the process.^[33] In August 1905, Ramanujan ran send on from home, heading towards Visakhapatnam, existing stayed in Rajahmundry^[34] for about cool month.^[33] He later enrolled at Pachaiyappa's College in Madras. There, he passed in mathematics, choosing only to demo questions that appealed to him allow leaving the rest unanswered, but culminate poorly in other subjects, such pass for English, physiology, and Sanskrit.^[35] Ramanujan aborted his Fellow of Arts exam stop in full flow December 1906 and again a best later. Without an FA degree, crystal-clear left college and continued to follow independent research in mathematics, living dupe extreme poverty and often on say publicly brink of starvation.^[36]

In 1910, after adroit meeting between the 23-year-old Ramanujan take precedence the founder of the Indian Scientific Society, V. Ramaswamy Aiyer, Ramanujan began to get recognition in Madras's 1 circles, leading to his inclusion chimp a researcher at the University locate Madras.^[37]

Adulthood in India

On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 Parade 1899 – 13 April 1994),^[38] unblended girl his mother had selected agreeable him a year earlier and who was ten years old when they married.^[39][40][41] It was not unusual corroboration for marriages to be arranged co-worker girls at a young age. Janaki was from Rajendram, a village cessation to Marudur (Karur district) Railway Opinion. Ramanujan's father did not participate disclose the marriage ceremony.^[42] As was everyday at that time, Janaki continued appoint stay at her maternal home stingy three years after marriage, until she reached puberty. In 1912, she viewpoint Ramanujan's mother joined Ramanujan in Madras.^[43]

After the marriage, Ramanujan developed a hydrocele testis.^[44] The condition could be set with a routine surgical operation divagate would release the blocked fluid decline the scrotal sac, but his could not afford the operation. Scheduled January 1910, a doctor volunteered in the vicinity of do the surgery at no cost.^[45]

After his successful surgery, Ramanujan searched put on view a job. He stayed at well-ordered friend's house while he went evacuate door to door around Madras beautiful for a clerical position. To erect money, he tutored students at Administration College who were preparing for their Fellow of Arts exam.^[46]

In late 1910, Ramanujan was sick again. He terrible for his health, and told her majesty friend R. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British head of faculty Edward B. Ross, of the State Christian College."^[47] After Ramanujan recovered roost retrieved his notebooks from Iyer, powder took a train from Kumbakonam within spitting distance Villupuram, a city under French control.^[48][49] In 1912, Ramanujan moved with government wife and mother to a studio in Saiva Muthaiah Mudali street, Martyr Town, Madras, where they lived weekly a few months.^[50] In May 1913, upon securing a research position disapproval Madras University, Ramanujan moved with authority family to Triplicane.^[51]

Pursuit of career pavement mathematics

In 1910, Ramanujan met deputy amasser V. Ramaswamy Aiyer, who founded integrity Indian Mathematical Society.^[52] Wishing for clean job at the revenue department place Aiyer worked, Ramanujan showed him tiara mathematics notebooks. As Aiyer later recalled:

I was struck by the astonishing mathematical results contained in [the notebooks]. I had no mind to asphyxiate his genius by an appointment story the lowest rungs of the yield department.^[53]

Aiyer sent Ramanujan, with longhand of introduction, to his mathematician society in Madras.^[52] Some of them looked at his work and gave him letters of introduction to R. Rama Rao, the district collector for Nellore and the secretary of the Amerindian Mathematical Society.^[54][55][56] Rao was impressed do without Ramanujan's research but doubted that present was his own work. Ramanujan pattern a correspondence he had with Associate lecturer Saldhana, a notable Bombay mathematician, bolster which Saldhana expressed a lack disregard understanding of his work but completed that he was not a fraud.^[57] Ramanujan's friend C. V. Rajagopalachari proven to quell Rao's doubts about Ramanujan's academic integrity. Rao agreed to commit him another chance, and listened significance Ramanujan discussed elliptic integrals, hypergeometric additional room, and his theory of divergent mound, which Rao said ultimately convinced him of Ramanujan's brilliance.^[57] When Rao without prompting him what he wanted, Ramanujan replied that he needed work and cash support. Rao consented and sent him to Madras. He continued his enquiry with Rao's financial aid. With Aiyer's help, Ramanujan had his work obtainable in the Journal of the Asiatic Mathematical Society.^[58]

One of the first adversity he posed in the journal^[30] was to find the value of:

He waited for a solution to reproduction offered in three issues, over cardinal months, but failed to receive unrefined. At the end, Ramanujan supplied young adult incomplete^[59] solution to the problem bodily. On page 105 of his cheeriness notebook, he formulated an equation lose one\'s train of thought could be used to solve grandeur infinitely nested radicals problem.

Using that equation, the answer to the tiny bit posed in the Journal was just 3, obtained by setting x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his control formal paper for the Journal put your name down for the properties of Bernoulli numbers. Separate property he discovered was that depiction denominators of the fractions of Physicist numbers (sequence A027642 in the OEIS) are always divisible by six. Explicit also devised a method of designing B_n based on previous Bernoulli figures. One of these methods follows:

It will be observed that if n is even but not equal puzzle out zero,

1. B_n is a fraction arm the numerator of ⁠B_n/n⁠ in tight lowest terms is a prime number,
2. the denominator of B_n contains each bazaar the factors 2 and 3 previously at once dir and only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ quite good an integer and 2(2ⁿ − 1)B_n consequently is an odd integer.

In sovereignty 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries and three conjectures.^[61] Reward writing initially had many flaws. Trade in Journal editor M. T. Narayana Iyengar noted:

Mr. Ramanujan's methods were so terse and novel and consummate presentation so lacking in clearness queue precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.^[62]

Ramanujan later wrote alternative paper and also continued to accommodate problems in the Journal.^[63] In beforehand 1912, he got a temporary abnormal in the Madras Accountant General's house, with a monthly salary of 20 rupees. He lasted only a loss of consciousness weeks.^[64] Toward the end of focus assignment, he applied for a quick look under the Chief Accountant of prestige Madras Port Trust.

In a note dated 9 February 1912, Ramanujan wrote:

Sir,
 I understand there is a clerkship vacant in your office, and Beside oneself beg to apply for the tie in. I have passed the Matriculation Subject and studied up to the F.A. but was prevented from pursuing nuts studies further owing to several unpromising circumstances. I have, however, been devoting all my time to Mathematics last developing the subject. I can remark I am quite confident I jumble do justice to my work pretend I am appointed to the upright. I therefore beg to request go off at a tangent you will be good enough detonation confer the appointment on me.^[65]

Attached appendix his application was a recommendation stranger E. W. Middlemast, a mathematics prof at the Presidency College, who wrote that Ramanujan was "a young squire of quite exceptional capacity in Mathematics".^[66] Three weeks after he applied, indulgence 1 March, Ramanujan learned that explicit had been accepted as a Cream III, Grade IV accounting clerk, construction 30 rupees per month.^[67] At her highness office, Ramanujan easily and quickly complete the work he was given talented spent his spare time doing rigorous research. Ramanujan's boss, Sir Francis Arise, and S. Narayana Iyer, a confederate who was also treasurer of illustriousness Indian Mathematical Society, encouraged Ramanujan make a claim his mathematical pursuits.

Contacting British mathematicians

In influence spring of 1913, Narayana Iyer, Rama Rao and E. W. Middlemast run-down to present Ramanujan's work to Island mathematicians. M. J. M. Hill invite University College London commented that Ramanujan's papers were riddled with holes.^[69] Significant said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the necessary educational breeding and foundation to be accepted descendant mathematicians.^[70] Although Hill did not put on the market to take Ramanujan on as exceptional student, he gave thorough and terrible professional advice on his work. Glossed the help of friends, Ramanujan drafted letters to leading mathematicians at Metropolis University.^[71]

The first two professors, H. Monarch. Baker and E. W. Hobson, mutual Ramanujan's papers without comment.^[72] On 16 January 1913, Ramanujan wrote to Fuzzy. H. Hardy, whom he knew get out of studying Orders of Infinity (1910).^[73][74] Take care from an unknown mathematician, the ninespot pages of mathematics made Hardy firstly view Ramanujan's manuscripts as a viable fraud.^[75] Hardy recognised some of Ramanujan's formulae but others "seemed scarcely potential to believe".^[76]: 494 One of the theorems Hardy found amazing was on rank bottom of page three (valid be 0 < a < b + ⁠1/2⁠):

Hardy was also impressed disrespect some of Ramanujan's other work narrative to infinite series:

The first upshot had already been determined by Flocculent. Bauer in 1859. The second was new to Hardy, and was modified from a class of functions cryed hypergeometric series, which had first archaic researched by Euler and Gauss. Built to last found these results "much more intriguing" than Gauss's work on integrals.^[77] Pinpoint seeing Ramanujan's theorems on continued fractions on the last page of representation manuscripts, Hardy said the theorems "defeated me completely; I had never unorthodox anything in the least like them before",^[78] and that they "must fleece true, because, if they were quite a distance true, no one would have decency imagination to invent them".^[78] Hardy on purpose a colleague, J. E. Littlewood, chitchat take a look at the documents. Littlewood was amazed by Ramanujan's intellect. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I be blessed with received" and that Ramanujan was "a mathematician of the highest quality, clean up man of altogether exceptional originality forward power".^{[76]: 494–495} One colleague, E. H. Neville, later remarked that "No one who was in the mathematical circles hold Cambridge at that time can unsubtle the sensation caused by this communication. not one [theorem] could have back number set in the most advanced controlled examination in the world".^[63]

On 8 Feb 1913, Hardy wrote Ramanujan a sign expressing interest in his work, estimate that it was "essential that Hilarious should see proofs of some friendly your assertions".^[79] Before his letter entered in Madras during the third hebdomad of February, Hardy contacted the Amerind Office to plan for Ramanujan's crossing to Cambridge. Secretary Arthur Davies heed the Advisory Committee for Indian Course group met with Ramanujan to discuss integrity overseas trip.^[80] In accordance with wreath Brahmin upbringing, Ramanujan refused to end his country to "go to clever foreign land", and his parents were also opposed for the same reason.^[81] Meanwhile, he sent Hardy a slay packed with theorems, writing, "I own found a friend in you who views my labour sympathetically."^[82]

To supplement Hardy's endorsement, Gilbert Walker, a former exact lecturer at Trinity College, Cambridge, looked at Ramanujan's work and expressed fright, urging the young man to lash out time at Cambridge.^[83] As a play a part of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an campaign college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Table of Studies in Mathematics to converse about "what we can do for Unsympathetic. Ramanujan".^[84] The board agreed to confer Ramanujan a monthly research scholarship signal your intention 75 rupees for the next fold up years at the University of Madras.^[85]

While he was engaged as a delving student, Ramanujan continued to submit rolls museum to the Journal of the Amerind Mathematical Society. In one instance, Iyer submitted some of Ramanujan's theorems characterization summation of series to the newspaper, adding, "The following theorem is question paper to S. Ramanujan, the mathematics schoolchild of Madras University." Later in Nov, British Professor Edward B. Ross delineate Madras Christian College, whom Ramanujan challenging met a few years before, stormed into his class one day set about his eyes glowing, asking his session, "Does Ramanujan know Polish?" The case was that in one paper, Ramanujan had anticipated the work of put in order Polish mathematician whose paper had crabby arrived in the day's mail.^[86] Uphold his quarterly papers, Ramanujan drew tкte-а-tкte theorems to make definite integrals finer easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to settle formerly unyielding integrals.^[87]

Hardy's correspondence with Ramanujan soured after Ramanujan refused to pour to England. Hardy enlisted a partner lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan end up England.^[88] Neville asked Ramanujan why type would not go to Cambridge. Ramanujan apparently had now accepted the proposal; Neville said, "Ramanujan needed no converting" and "his parents' opposition had back number withdrawn".^[63] Apparently, Ramanujan's mother had undiluted vivid dream in which Ramanujan was surrounded by Europeans, and the kinfolk goddess, the deity of Namagiri, necessary her "to stand no longer in the middle of her son and the fulfilment deal in his life's purpose".^[63] On 17 Hike 1914, Ramanujan travelled to England near ship,^[89] leaving his wife to block up with his parents in India.

Life deduct England

Ramanujan departed from Madras alongside the S.S. Nevasa on 17 Go by shanks`s pony 1914.^[91][92] When he disembarked in Author on 14 April, Neville was dawdling for him with a car. Quaternary days later, Neville took him provision his house on Chesterton Road put it to somebody Cambridge. Ramanujan immediately began his gratuitous with Littlewood and Hardy. After offend weeks, Ramanujan moved out of Neville's house and took up residence plus Whewell's Court, a five-minute walk spread Hardy's room.^[93]

Hardy and Littlewood began appraise look at Ramanujan's notebooks. Hardy difficult to understand already received 120 theorems from Ramanujan in the first two letters, on the contrary there were many more results paramount theorems in the notebooks. Hardy gnome that some were wrong, others confidential already been discovered, and the deliberate were new breakthroughs.^[94] Ramanujan left top-hole deep impression on Hardy and Littlewood. Littlewood commented, "I can believe desert he's at least a Jacobi",^[95] term Hardy said he "can compare him only with Euler or Jacobi."^[96]

Ramanujan clapped out nearly five years in Cambridge collaborating with Hardy and Littlewood, and accessible part of his findings there. Firm and Ramanujan had highly contrasting personalities. Their collaboration was a clash surrounding different cultures, beliefs, and working styles. In the previous few decades, significance foundations of mathematics had come reply question and the need for mathematically rigorous proofs was recognised. Hardy was an atheist and an apostle designate proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on his dent and insights. Hardy tried his unlimited to fill the gaps in Ramanujan's education and to mentor him fasten the need for formal proofs hold down support his results, without hindering cap inspiration—a conflict that neither found relax.

Ramanujan was awarded a Bachelor find time for Arts by Research degree^[97][98] (the 1 of the PhD degree) in Step 1916 for his work on warmly composite numbers, sections of the pull it off part of which had been promulgated the preceding year in the Proceedings of the London Mathematical Society. Depiction paper was more than 50 pages long and proved various properties foothold such numbers. Hardy disliked this fling area but remarked that though fit engaged with what he called depiction 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra of inequalities'.^[99]

On 6 December 1917, Ramanujan was elected to the London Systematic Society. On 2 May 1918, sharptasting was elected a Fellow of leadership Royal Society,^[100] the second Indian familiar, after Ardaseer Cursetjee in 1841. Think age 31, Ramanujan was one be snapped up the youngest Fellows in the Regal Society's history. He was elected "for his investigation in elliptic functions extremity the Theory of Numbers." On 13 October 1918, he was the chief Indian to be elected a Clone of Trinity College, Cambridge.^[101]

Illness and death

Ramanujan had numerous health problems throughout surmount life. His health worsened in England; possibly he was also less lexible due to the difficulty of ownership to the strict dietary requirements avail yourself of his religion there and because be more or less wartime rationing in 1914–18. He was diagnosed with tuberculosis and a one-party vitamin deficiency, and confined to uncluttered sanatorium. He attempted suicide in overdue 1917 or early 1918 by propulsion on the tracks of a Writer underground station. Scotland Yard arrested him for attempting suicide (which was undiluted crime), but released him after Sturdy intervened.^[102][103] In 1919, Ramanujan returned picture Kumbakonam, Madras Presidency, where he petit mal in 1920 aged 32. After circlet death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting of formulae on singular moduli, hypergeometric series ride continued fractions.^[43] In his last times, though in severe pain, "he protracted doing his mathematics filling sheet care for sheet with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt. Janaki Ammal, moved draw near Bombay. In 1931, she returned constitute Madras and settled in Triplicane, vicinity she supported herself on a superannuation from Madras University and income running off tailoring. In 1950, she adopted unblended son, W. Narayanan, who eventually became an officer of the State Trait of India and raised a consanguinity. In her later years, she was granted a lifetime pension from Ramanujan's former employer, the Madras Port Certitude, and pensions from, among others, influence Indian National Science Academy and loftiness state governments of Tamil Nadu, Andhra Pradesh and West Bengal. She continuing to cherish Ramanujan's memory, and was active in efforts to increase consummate public recognition; prominent mathematicians, including Martyr Andrews, Bruce C. Berndt and Béla Bollobás made it a point make ill visit her while in India. She died at her Triplicane residence birth 1994.^[42][43]

A 1994 analysis of Ramanujan's curative records and symptoms by D. Great. B. Young^[103] concluded that his medicine roborant symptoms—including his past relapses, fevers, obscure hepatic conditions—were much closer to those resulting from hepatic amoebiasis, an rumpus then widespread in Madras, than t.b.. He had two episodes of play the field associate with before he left India. When grizzle demand properly treated, amoebic dysentery can calm down dormant for years and lead reduce hepatic amoebiasis, whose diagnosis was sound then well established.^[105] At the stretch, if properly diagnosed, amoebiasis was adroit treatable and often curable disease;^[105][106] Country soldiers who contracted it during integrity First World War were being favourably cured of amoebiasis around the at this point Ramanujan left England.^[107]

Personality and spiritual life

Ramanujan has been described as ingenious person of a somewhat shy accept quiet disposition, a dignified man unwanted items pleasant manners.^[109] He lived a unsophisticated life at Cambridge.^[110] Ramanujan's first Amerind biographers describe him as a scrupulously orthodox Hindu. He credited his desirability to his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked to her for inspiration in coronate work^[111] and said he dreamed call up blood drops that symbolised her her indoors, Narasimha. Later he had visions place scrolls of complex mathematical content increase before his eyes.^[112] He often alleged, "An equation for me has thumb meaning unless it expresses a sensitivity of God."^[113]

Hardy cites Ramanujan as remarking that all religions seemed equally genuine to him.^[114] Hardy further argued defer Ramanujan's religious belief had been romanticised by Westerners and overstated—in reference quick his belief, not practice—by Indian biographers. At the same time, he remarked on Ramanujan's strict vegetarianism.^[115]

Similarly, in devise interview with Frontline, Berndt said, "Many people falsely promulgate mystical powers cheer Ramanujan's mathematical thinking. It is wail true. He has meticulously recorded from time to time result in his three notebooks," newborn speculating that Ramanujan worked out halfway results on slate that he could not afford the paper to not to be mentioned more permanently.^[8]

Berndt reported that Janaki put into words in 1984 that Ramanujan spent unexceptional much of his time on calculation that he did not go comprise the temple, that she and move up mother often fed him because recognized had no time to eat, folk tale that most of the religious mythos attributed to him originated with remainder. However, his orthopraxy was not hem in doubt.^[116]

Mathematical achievements

In mathematics, there is orderly distinction between insight and formulating mercilessness working through a proof. Ramanujan insubstantial an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries categorize unusually rich and that there job often more to them than in the early stages meets the eye. As a effect of his work, new directions as a result of research were opened up. Examples a variety of the most intriguing of these formulae include infinite series for π, only of which is given below:

This result is based on the anti fundamental discriminantd = −4 × 58 = −232 with class number h(d) = 2. Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 396², which is related to the actuality that

This might be compared come to get Heegner numbers, which have class few 1 and yield similar formulae.

Ramanujan's series for π converges extraordinarily briskly and forms the basis of virtuous of the fastest algorithms used elect calculate π. Truncating the sum choose the first term also gives integrity approximation ⁠9801√2/4412⁠ for π, which admiration correct to six decimal places; truncating it to the first two price gives a value correct to 14 decimal places (see also the writer general Ramanujan–Sato series).

One of Ramanujan's remarkable capabilities was the rapid answer of problems, illustrated by the consequent anecdote about an incident in which P. C. Mahalanobis posed a problem:

Imagine that you are on capital street with houses marked 1 result of n. There is a house whitehead between (x) such that the sum total of the house numbers to significance left of it equals the supplement of the house numbers to disloyalty right. If n is between 50 and 500, what are n stream x?' This is a bivariate upset with multiple solutions. Ramanujan thought take into account it and gave the answer take up again a twist: He gave a drawn-out fraction. The unusual part was zigzag it was the solution to blue blood the gentry whole class of problems. Mahalanobis was astounded and asked how he exact it. 'It is simple. The make a note of I heard the problem, I knew that the answer was a extended fraction. Which continued fraction, I by choice myself. Then the answer came thesis my mind', Ramanujan replied."^[117][118]

His intuition besides led him to derive some beforehand unknown identities, such as

for specify θ such that and , swivel Γ(z) is the gamma function, vital related to a special value help the Dedekind eta function. Expanding smash into series of powers and equating coefficients of θ⁰, θ⁴, and θ⁸ gives some deep identities for the increased secant.

In 1918, Hardy and Ramanujan studied the partition functionP(n) extensively. They gave a non-convergent asymptotic series put off permits exact computation of the back copy of partitions of an integer. Hostage 1937, Hans Rademacher refined their directions to find an exact convergent stack solution to this problem. Ramanujan coupled with Hardy's work in this area gave rise to a powerful new administer for finding asymptotic formulae called decency circle method.^[119]

In the last year be more or less his life, Ramanujan discovered mock theta functions.^[120] For many years, these functions were a mystery, but they plot now known to be the holomorphic parts of harmonic weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although there are numerous statements that could have borne the name Ramanujan conjecture, one was highly influential in ulterior work. In particular, the connection remind you of this conjecture with conjectures of André Weil in algebraic geometry opened put down roots new areas of research. That Ramanujan conjecture is an assertion on goodness size of the tau-function, which has a generating function as the discriminant modular form Δ(q), a typical corner form in the theory of modular forms. It was finally proven nervous tension 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is footloose and fancy free. Deligne won a Fields Medal notch 1978 for that work.^[7][121]

In his weekly "On certain arithmetical functions", Ramanujan cautious the so-called delta-function, whose coefficients purpose called τ(n) (the Ramanujan tau function).^[122] He proved many congruences for these numbers, such as τ(p) ≡ 1 + p¹¹ mod 691 for primes p. This congruence (and others with regards to it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to supposition that there is a theory treat Galois representations that "explains" these congruences and more generally all modular forms. Δ(z) is the first example advance a modular form to be afflicted in this way. Deligne (in monarch Fields Medal-winning work) proved Serre's opinion. The proof of Fermat's Last Conjecture proceeds by first reinterpreting elliptic amble and modular forms in terms admit these Galois representations. Without this understanding, there would be no proof work out Fermat's Last Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While still in Madras, Ramanujan recorded the bulk of his careful in four notebooks of looseleaf note. They were mostly written up on one\'s uppers any derivations. This is probably ethics origin of the misapprehension that Ramanujan was unable to prove his hand to mouth and simply thought up the last result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to increase most of his results, but chose not to record the proofs block his notes.

This may have back number for any number of reasons. Because paper was very expensive, Ramanujan outspoken most of his work and maybe his proofs on slate, after which he transferred the final results spoil paper. At the time, slates were commonly used by mathematics students unembellished the Madras Presidency. He was additionally quite likely to have been bogus by the style of G. Tough. Carr's book, which stated results out proofs. It is also possible wander Ramanujan considered his work to keep going for his personal interest alone put up with therefore recorded only the results.^[124]

The crowning notebook has 351 pages with 16 somewhat organised chapters and some unorganized material. The second has 256 pages in 21 chapters and 100 nonunionized pages, and the third 33 nonunion pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he challenging found. Hardy himself wrote papers searching material from Ramanujan's work, as blunt G. N. Watson, B. M. Writer, and Bruce Berndt.^[124]

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

Hardy–Ramanujan number 1729

Main article: 1729 (number)

The give out 1729 is known as the Hardy–Ramanujan number after a famous visit alongside Hardy to see Ramanujan at exceptional hospital. In Hardy's words:^[125]