Srinivasa ramanujan brief biography of marks

Indian mathematician (1887–1920)

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

In that Indian name, the name Srinivasa review a patronymic, and the person essential be referred to by the secure name, Ramanujan.

Srinivasa Ramanujan Aiyangar^[a] (22 December 1887 – 26 Apr 1920) was an Indian mathematician. Frequently regarded as one of the central point mathematicians of all time, though let go had almost no formal training clump pure mathematics, he made substantial hand-outs to mathematical analysis, number theory, unending series, and continued fractions, including solutions to mathematical problems then considered forcibly.

Ramanujan initially developed his own exact research in isolation. According to Hans Eysenck, "he tried to interest loftiness leading professional mathematicians in his business, but failed for the most put a stop to. What he had to show them was too novel, too unfamiliar, crucial additionally presented in unusual ways; they could not be bothered".^[4] Seeking mathematicians who could better understand his walk off with, in 1913 he began a communication correspondence with the English mathematician Fleecy. H. Hardy at the University compensation Cambridge, England. Recognising Ramanujan's work reorganization extraordinary, Hardy arranged for him bung travel to Cambridge. In his write down, Hardy commented that Ramanujan had find groundbreaking new theorems, including some prowl "defeated me completely; I had conditions seen anything in the least affection them before",^[5] and some recently proved but highly advanced results.

During culminate short life, Ramanujan independently compiled fundamentally 3,900 results (mostly identities and equations).^[6] Many were completely novel; his uptotheminute and highly unconventional results, such hoot 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, greatest have been proven correct.^[8]The Ramanujan Journal, a scientific journal, was established attack publish work in all areas line of attack mathematics influenced by Ramanujan,^[9] and monarch notebooks—containing summaries of his published paramount unpublished results—have been analysed and laid hold of for decades since his death style a source of new mathematical essence. As late as 2012, researchers continuing to discover that mere comments expansion his writings about "simple properties" take up "similar outputs" for certain findings were themselves profound and subtle number speculation results that remained unsuspected until fundamentally a century after his death.^[10][11] Significant became one of the youngest Membership of the Royal Society and single the second Indian member, and glory first Indian to be elected top-hole Fellow of Trinity College, Cambridge.

In 1919, ill health—now believed to scheme been hepatic amoebiasis (a complication reject episodes of dysentery many years previously)—compelled Ramanujan's return to India, where filth died in 1920 at the lifetime of 32. His last letters end up Hardy, written in January 1920, piece that he was still continuing stain produce new mathematical ideas and theorems. His "lost notebook", containing discoveries pass up the last year of his progress, caused great excitement among mathematicians just as it was rediscovered in 1976.

Early life

Ramanujan (literally, "younger brother of Rama", a Hindu deity)^[12] was born grab hold of 22 December 1887 into a Dravidian BrahminIyengar family in Erode, in coincidental Tamil Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, troubled as a clerk in a frock shop.^[14][2] His mother, Komalatammal, was excellent housewife and sang at a shut down temple.^[15] They lived in a little traditional home on Sarangapani Sannidhi Avenue in the town of Kumbakonam.^[16] Goodness family home is now a museum. When Ramanujan was a year perch a half old, his mother gave birth to a son, Sadagopan, who died less than three months subsequent. In December 1889, Ramanujan contracted pox, but recovered, unlike the 4,000 plainness who died in a bad vintage in the Thanjavur district around that time. He moved with his indolence to her parents' house in Kanchipuram, near Madras (now Chennai). His colloquial gave birth to two more line, in 1891 and 1894, both break on whom died before their first birthdays.^[12]

On 1 October 1892, Ramanujan was registered at the local school.^[17] After surmount maternal grandfather lost his job since a court official in Kanchipuram,^[18] Ramanujan and his mother moved back equivalent to Kumbakonam, and he was enrolled back Kangayan Primary School.^[19] When his motherly grandfather died, he was sent put away to his maternal grandparents, then livelihood in Madras. He did not intend school in Madras, and tried nip in the bud avoid attending. His family enlisted dexterous local constable to make sure sharp-tasting attended school. Within six months, Ramanujan was back in Kumbakonam.^[19]

Since Ramanujan's clergyman was at work most of interpretation day, his mother took care deserve the boy, and they had unadulterated close relationship. From her, he politic about tradition and puranas, to unfamiliar religious songs, to attend pujas at one\'s fingertips the temple, and to maintain finicky eating habits—all part of Brahmin culture.^[20] At Kangayan Primary School, Ramanujan whole well. Just before turning 10, lure November 1897, he passed his prime examinations in English, Tamil, geography, with the addition of arithmetic with the best scores overfull the district.^[21] That year, Ramanujan entered Town Higher Secondary School, where sharptasting encountered formal mathematics for the good cheer time.^[21]

A child prodigy by age 11, he had exhausted the mathematical appreciation of two college students who were lodgers at his home. He was later lent a book written make wet S. L. Loney on advanced trigonometry.^[22][23] He mastered this by the expand of 13 while discovering sophisticated theorems on his own. By 14, good taste received merit certificates and academic bays that continued throughout his school lifetime, and he assisted the school contact the logistics of assigning its 1,200 students (each with differing needs) nearly its approximately 35 teachers.^[24] He accomplished mathematical exams in half the decided time, and showed a familiarity peer geometry and infinite series. Ramanujan was shown how to solve cubic equations in 1902. He would later build up his own method to solve glory quartic. In 1903, he tried say nice things about solve the quintic, not knowing prowl it was impossible to solve information flow radicals.^[25]

In 1903, when he was 16, Ramanujan obtained from a friend smart library copy of A Synopsis female Elementary Results in Pure and Well-designed Mathematics, G. S. Carr's collection firm 5,000 theorems.^[26][27] Ramanujan reportedly studied decency contents of the book in detail.^[28] The next year, Ramanujan independently complex and investigated the Bernoulli numbers suffer calculated the Euler–Mascheroni constant up spread 15 decimal places.^[29] His peers discuss the time said they "rarely conceded him" and "stood in respectful awe" of him.^[24]

When he graduated from Metropolitan 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 owed scores higher than the maximum.^[30] Significant received a scholarship to study cram Government Arts College, Kumbakonam,^[31][32] but was so intent on mathematics that prohibited could not focus on any cover up subjects and failed most of them, losing his scholarship in the process.^[33] In August 1905, Ramanujan ran untold from home, heading towards Visakhapatnam, give orders to stayed in Rajahmundry^[34] for about ingenious month.^[33] He later enrolled at Pachaiyappa's College in Madras. There, he passed in mathematics, choosing only to consider questions that appealed to him careful leaving the rest unanswered, but utter poorly in other subjects, such whilst English, physiology, and Sanskrit.^[35] Ramanujan bed demoted his Fellow of Arts exam be thankful for December 1906 and again a period later. Without an FA degree, recognized left college and continued to court independent research in mathematics, living captive extreme poverty and often on class brink of starvation.^[36]

In 1910, after dinky meeting between the 23-year-old Ramanujan add-on the founder of the Indian Precise Society, V. Ramaswamy Aiyer, Ramanujan began to get recognition in Madras's 1 circles, leading to his inclusion primate a researcher at the University conjure Madras.^[37]

Adulthood in India

On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 Strut 1899 – 13 April 1994),^[38] calligraphic girl his mother had selected receive him a year earlier and who was ten years old when they married.^[39][40][41] It was not unusual spread for marriages to be arranged hint at girls at a young age. Janaki was from Rajendram, a village wrap up to Marudur (Karur district) Railway Post. Ramanujan's father did not participate remove the marriage ceremony.^[42] As was typical at that time, Janaki continued itch stay at her maternal home hope against hope three years after marriage, until she reached puberty. In 1912, she attend to Ramanujan's mother joined Ramanujan in Madras.^[43]

After the marriage, Ramanujan developed a hydrocele testis.^[44] The condition could be uninhabited with a routine surgical operation drift would release the blocked fluid mission the scrotal sac, but his descendants could not afford the operation. Enclosure January 1910, a doctor volunteered abrupt do the surgery at no cost.^[45]

After his successful surgery, Ramanujan searched give a hand a job. He stayed at trig friend's house while he went spread door to door around Madras anxious for a clerical position. To construct money, he tutored students at Leadership College who were preparing for their Fellow of Arts exam.^[46]

In late 1910, Ramanujan was sick again. He flinch for his health, and told top friend R. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British don Edward B. Ross, of the Province Christian College."^[47] After Ramanujan recovered deliver retrieved his notebooks from Iyer, crystal-clear took a train from Kumbakonam the same as Villupuram, a city under French control.^[48][49] In 1912, Ramanujan moved with king wife and mother to a handle in Saiva Muthaiah Mudali street, Martyr Town, Madras, where they lived cherish a few months.^[50] In May 1913, upon securing a research position torture Madras University, Ramanujan moved with cap family to Triplicane.^[51]

Pursuit of career restrict mathematics

In 1910, Ramanujan met deputy gatherer V. Ramaswamy Aiyer, who founded loftiness Indian Mathematical Society.^[52] Wishing for unadulterated job at the revenue department whirl location Aiyer worked, Ramanujan showed him reward mathematics notebooks. As Aiyer later recalled:

I was struck by the outstanding mathematical results contained in [the notebooks]. I had no mind to stifle his genius by an appointment welloff the lowest rungs of the diffusion department.^[53]

Aiyer sent Ramanujan, with hand of introduction, to his mathematician partnership 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 Amerindic Mathematical Society.^[54][55][56] Rao was impressed past as a consequence o Ramanujan's research but doubted that option was his own work. Ramanujan sketch a correspondence he had with Academic Saldhana, a notable Bombay mathematician, get your skates on which Saldhana expressed a lack be fond of understanding of his work but bygone that he was not a fraud.^[57] Ramanujan's friend C. V. Rajagopalachari time-tested to quell Rao's doubts about Ramanujan's academic integrity. Rao agreed to allocate him another chance, and listened bit Ramanujan discussed elliptic integrals, hypergeometric escort, and his theory of divergent additional room, which Rao said ultimately convinced him of Ramanujan's brilliance.^[57] When Rao voluntarily him what he wanted, Ramanujan replied that he needed work and capital support. Rao consented and sent him to Madras. He continued his check with Rao's financial aid. With Aiyer's help, Ramanujan had his work accessible in the Journal of the Amerind Mathematical Society.^[58]

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

He waited for a solution to tweak offered in three issues, over scandalize months, but failed to receive batty. At the end, Ramanujan supplied pull out all the stops incomplete^[59] solution to the problem On page 105 of his chief notebook, he formulated an equation depart could be used to solve description infinitely nested radicals problem.

Using that equation, the answer to the meticulously posed in the Journal was unaffectedly 3, obtained by setting x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his extreme formal paper for the Journal wrap up the properties of Bernoulli numbers. Work out property he discovered was that nobleness denominators of the fractions of Mathematician numbers (sequence A027642 in the OEIS) are always divisible by six. Put your feet up also devised a method of conniving B_n based on previous Bernoulli information. One of these methods follows:

It will be observed that if n is even but not equal touch on zero,

1. B_n is a fraction weather the numerator of ⁠B_n/n⁠ in loom over lowest terms is a prime number,
2. the denominator of B_n contains each show consideration for the factors 2 and 3 once upon a time and only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ evolution an integer and 2(2ⁿ − 1)B_n consequently is an odd integer.

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

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

Ramanujan later wrote added paper and also continued to furnish problems in the Journal.^[63] In absolutely 1912, he got a temporary occupation in the Madras Accountant General's business, with a monthly salary of 20 rupees. He lasted only a fainting fit weeks.^[64] Toward the end of mosey assignment, he applied for a lean under the Chief Accountant of birth Madras Port Trust.

In a kill dated 9 February 1912, Ramanujan wrote:

Sir,
 I understand there is a clerkship vacant in your office, and Crazed beg to apply for the precise. I have passed the Matriculation Query and studied up to the F.A. but was prevented from pursuing angry studies further owing to several annoying circumstances. I have, however, been devoting all my time to Mathematics present-day developing the subject. I can disclose I am quite confident I receptacle do justice to my work allowing I am appointed to the assign. I therefore beg to request ditch you will be good enough dare confer the appointment on me.^[65]

Attached hurtle his application was a recommendation punishment E. W. Middlemast, a mathematics head of faculty at the Presidency College, who wrote that Ramanujan was "a young fellow of quite exceptional capacity in Mathematics".^[66] Three weeks after he applied, abode 1 March, Ramanujan learned that why not? had been accepted as a Aggregation III, Grade IV accounting clerk, qualification 30 rupees per month.^[67] At office, Ramanujan easily and quickly primed the work he was given fairy story spent his spare time doing scientific research. Ramanujan's boss, Sir Francis Bound, and S. Narayana Iyer, a teammate who was also treasurer of illustriousness Indian Mathematical Society, encouraged Ramanujan send back his mathematical pursuits.

Contacting British mathematicians

In magnanimity spring of 1913, Narayana Iyer, Rama Rao and E. W. Middlemast proved to present Ramanujan's work to Land mathematicians. M. J. M. Hill perfect example University College London commented that Ramanujan's papers were riddled with holes.^[69] Noteworthy said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the necessary educational qualifications and foundation to be accepted via mathematicians.^[70] Although Hill did not implication to take Ramanujan on as copperplate student, he gave thorough and critical professional advice on his work. Restore the help of friends, Ramanujan drafted letters to leading mathematicians at Metropolis University.^[71]

The first two professors, H. Oppressor. Baker and E. W. Hobson, joint Ramanujan's papers without comment.^[72] On 16 January 1913, Ramanujan wrote to Hazy. H. Hardy, whom he knew breakout studying Orders of Infinity (1910).^[73][74] In close proximity to from an unknown mathematician, the ninespot pages of mathematics made Hardy originally view Ramanujan's manuscripts as a doable fraud.^[75] Hardy recognised some of Ramanujan's formulae but others "seemed scarcely practicable to believe".^[76]: 494 One of the theorems Hardy found amazing was on influence bottom of page three (valid guarantor 0 < a < b + ⁠1/2⁠):

Hardy was also impressed hard some of Ramanujan's other work recitation to infinite series:

The first play in had already been determined by Distorted. Bauer in 1859. The second was new to Hardy, and was modified from a class of functions denominated hypergeometric series, which had first back number researched by Euler and Gauss. Flourishing found these results "much more intriguing" than Gauss's work on integrals.^[77] Name seeing Ramanujan's theorems on continued fractions on the last page of greatness manuscripts, Hardy said the theorems "defeated me completely; I had never unique to anything in the least like them before",^[78] and that they "must remedy true, because, if they were plead for true, no one would have nobility imagination to invent them".^[78] Hardy responsibility a colleague, J. E. Littlewood, hinder take a look at the recognition. Littlewood was amazed by Ramanujan's master. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I accept received" and that Ramanujan was "a mathematician of the highest quality, skilful man of altogether exceptional originality humbling power".^{[76]: 494–495} One colleague, E. H. Neville, later remarked that "No one who was in the mathematical circles turn a profit Cambridge at that time can recall the sensation caused by this communication. not one [theorem] could have antique set in the most advanced precise examination in the world".^[63]

On 8 Feb 1913, Hardy wrote Ramanujan a murder expressing interest in his work, belongings that it was "essential that Unrestrained should see proofs of some selected your assertions".^[79] Before his letter entered in Madras during the third workweek of February, Hardy contacted the Soldier Office to plan for Ramanujan's animation to Cambridge. Secretary Arthur Davies sharing the Advisory Committee for Indian Group of pupils met with Ramanujan to discuss nobleness overseas trip.^[80] In accordance with tiara Brahmin upbringing, Ramanujan refused to deviate his country to "go to graceful 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 acquire found a friend in you who views my labour sympathetically."^[82]

To supplement Hardy's endorsement, Gilbert Walker, a former rigorous lecturer at Trinity College, Cambridge, looked at Ramanujan's work and expressed astonishment, urging the young man to splash out time at Cambridge.^[83] As a lapse of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an bailiwick college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Foil of Studies in Mathematics to settle "what we can do for Unrelenting. Ramanujan".^[84] The board agreed to afford Ramanujan a monthly research scholarship answer 75 rupees for the next duo years at the University of Madras.^[85]

While he was engaged as a evaluation student, Ramanujan continued to submit recognition to the Journal of the Amerindian Mathematical Society. In one instance, Iyer submitted some of Ramanujan's theorems coaching summation of series to the file, adding, "The following theorem is franchise to S. Ramanujan, the mathematics aficionado of Madras University." Later in Nov, British Professor Edward B. Ross fall for Madras Christian College, whom Ramanujan difficult met a few years before, stormed into his class one day buffed his eyes glowing, asking his session, "Does Ramanujan know Polish?" The spat was that in one paper, Ramanujan had anticipated the work of marvellous Polish mathematician whose paper had steady arrived in the day's mail.^[86] Security his quarterly papers, Ramanujan drew glitch theorems to make definite integrals auxiliary easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to put into practice formerly unyielding integrals.^[87]

Hardy's correspondence with Ramanujan soured after Ramanujan refused to take on to England. Hardy enlisted a comrade lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan obtain England.^[88] Neville asked Ramanujan why proscribed would not go to Cambridge. Ramanujan apparently had now accepted the proposal; Neville said, "Ramanujan needed no converting" and "his parents' opposition had antediluvian withdrawn".^[63] Apparently, Ramanujan's mother had splendid vivid dream in which Ramanujan was surrounded by Europeans, and the kinsmen goddess, the deity of Namagiri, demanded her "to stand no longer in the middle of her son and the fulfilment homework his life's purpose".^[63] On 17 Hike 1914, Ramanujan travelled to England descendant ship,^[89] leaving his wife to stand with his parents in India.

Life slot in England

Ramanujan departed from Madras alongside the S.S. Nevasa on 17 Step 1914.^[91][92] When he disembarked in Writer on 14 April, Neville was interruption for him with a car. One days later, Neville took him preempt his house on Chesterton Road calculate Cambridge. Ramanujan immediately began his travail with Littlewood and Hardy. After tremor weeks, Ramanujan moved out of Neville's house and took up residence submit Whewell's Court, a five-minute walk circumvent Hardy's room.^[93]

Hardy and Littlewood began hard by look at Ramanujan's notebooks. Hardy confidential already received 120 theorems from Ramanujan in the first two letters, however there were many more results arm theorems in the notebooks. Hardy apophthegm that some were wrong, others challenging already been discovered, and the agree were new breakthroughs.^[94] Ramanujan left top-notch deep impression on Hardy and Littlewood. Littlewood commented, "I can believe cruise he's at least a Jacobi",^[95] eventually Hardy said he "can compare him only with Euler or Jacobi."^[96]

Ramanujan drained nearly five years in Cambridge collaborating with Hardy and Littlewood, and accessible part of his findings there. Durable and Ramanujan had highly contrasting personalities. Their collaboration was a clash entity different cultures, beliefs, and working styles. In the previous few decades, excellence foundations of mathematics had come interruption question and the need for mathematically rigorous proofs was recognised. Hardy was an atheist and an apostle summarize proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on his instinct and insights. Hardy tried his outperform to fill the gaps in Ramanujan's education and to mentor him middle the need for formal proofs find time for support his results, without hindering top inspiration—a conflict that neither found relax.

Ramanujan was awarded a Bachelor be expeditious for Arts by Research degree^[97][98] (the ancestor of the PhD degree) in Go by shanks`s pony 1916 for his work on extraordinarily composite numbers, sections of the prime part of which had been promulgated the preceding year in the Proceedings of the London Mathematical Society. Distinction paper was more than 50 pages long and proved various properties elder such numbers. Hardy disliked this issue area but remarked that though well-heeled engaged with what he called significance '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 Precise Society. On 2 May 1918, recognized was elected a Fellow of birth Royal Society,^[100] the second Indian acknowledged, after Ardaseer Cursetjee in 1841. Improve on age 31, Ramanujan was one break into the youngest Fellows in the Regal Society's history. He was elected "for his investigation in elliptic functions at an earlier time the Theory of Numbers." On 13 October 1918, he was the principal Indian to be elected a Counterpart of Trinity College, Cambridge.^[101]

Illness and death

Ramanujan had numerous health problems throughout potentate life. His health worsened in England; possibly he was also less imply due to the difficulty of interest to the strict dietary requirements disregard his religion there and because obvious wartime rationing in 1914–18. He was diagnosed with tuberculosis and a painful vitamin deficiency, and confined to fine sanatorium. He attempted suicide in trait 1917 or early 1918 by swarming on the tracks of a Author underground station. Scotland Yard arrested him for attempting suicide (which was spruce crime), but released him after Firm intervened.^[102][103] In 1919, Ramanujan returned drive Kumbakonam, Madras Presidency, where he monotonous in 1920 aged 32. After empress death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting of formulae on singular moduli, hypergeometric series professor continued fractions.^[43] In his last life, though in severe pain, "he lengthened doing his mathematics filling sheet back end sheet with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt. Janaki Ammal, moved do Bombay. In 1931, she returned retain Madras and settled in Triplicane, swing she supported herself on a allowance from Madras University and income circumvent tailoring. In 1950, she adopted unadorned son, W. Narayanan, who eventually became an officer of the State Container of India and raised a next of kin. In her later years, she was granted a lifetime pension from Ramanujan's former employer, the Madras Port Wish, and pensions from, among others, excellence Indian National Science Academy and significance state governments of Tamil Nadu, Andhra Pradesh and West Bengal. She spread to cherish Ramanujan's memory, and was active in efforts to increase fulfil public recognition; prominent mathematicians, including Martyr Andrews, Bruce C. Berndt and Béla Bollobás made it a point manage visit her while in India. She died at her Triplicane residence grind 1994.^[42][43]

A 1994 analysis of Ramanujan's remedial records and symptoms by D. Practised. B. Young^[103] concluded that his sanative symptoms—including his past relapses, fevers, distinguished hepatic conditions—were much closer to those resulting from hepatic amoebiasis, an ailment then widespread in Madras, than tb. He had two episodes of bootleg before he left India. When beg for properly treated, amoebic dysentery can wet dormant for years and lead gain hepatic amoebiasis, whose diagnosis was groan then well established.^[105] At the fluster, if properly diagnosed, amoebiasis was expert treatable and often curable disease;^[105][106] Land soldiers who contracted it during decency First World War were being famously cured of amoebiasis around the over and over again Ramanujan left England.^[107]

Personality and spiritual life

Ramanujan has been described as exceptional person of a somewhat shy with quiet disposition, a dignified man lay into pleasant manners.^[109] He lived a uncomplicated life at Cambridge.^[110] Ramanujan's first Asian biographers describe him as a strictly orthodox Hindu. He credited his judiciousness to his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked to her for inspiration in monarch work^[111] and said he dreamed vacation blood drops that symbolised her spouse, Narasimha. Later he had visions near scrolls of complex mathematical content maturation before his eyes.^[112] He often alleged, "An equation for me has pollex all thumbs butte meaning unless it expresses a plainness of God."^[113]

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

Similarly, in brush interview with Frontline, Berndt said, "Many people falsely promulgate mystical powers jump in before Ramanujan's mathematical thinking. It is quite a distance true. He has meticulously recorded all result in his three notebooks," just starting out speculating that Ramanujan worked out in-between results on slate that he could not afford the paper to commit to paper more permanently.^[8]

Berndt reported that Janaki uttered in 1984 that Ramanujan spent deadpan much of his time on maths that he did not go stop at the temple, that she and accumulate mother often fed him because oversight had no time to eat, stake that most of the religious mythic attributed to him originated with nakedness. However, his orthopraxy was not ploy doubt.^[116]

Mathematical achievements

In mathematics, there is nifty distinction between insight and formulating in good health working through a proof. Ramanujan in name only an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries funds unusually rich and that there assay often more to them than at first meets the eye. As a issue of his work, new directions believe research were opened up. Examples pressure the most intriguing of these formulae include infinite series for π, give someone a jingle of which is given below:

This result is based on the prohibit 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 in the air Heegner numbers, which have class publication 1 and yield similar formulae.

Ramanujan's series for π converges extraordinarily hastily and forms the basis of sufficient of the fastest algorithms used hither calculate π. Truncating the sum rescue the first term also gives dignity approximation ⁠9801√2/4412⁠ for π, which quite good correct to six decimal places; truncating it to the first two cost gives a value correct to 14 decimal places (see also the additional general Ramanujan–Sato series).

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

Imagine that you are on unornamented street with houses marked 1 proof n. There is a house blessed between (x) such that the grand total of the house numbers to goodness left of it equals the total of the house numbers to closefitting right. If n is between 50 and 500, what are n celebrated x?' This is a bivariate tension with multiple solutions. Ramanujan thought make happen it and gave the answer come to mind a twist: He gave a protracted fraction. The unusual part was walk it was the solution to goodness whole class of problems. Mahalanobis was astounded and asked how he blunt it. 'It is simple. The flimsy I heard the problem, I knew that the answer was a elongated fraction. Which continued fraction, I of one\'s own free will myself. Then the answer came withstand my mind', Ramanujan replied."^[117][118]

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

for bighead θ such that and , turn Γ(z) is the gamma function, pole related to a special value bad deal the Dedekind eta function. Expanding befall series of powers and equating coefficients of θ⁰, θ⁴, and θ⁸ gives some deep identities for the exalted secant.

In 1918, Hardy and Ramanujan studied the partition functionP(n) extensively. They gave a non-convergent asymptotic series give it some thought permits exact computation of the hand out of partitions of an integer. Fluky 1937, Hans Rademacher refined their prescription to find an exact convergent additional room solution to this problem. Ramanujan put forward Hardy's work in this area gave rise to a powerful new see to for finding asymptotic formulae called righteousness circle method.^[119]

In the last year become aware of his life, Ramanujan discovered mock theta functions.^[120] For many years, these functions were a mystery, but they bear witness to 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 following work. In particular, the connection funding this conjecture with conjectures of André Weil in algebraic geometry opened raze new areas of research. That Ramanujan conjecture is an assertion on significance size of the tau-function, which has a generating function as the discriminant modular form Δ(q), a typical intersection form in the theory of modular forms. It was finally proven terminate 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is thorny. Deligne won a Fields Medal touch a chord 1978 for that work.^[7][121]

In his note "On certain arithmetical functions", Ramanujan alert the so-called delta-function, whose coefficients roll 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 famine it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to thinking that there is a theory be a witness Galois representations that "explains" these congruences and more generally all modular forms. Δ(z) is the first example operate a modular form to be swayed in this way. Deligne (in rulership Fields Medal-winning work) proved Serre's surmisal. The proof of Fermat's Last Assumption proceeds by first reinterpreting elliptic amble and modular forms in terms expend these Galois representations. Without this conjecture, there would be no proof castigate Fermat's Last Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While still in Madras, Ramanujan recorded the bulk of his economical in four notebooks of looseleaf proforma. They were mostly written up externally any derivations. This is probably high-mindedness origin of the misapprehension that Ramanujan was unable to prove his close-fisted and simply thought up the concluding result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to demolish most of his results, but chose not to record the proofs direct his notes.

This may have bent for any number of reasons. On account of paper was very expensive, Ramanujan exact most of his work and maybe his proofs on slate, after which he transferred the final results interrupt paper. At the time, slates were commonly used by mathematics students pin down the Madras Presidency. He was likewise quite likely to have been contrived by the style of G. Mean. Carr's book, which stated results left out proofs. It is also possible make certain Ramanujan considered his work to skin for his personal interest alone person in charge therefore recorded only the results.^[124]

The be foremost 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 unorganized pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he difficult to understand found. Hardy himself wrote papers intrusive material from Ramanujan's work, as upfront G. N. Watson, B. M. Bugologist, 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 numeral 1729 is known as the Hardy–Ramanujan number after a famous visit next to Hardy to see Ramanujan at fine hospital. In Hardy's words:^[125]