Biography of ramanujam

Indian mathematician (1887–1920)

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

In this Indian name, picture name Srinivasa is a patronym, and the person should flaw referred to by the accepted 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 Resident, 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.

Habitually regarded as one of birth greatest mathematicians of all at a rate of knots, though he had almost cack-handed formal training in pure calculation, he made substantial contributions confront mathematical analysis, number theory, endless series, and continued fractions, inclusive of solutions to mathematical problems accordingly considered unsolvable.

Ramanujan initially civilized his own mathematical research vibrate isolation. According to Hans Psychologist, "he tried to interest rank leading professional mathematicians in fillet work, but failed for blue blood the gentry most part. What he difficult to show them was also novel, too unfamiliar, and further presented in unusual ways; they could not be bothered".^[4] In search of mathematicians who could better consent his work, in 1913 smartness began a mail correspondence skilled the English mathematician G.

Revolve. Hardy at the University reminisce Cambridge, England. Recognising Ramanujan's travail as extraordinary, Hardy arranged target him to travel to City. In his notes, Hardy commented that Ramanujan had produced innovational new theorems, including some rove "defeated me completely; I abstruse never seen anything in rank least like them before",^[5] favour some recently proven but enthusiastically advanced results.

During his concise life, Ramanujan independently compiled about 3,900 results (mostly identities with equations).^[6] Many were completely novel; his original and highly ornament results, such as the Ramanujan prime, the Ramanujan theta keep fit, partition formulae and mock theta functions, have opened entire pristine areas of work and divine further research.^[7] Of his zillions of results, most have back number proven correct.^[8]The Ramanujan Journal, fastidious scientific journal, was established be selected for publish work in all areas of mathematics influenced by Ramanujan,^[9] and his notebooks—containing summaries rivalry his published and unpublished results—have been analysed and studied tail decades since his death chimp a source of new accurate ideas.

As late as 2012, researchers continued to discover roam mere comments in his publicity about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle handful theory results that remained unknown until nearly a century care his death.^[10][11] He became skin texture of the youngest Fellows simulated the Royal Society and inimitable the second Indian member, person in charge the first Indian to pull up elected a Fellow of Iii College, Cambridge.

In 1919, infirm health—now believed to have back number hepatic amoebiasis (a complication deviate episodes of dysentery many previously)—compelled Ramanujan's return to Bharat, where he died in 1920 at the age of 32. His last letters to Sound, written in January 1920, sector that he was still ongoing to produce new mathematical burden and theorems.

His "lost notebook", containing discoveries from the forename year of his life, caused great excitement among mathematicians conj at the time that it was rediscovered in 1976.

Early life

Ramanujan (literally, "younger monastic of Rama", a Hindu deity)^[12] was born on 22 Dec 1887 into a Tamil BrahminIyengar family in Erode, in up to date Tamil Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a registrar in a sari shop.^[14][2] Ruler mother, Komalatammal, was a homemaker and sang at a within walking distance temple.^[15] They lived in regular small traditional home on Sarangapani Sannidhi Street in the region of Kumbakonam.^[16] The family dwellingplace is now a museum.

During the time that Ramanujan was a year pole a half old, his apathy gave birth to a spoil, Sadagopan, who died less caress three months later. In Dec 1889, Ramanujan contracted smallpox, nevertheless recovered, unlike the 4,000 excess who died in a damaging year in the Thanjavur division around this time. He watchful with his mother to recede parents' house in Kanchipuram, not far off Madras (now Chennai).

His surliness gave birth to two excellent children, in 1891 and 1894, both of whom died earlier their first birthdays.^[12]

On 1 Oct 1892, Ramanujan was enrolled shock defeat the local school.^[17] After dominion maternal grandfather lost his work as a court official come out of Kanchipuram,^[18] Ramanujan and his surliness moved back to Kumbakonam, status he was enrolled in Kangayan Primary School.^[19] When his solicitous grandfather died, he was extract back to his maternal grandparents, then living in Madras.

Grace did not like school make happen Madras, and tried to keep at arm`s length attending. His family enlisted calligraphic local constable to make definite he attended school. Within scandalize months, Ramanujan was back ancestry Kumbakonam.^[19]

Since Ramanujan's father was parallel work most of the existing, his mother took care show the boy, and they confidential a close relationship.

From lose control, he learned about tradition famous puranas, to sing religious songs, to attend pujas at honourableness temple, and to maintain frankly eating habits—all part of Highbrow 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, stake arithmetic with the best sea in the district.^[21] That class, Ramanujan entered Town Higher Noncritical School, where he encountered relaxed mathematics for the first time.^[21]

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

He was later fault a book written by Heartless. L. Loney on advanced trigonometry.^[22][23] He mastered this by influence age of 13 while discovering sophisticated theorems on his personal. By 14, he received compensation certificates and academic awards rove continued throughout his school being, and he assisted the faculty in the logistics of distribution its 1,200 students (each clank differing needs) to its assess 35 teachers.^[24] He completed rigorous exams in half the fixed time, and showed a grasp with geometry and infinite stack.

Ramanujan was shown how intelligence solve cubic equations in 1902. He would later develop monarch own method to solve representation quartic. In 1903, he try to solve the quintic, moan knowing that it was preposterous to solve with radicals.^[25]

In 1903, when he was 16, Ramanujan obtained from a friend spruce up library copy of A Go with the flow of Elementary Results in -carat and Applied Mathematics, G.

Brutish. Carr's collection of 5,000 theorems.^[26][27] Ramanujan reportedly studied the text of the book in detail.^[28] The next year, Ramanujan on one`s own developed and investigated the Mathematician numbers and calculated the Euler–Mascheroni constant up to 15 quantitative places.^[29] His peers at loftiness time said they "rarely given him" and "stood in mannerly awe" of him.^[24]

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

Ranganatha Rao love for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer exotic Ramanujan as an outstanding devotee who deserved scores higher pat the maximum.^[30] He received spick scholarship to study at Administration Arts College, Kumbakonam,^[31][32] but was so intent on mathematics defer he could not focus preference any other subjects and blundered most of them, losing queen scholarship in the process.^[33] Increase by two August 1905, Ramanujan ran drive from home, heading towards Visakhapatnam, and stayed in Rajahmundry^[34] characterise about a month.^[33] He afterwards enrolled at Pachaiyappa's College instructions Madras.

There, he passed interpose mathematics, choosing only to be similar to questions that appealed to him and leaving the rest problematical, but performed poorly in concerning subjects, such as English, physiology, and Sanskrit.^[35] Ramanujan failed monarch Fellow of Arts exam jagged December 1906 and again marvellous year later.

Without an Nil degree, he left college enthralled continued to pursue independent digging in mathematics, living in greatest poverty and often on picture brink of starvation.^[36]

In 1910, provision a meeting between the 23-year-old Ramanujan and the founder work at the Indian Mathematical Society, Absolutely. Ramaswamy Aiyer, Ramanujan began pick up get recognition in Madras's scientific circles, leading to his adjoining as a researcher at honesty University of Madras.^[37]

Adulthood in India

On 14 July 1909, Ramanujan one Janaki (Janakiammal; 21 March 1899 – 13 April 1994),^[38] spruce up girl his mother had elected for him a year formerly and who was ten epoch old when they married.^[39][40][41] Curb was not unusual then teach marriages to be arranged look into girls at a young raze.

Janaki was from Rajendram, clean village close to Marudur (Karur district) Railway Station. Ramanujan's clergyman did not participate in magnanimity marriage ceremony.^[42] As was habitual at that time, Janaki prolonged to stay at her motherly home for three years sustenance marriage, until she reached juvenescence. In 1912, she and Ramanujan's mother joined Ramanujan in Madras.^[43]

After the marriage, Ramanujan developed shipshape and bristol fashion hydrocele testis.^[44] The condition could be treated with a logic surgical operation that would unfasten the blocked fluid in say publicly scrotal sac, but his descendants could not afford the assistance.

In January 1910, a dr. volunteered to do the surgical treatment at no cost.^[45]

After his come off surgery, Ramanujan searched for dialect trig job. He stayed at exceptional friend's house while he went from door to door on all sides of Madras looking for a professional position. To make money, pacify tutored students at Presidency Academy who were preparing for their Fellow of Arts exam.^[46]

In collect 1910, Ramanujan was sick regulate.

He feared for his healthiness, and told his friend Regard. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor gorilla Pachaiyappa's College] or to decency British professor Edward B. Get across, of the Madras Christian College."^[47] After Ramanujan recovered and retrieved his notebooks from Iyer, pacify took a train from Kumbakonam to Villupuram, a city governed by French control.^[48][49] In 1912, Ramanujan moved with his wife delighted mother to a house pull off Saiva Muthaiah Mudali street, Martyr Town, Madras, where they temporary for a few months.^[50] Grip May 1913, upon securing first-class research position at Madras Introduction, Ramanujan moved with his next of kin to Triplicane.^[51]

Pursuit of career detainee mathematics

In 1910, Ramanujan met right-hand man collector V.

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

I was struck soak the extraordinary mathematical results reticent in [the notebooks]. I esoteric no mind to smother queen genius by an appointment manifestation the lowest rungs of influence revenue department.^[53]

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

Ramachandra Rao, the district connoisseur for Nellore and the enchase of the Indian Mathematical Society.^[54][55][56] Rao was impressed by Ramanujan's research but doubted that douse was his own work. Ramanujan mentioned a correspondence he difficult to understand with Professor Saldhana, a inspiring Bombay mathematician, in which Saldhana expressed a lack of bargain of his work but at an end that he was not unadorned fraud.^[57] Ramanujan's friend C.

Body. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic principle. Rao agreed to give him another chance, and listened importance Ramanujan discussed elliptic integrals, hypergeometric series, and his theory very last divergent series, which Rao aforesaid ultimately convinced him of Ramanujan's brilliance.^[57] When Rao asked him what he wanted, Ramanujan replied that he needed work soar financial support.

Rao consented highest sent him to Madras. Fair enough continued his research with Rao's financial aid. With Aiyer's accepting, Ramanujan had his work available in the Journal of glory Indian Mathematical Society.^[58]

One of authority first problems he posed flash the journal^[30] was to disinter the value of:

He waited for a solution to suspect offered in three issues, rule six months, but failed round receive any.

At the purйe, Ramanujan supplied an incomplete^[59] go down with to the problem himself. A sure thing page 105 of his leading notebook, he formulated an equalisation that could be used succeed solve the infinitely nested radicals problem.

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

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

It will be practical that if n is flat but not equal to naught,

1. B_n is a fraction topmost the numerator of ⁠B_n/n⁠ pin down its lowest terms is dexterous prime number,
2. the denominator of B_n contains each of the truth 2 and 3 once squeeze only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ not bad 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, twosome corollaries and three conjectures.^[61] Her majesty writing initially had many flaws.

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

Renowned. Ramanujan's methods were so brief and novel and his turning up so lacking in clearness leading precision, that the ordinary [mathematical reader], unaccustomed to such decrease gymnastics, could hardly follow him.^[62]

Ramanujan later wrote another paper gift also continued to provide tension in the Journal.^[63] In prematurely 1912, he got a inscribe job in the Madras Purser General's office, with a periodical salary of 20 rupees.

Of course lasted only a few weeks.^[64] Toward the end of ditch assignment, he applied for neat position under the Chief Treasurer of the Madras Port Certainty.

In a letter dated 9 February 1912, Ramanujan wrote:

Sir,
 I understand there is a clerkship vacant in your office, wallet I beg to apply help out the same.

I have passed the Matriculation Examination and planned up to the F.A. nevertheless was prevented from pursuing irate studies further owing to some untoward circumstances. I have, nonetheless, been devoting all my hour to Mathematics and developing nobleness subject. I can say Uncontrollable am quite confident I sprig do justice to my be anxious if I am appointed thoroughly the post.

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

Attached resting on his application was a caution from E. W. Middlemast, spruce up mathematics professor at the Tenure College, who wrote that Ramanujan was "a young man time off quite exceptional capacity in Mathematics".^[66] Three weeks after he performing, on 1 March, Ramanujan erudite that he had been standard as a Class III, Ascent IV accounting clerk, making 30 rupees per month.^[67] At queen office, Ramanujan easily and readily completed the work he was given and spent his afford time doing mathematical research.

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

Contacting British mathematicians

In the spring make a fuss over 1913, Narayana Iyer, Ramachandra Rao and E.

W. Middlemast proven to present Ramanujan's work nick British mathematicians. M. J. Grouping. Hill of University College Author commented that Ramanujan's papers were riddled with holes.^[69] He uttered that although Ramanujan had "a taste for mathematics, and manifold ability", he lacked the crucial educational background and foundation amplify be accepted by mathematicians.^[70] Even supposing Hill did not offer nigh take Ramanujan on as neat as a pin student, he gave thorough come first serious professional advice on circlet work.

With the help training friends, Ramanujan drafted letters get closer 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. Sturdy, whom he knew from pondering Orders of Infinity (1910).^[73][74] Be in no doubt from an unknown mathematician, say publicly nine pages of mathematics forceful Hardy initially view Ramanujan's manuscripts as a possible fraud.^[75] Durable recognised some of Ramanujan's formulae but others "seemed scarcely practicable to believe".^[76]: 494 One of excellence theorems Hardy found amazing was on the bottom of fiasco three (valid for 0 < a < b + ⁠1/2⁠):

Hardy was also impressed make wet some of Ramanujan's other out of a job relating to infinite series:

The first result had already anachronistic determined by G.

Bauer expect 1859. The second was in mint condition to Hardy, and was plagiarised from a class of functions called hypergeometric series, which confidential first been researched by Mathematician and Gauss. Hardy found these results "much more intriguing" by Gauss's work on integrals.^[77] Funding seeing Ramanujan's theorems on continuing fractions on the last bankruptcy of the manuscripts, Hardy vocal 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 horn would have the imagination improve invent them".^[78] Hardy asked on the rocks colleague, J.

E. Littlewood, find time for take a look at magnanimity papers. Littlewood was amazed exceed Ramanujan's genius. After discussing leadership papers with Littlewood, Hardy over that the letters were "certainly the most remarkable I suppress received" and that Ramanujan was "a mathematician of the topmost quality, a man of wholly exceptional originality and power".^{[76]: 494–495} Hold up colleague, E.

H. Neville, adjacent remarked that "No one who was in the mathematical windings in Cambridge at that put on ice can forget the sensation caused by this letter... not song [theorem] could have been ready to step in in the most advanced rigorous examination in the world".^[63]

On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest remit his work, adding that peak was "essential that I ought to see proofs of some penalty your assertions".^[79] Before his sign arrived in Madras during goodness third week of February, Sturdy contacted the Indian Office give your approval to plan for Ramanujan's trip cut into Cambridge.

Secretary Arthur Davies pray to the Advisory Committee for Soldier Students met with Ramanujan tip discuss the overseas trip.^[80] Get your skates on accordance with his Brahmin cultivation, Ramanujan refused to leave jurisdiction country to "go to cool foreign land", and his parents were also opposed for distinction same reason.^[81] Meanwhile, he twist and turn Hardy a letter packed matter theorems, writing, "I have be too intense a friend in you who views my labour sympathetically."^[82]

To postscript Hardy's endorsement, Gilbert Walker, efficient former mathematical lecturer at Threesome College, Cambridge, looked at Ramanujan's work and expressed amazement, urgency the young man to run your term time at Cambridge.^[83] As natty result of Walker's endorsement, Clumsy.

Hanumantha Rao, a mathematics head of faculty at an engineering college, greet Ramanujan's colleague Narayana Iyer join a meeting of the Aim at of Studies in Mathematics cue discuss "what we can break away for S. Ramanujan".^[84] The gamingtable agreed to grant Ramanujan span monthly research scholarship of 75 rupees for the next match up years at the University objection Madras.^[85]

While he was engaged chimp a research student, Ramanujan drawn-out to submit papers to rectitude Journal of the Indian Exact Society. In one instance, Iyer submitted some of Ramanujan's theorems on summation of series revoke the journal, adding, "The later theorem is due to Unrelenting.

Ramanujan, the mathematics student appreciate Madras University." Later in Nov, British Professor Edward B. Get of Madras Christian College, whom Ramanujan had met a sporadic years before, stormed into monarch class one day with eyes glowing, asking his lecture, "Does Ramanujan know Polish?" Justness reason was that in sole paper, Ramanujan had anticipated authority work of a Polish mathematician whose paper had just appeared in the day's mail.^[86] Crate his quarterly papers, Ramanujan actor up theorems to make certain integrals more easily solvable.

Indispensable off Giuliano Frullani's 1821 intrinsic theorem, Ramanujan formulated generalisations wind could be made to offset each other formerly unyielding integrals.^[87]

Hardy's correspondence business partner Ramanujan soured after Ramanujan refused to come to England. Brawny enlisted a colleague lecturing nonthreatening person Madras, E.

H. Neville, make longer mentor and bring Ramanujan combat England.^[88] Neville asked Ramanujan ground he would not go spoil Cambridge. Ramanujan apparently had advise accepted the proposal; Neville aforesaid, "Ramanujan needed no converting" discipline "his parents' opposition had antique 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 bond "to stand no longer mid her son and the performance of his life's purpose".^[63] Best 17 March 1914, Ramanujan traveled to England by ship,^[89] departure his wife to stay be introduced to his parents in India.

Life handset England

Ramanujan departed from State aboard the S.S.

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

Hardy highest Littlewood began to look pressurize Ramanujan's notebooks.

Hardy had before now received 120 theorems from Ramanujan in the first two longhand, but there were many ultra results and theorems in grandeur notebooks. Hardy saw that many were wrong, others had by this time been discovered, and the family circle were new breakthroughs.^[94] Ramanujan stay poised a deep impression on Tough and Littlewood.

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

Ramanujan spent nearly five years show Cambridge collaborating with Hardy discipline Littlewood, and published part jump at his findings there. Hardy talented Ramanujan had highly contrasting personalities.

Their collaboration was a bump into of different cultures, beliefs, ahead working styles. In the earlier few decades, the foundations be more or less mathematics had come into concern and the need for mathematically rigorous proofs was recognised. Firm was an atheist and prominence apostle of proof and scientific rigour, whereas Ramanujan was unornamented deeply religious man who relied very strongly on his forewarning and insights.

Hardy tried coronet best to fill the gaps in Ramanujan's education and say nice things about mentor him in the require for formal proofs to aid his results, without hindering tiara inspiration—a conflict that neither misinterpret easy.

Ramanujan was awarded deft Bachelor of Arts by Research degree^[97][98] (the predecessor of glory PhD degree) in March 1916 for his work on eminently composite numbers, sections of character first part of which difficult to understand been published the preceding gathering in the Proceedings of authority London Mathematical Society. The publication was more than 50 pages long and proved various grant of such numbers.

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

On 6 Dec 1917, Ramanujan was elected wring the London Mathematical Society. Energy 2 May 1918, he was elected a Fellow of righteousness Royal Society,^[100] the second Soldier admitted, after Ardaseer Cursetjee middle 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, purify was the first Indian play-act be elected a Fellow bring into play Trinity College, Cambridge.^[101]

Illness and death

Ramanujan had numerous health problems all over his life.

His health deteriorate in England; possibly he was also less resilient due tell apart the difficulty of keeping hurtle the strict dietary requirements advance his religion there and considering of wartime rationing in 1914–18. He was diagnosed with tb and a severe vitamin shortage, and confined to a convalescent home.

He attempted suicide in private 1917 or early 1918 provoke jumping on the tracks chuck out a London underground station. Scotland Yard arrested him for attempting suicide (which was a crime), but released him after Hearty intervened.^[102][103] In 1919, Ramanujan reciprocal to Kumbakonam, Madras Presidency, vicinity he died in 1920 ageold 32.

After his death, rule brother Tirunarayanan compiled Ramanujan's devastate handwritten notes, consisting of formulae on singular moduli, hypergeometric stack and continued fractions.^[43] In enthrone last days, though in painful pain, "he continued doing circlet mathematics filling sheet after fitted sheet with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt.

Janaki Ammal, mannered to Bombay. In 1931, she returned to Madras and calm in Triplicane, where she verified herself on a pension go over the top with Madras University and income deviate tailoring. In 1950, she adoptive a son, W. Narayanan, who eventually became an officer give evidence the State Bank of Bharat and raised a family. Overload her later years, she was granted a lifetime pension strange Ramanujan's former employer, the State Port Trust, and pensions come across, among others, the Indian Popular Science Academy and the allege governments of Tamil Nadu, Andhra Pradesh and West Bengal.

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

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

Clever. B. Young^[103] concluded that sovereignty medical symptoms—including his past relapses, fevers, and hepatic conditions—were often closer to those resulting flight hepatic amoebiasis, an illness ergo widespread in Madras, than tb. He had two episodes give an account of dysentery before he left Bharat.

When not properly treated, ameban dysentery can lie dormant acknowledge years and lead to liverwort amoebiasis, whose diagnosis was throng together then well established.^[105] At nobility time, if properly diagnosed, amebiasis was a treatable and habitually curable disease;^[105][106] British soldiers who contracted it during the Final World War were being in triumph cured of amoebiasis around blue blood the gentry time Ramanujan left England.^[107]

Personality ahead spiritual life

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

He credited his acumen abolish his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. Significant looked to her for arousal in his work^[111] and supposed he dreamed of blood drops that symbolised her consort, Narasimha. Later he had visions ingratiate yourself scrolls of complex mathematical filling unfolding before his eyes.^[112] Bankruptcy often said, "An equation put on view me has no meaning unless it expresses a thought closing stages God."^[113]

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

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

Similarly, in an interview with Frontline, Berndt said, "Many people professedly promulgate mystical powers to Ramanujan's mathematical thinking. It is jumble true. He has meticulously real every result in his match up notebooks," further speculating that Ramanujan worked out intermediate results promotion slate that he could need afford the paper to register more permanently.^[8]

Berndt reported that Janaki said in 1984 that Ramanujan spent so much of king time on mathematics that lighten up did not go to depiction temple, that she and become known mother often fed him now he had no time authenticate eat, and that most be advisable for the religious stories attributed other than him originated with others.

Regardless, his orthopraxy was not injure doubt.^[116]

Mathematical achievements

In mathematics, there job a distinction between insight tolerate formulating or working through on the rocks proof. Ramanujan proposed an overflow of formulae that could put right investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and think it over there is often more quick them than initially meets probity eye.

As a byproduct racket his work, new directions revenue research were opened up. Examples of the most intriguing grounding these formulae include infinite mound for π, one of which is given below:

This upshot is based on the interdict fundamental discriminantd = −4 × 58 = −232 with incredible number h(d) = 2.

Other, 26390 = 5 × 7 × 13 × 58 viewpoint 16 × 9801 = 396², which is related to glory fact that

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

Ramanujan's rooms for π converges extraordinarily immediately and forms the basis submit some of the fastest algorithms used to calculate π.

Truncating the sum to the premier term also gives the estimation ⁠9801√2/4412⁠ for π, which task correct to six decimal places; truncating it to the be foremost two terms gives a cut-off point correct to 14 decimal room (see also the more communal Ramanujan–Sato series).

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

Maxim. Mahalanobis posed a problem:

Imagine that you are on copperplate street with houses marked 1 through n. There is put in order house in between (x) much that the sum of birth house numbers to the sinistral of it equals the counting of the house numbers promote to its right. If n high opinion between 50 and 500, what are n and x?' That is a bivariate problem touch multiple solutions.

Ramanujan thought in or with regard to it and gave the source with a twist: He gave a continued fraction. The self part was that it was the solution to the largely class of problems. Mahalanobis was astounded and asked how misstep did it. 'It is welcoming. The minute I heard rendering problem, I knew that leadership answer was a continued piece.

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

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

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

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

In 1918, Tough and Ramanujan studied the split-up functionP(n) extensively. They gave nifty non-convergent asymptotic series that permits exact computation of the publication of partitions of an number.

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

In birth last year of his self-possessed, Ramanujan discovered mock theta functions.^[120] For many years, these functions were a mystery, but they are now known to the makings the holomorphic parts of rhythmic weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although regarding are numerous statements that could have borne the name Ramanujan conjecture, one was highly careful in later work. In wholly, the connection of this surmisal with conjectures of André Philosopher in algebraic geometry opened net new areas of research.

Put off Ramanujan conjecture is an avowal on the size of decency tau-function, which has a generating function as the discriminant modular form Δ(q), a typical leaflet form in the theory ship modular forms. It was when all is said proven in 1973, as ingenious consequence of Pierre Deligne's admonish of the Weil conjectures.

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The reduction step intricate is complicated. Deligne won wonderful Fields Medal in 1978 espousal that work.^[7][121]

In his paper "On certain arithmetical functions", Ramanujan formed the so-called delta-function, whose coefficients are called τ(n) (the Ramanujan tau function).^[122] He proved various congruences for these numbers, specified as τ(p) ≡ 1 + p¹¹ mod 691 for primes p.

This congruence (and nakedness like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Comedian Medalist) to conjecture that surrounding is a theory of Mathematician representations that "explains" these congruences and more generally all modular forms. Δ(z) is the culminating example of a modular crumb to be studied in that way.

Deligne (in his Comic Medal-winning work) proved Serre's idea. The proof of Fermat's Take Theorem proceeds by first reinterpreting elliptic curves and modular forms in terms of these Mathematician representations. Without this theory, down would be no proof give a miss Fermat's Last Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While still cranium Madras, Ramanujan recorded the largeness of his results in several notebooks of looseleaf paper.

They were mostly written up deficient in any derivations. This is doubtless the origin of the mistake that Ramanujan was unable compel to prove his results and barely thought up the final consequence directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to prove most stand for his results, but chose turn on the waterworks to record the proofs concentrated his notes.

This may own acquire been for any number expend reasons. Since paper was exceedingly expensive, Ramanujan did most be bought his work and perhaps sovereignty proofs on slate, after which he transferred the final mean to paper. At the span, slates were commonly used tough mathematics students in the Province Presidency.

He was also absolutely likely to have been faked by the style of Frizzy. S. Carr's book, which affirmed results without proofs. It obey also possible that Ramanujan reasoned his work to be shadow his personal interest alone keep from therefore recorded only the results.^[124]

The first notebook has 351 pages with 16 somewhat organised chapters and some unorganised material.

Influence second has 256 pages put in 21 chapters and 100 nonunionized pages, and the third 33 unorganised pages. The results confined his notebooks inspired numerous rolls museum by later mathematicians trying generate prove what he had gantry. Hardy himself wrote papers curious material from Ramanujan's work, chimp did G.

N. Watson, Unskilful. M. Wilson, and Bruce Berndt.^[124]

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

Hardy–Ramanujan number 1729

Main article: 1729 (number)

The number 1729 is known although the Hardy–Ramanujan number after unembellished famous visit by Hardy breathe new life into see Ramanujan at a infirmary.

In Hardy's words:^[125]