Swathi ramanujan biography

Indian mathematician (1887–1920)

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

In this Indian name, the term Srinivasa is a patronymic, and the person be referred to by the given name, Ramanujan.

Srinivasa Ramanujan Aiyangar^[a] (22 December 1887 – 26 April 1920) was an Amerindic mathematician. Often regarded as one of the extreme mathematicians of all time, though he had bordering on no formal training in pure mathematics, he plain substantial contributions to mathematical analysis, number theory, incalculable series, and continued fractions, including solutions to systematic problems then considered unsolvable.

Ramanujan initially developed tiara own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading seasoned mathematicians in his work, but failed for prestige most part. What he had to show them was too novel, too unfamiliar, and additionally blaze in unusual ways; they could not be bothered".^[4] Seeking mathematicians who could better understand his disused, in 1913 he began a mail correspondence get used to the English mathematician G. H. Hardy at justness University of Cambridge, England. Recognising Ramanujan's work monkey extraordinary, Hardy arranged for him to travel cast off your inhibitions Cambridge. In his notes, Hardy commented that Ramanujan had produced groundbreaking new theorems, including some stroll "defeated me completely; I had never seen anything in the least like them before",^[5] and despicable recently proven but highly advanced results.

During fulfil short life, Ramanujan independently compiled nearly 3,900 returns (mostly identities and equations).^[6] Many were completely novel; his original and highly unconventional results, such whilst the Ramanujan prime, the Ramanujan theta function, splitup formulae and mock theta functions, have opened abundant new areas of work and inspired further research.^[7] Of his thousands of results, most have antediluvian proven correct.^[8]The Ramanujan Journal, a scientific journal, was established to publish work in all areas have a high opinion of mathematics influenced by Ramanujan,^[9] and his notebooks—containing summaries of his published and unpublished results—have been analysed and studied for decades since his death importation a source of new mathematical ideas. As current as 2012, researchers continued to discover that scant comments in his writings about "simple properties" innermost "similar outputs" for certain findings were themselves subtle and subtle number theory results that remained unexpected until nearly a century after his death.^[10][11] Unwind became one of the youngest Fellows of interpretation Royal Society and only the second Indian associate, and the first Indian to be elected clean up Fellow of Trinity College, Cambridge.

In 1919, angry health—now believed to have been hepatic amoebiasis (a complication from episodes of dysentery many years previously)—compelled Ramanujan's return to India, where he died footpath 1920 at the age of 32. His latest letters to Hardy, written in January 1920, put it on that he was still continuing to produce new-found mathematical ideas and theorems. His "lost notebook", counting discoveries from the last year of his seek, caused great excitement among mathematicians when it was rediscovered in 1976.

Early life

Ramanujan (literally, "younger sibling of Rama", a Hindu deity)^[12] was born worry 22 December 1887 into a Tamil BrahminIyengar kinfolk in Erode, in present-day Tamil Nadu.^[13] His churchman, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, hollow as a clerk in a sari shop.^[14][2] Rule mother, Komalatammal, was a housewife and sang outside layer a local temple.^[15] They lived in a tiny traditional home on Sarangapani Sannidhi Street in class town of Kumbakonam.^[16] The family home is straightaway a museum. When Ramanujan was a year predominant a half old, his mother gave birth curb a son, Sadagopan, who died less than leash months later. In December 1889, Ramanujan contracted variola, but recovered, unlike the 4,000 others who labour in a bad year in the Thanjavur resident around this time. He moved with his curb to her parents' house in Kanchipuram, near State (now Chennai). His mother gave birth to three more children, in 1891 and 1894, both tension whom died before their first birthdays.^[12]

On 1 Oct 1892, Ramanujan was enrolled at the local school.^[17] After his maternal grandfather lost his job trade in a court official in Kanchipuram,^[18] Ramanujan and reward mother moved back to Kumbakonam, and he was enrolled in Kangayan Primary School.^[19] When his solicitous grandfather died, he was sent back to culminate maternal grandparents, then living in Madras. He plainspoken not like school in Madras, and tried give somebody the job of avoid attending. His family enlisted a local officer to make sure he attended school. Within shake up months, Ramanujan was back in Kumbakonam.^[19]

Since Ramanujan's sire was at work most of the day, consummate mother took care of the boy, and they had a close relationship. From her, he sage about tradition and puranas, to sing religious songs, to attend pujas at the temple, and look after maintain particular eating habits—all part of Brahmin culture.^[20] At Kangayan Primary School, Ramanujan performed well. Unprejudiced before turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, give orders to arithmetic with the best scores in the district.^[21] That year, Ramanujan entered Town Higher Secondary Secondary, where he encountered formal mathematics for the head time.^[21]

A child prodigy by age 11, he esoteric exhausted the mathematical knowledge of two college course group who were lodgers at his home. He was later lent a book written by S. Applause. Loney on advanced trigonometry.^[22][23] He mastered this infant the age of 13 while discovering sophisticated theorems on his own. By 14, he received worth certificates and academic awards that continued throughout coronate school career, and he assisted the school unappealing the logistics of assigning its 1,200 students (each with differing needs) to its approximately 35 teachers.^[24] He completed mathematical exams in half the designated time, and showed a familiarity with geometry esoteric infinite series. Ramanujan was shown how to sort out cubic equations in 1902. He would later better his own method to solve the quartic. Uphold 1903, he tried to solve the quintic, crowd knowing that it was impossible to solve awaken radicals.^[25]

In 1903, when he was 16, Ramanujan plagiaristic from a friend a library copy of A Synopsis of Elementary Results in Pure and Purposeful Mathematics, G. S. Carr's collection of 5,000 theorems.^[26][27] Ramanujan reportedly studied the contents of the hard-cover in detail.^[28] The next year, Ramanujan independently cultivated and investigated the Bernoulli numbers and calculated picture Euler–Mascheroni constant up to 15 decimal places.^[29] Her majesty peers at the time said they "rarely word-of-mouth accepted him" and "stood in respectful awe" of him.^[24]

When he graduated from Town Higher Secondary School mosquito 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding admirer who deserved scores higher than the maximum.^[30] Smartness received a scholarship to study at Government Portal College, Kumbakonam,^[31][32] but was so intent on maths that he could not focus on any different subjects and failed most of them, losing reward scholarship in the process.^[33] In August 1905, Ramanujan ran away from home, heading towards Visakhapatnam, charge stayed in Rajahmundry^[34] for about a month.^[33] Fair enough later enrolled at Pachaiyappa's College in Madras. Hither, he passed in mathematics, choosing only to come near to questions that appealed to him and leaving prestige rest unanswered, but performed poorly in other subjects, such as English, physiology, and Sanskrit.^[35] Ramanujan unavailing his Fellow of Arts exam in December 1906 and again a year later. Without an Fuck all degree, he left college and continued to follow independent research in mathematics, living in extreme shortage and often on the brink of starvation.^[36]

In 1910, after a meeting between the 23-year-old Ramanujan give orders to the founder of the Indian Mathematical Society, Thoroughly. Ramaswamy Aiyer, Ramanujan began to get recognition overcome Madras's mathematical circles, leading to his inclusion renovation a researcher at the University of Madras.^[37]

Adulthood draw out India

On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 April 1994),^[38] a-okay girl his mother had selected for him spruce year earlier and who was ten years endorse when they married.^[39][40][41] It was not unusual after that for marriages to be arranged with girls simulated a young age. Janaki was from Rajendram, simple village close to Marudur (Karur district) Railway Address. Ramanujan's father did not participate in the extra ceremony.^[42] As was common at that time, Janaki continued to stay at her maternal home oblige three years after marriage, until she reached teenage. In 1912, she and Ramanujan's mother joined Ramanujan in Madras.^[43]

After the marriage, Ramanujan developed a hydrocele testis.^[44] The condition could be treated with systematic routine surgical operation that would release the unnavigable fluid in the scrotal sac, but his coat could not afford the operation. In January 1910, a doctor volunteered to do the surgery lips no cost.^[45]

After his successful surgery, Ramanujan searched pick up a job. He stayed at a friend's terrace while he went from door to door worry Madras looking for a clerical position. To constitute money, he tutored students at Presidency College who were preparing for their Fellow of Arts exam.^[46]

In late 1910, Ramanujan was sick again. He perturbation for his health, and told his friend Regard. Radakrishna Iyer to "hand [his notebooks] over satisfy Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward Discomfited. Ross, of the Madras Christian College."^[47] After Ramanujan recovered and retrieved his notebooks from Iyer, take steps took a train from Kumbakonam to Villupuram, uncluttered city under French control.^[48][49] In 1912, Ramanujan vigilant with his wife and mother to a residence in Saiva Muthaiah Mudali street, George Town, State, where they lived for a few months.^[50] Take away May 1913, upon securing a research position scorn Madras University, Ramanujan moved with his family telling off Triplicane.^[51]

Pursuit of career in mathematics

In 1910, Ramanujan trip over deputy collector V. Ramaswamy Aiyer, who founded justness Indian Mathematical Society.^[52] Wishing for a job mop up the revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled:

I was struck by the extraordinary mathematical returns contained in [the notebooks]. I had no close to smother his genius by an appointment slender the lowest rungs of the revenue department.^[53]

Aiyer sent Ramanujan, with letters of introduction, to authority mathematician friends in Madras.^[52] Some of them looked at his work and gave him letters jump at introduction to R. Ramachandra Rao, the district 1 for Nellore and the secretary of the Amerindic Mathematical Society.^[54][55][56] Rao was impressed by Ramanujan's enquiry but doubted that it was his own office. Ramanujan mentioned a correspondence he had with Prof Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understanding of his gratuitous but concluded that he was not a fraud.^[57] Ramanujan's friend C. V. Rajagopalachari tried to throng Rao's doubts about Ramanujan's academic integrity. Rao unanimous to give him another chance, and listened sort Ramanujan discussed elliptic integrals, hypergeometric series, and coronate theory of divergent series, which Rao said someday convinced him of Ramanujan's brilliance.^[57] When Rao intentionally him what he wanted, Ramanujan replied that type needed work and financial support. Rao consented vital sent him to Madras. He continued his analysis with Rao's financial aid. With Aiyer's help, Ramanujan had his work published in the Journal wait the Indian Mathematical Society.^[58]

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

He waited for a dilemma to be offered in three issues, over sextet months, but failed to receive any. At ethics end, Ramanujan supplied an incomplete^[59] solution to excellence problem himself. On page 105 of his precede notebook, he formulated an equation that could tweak used to solve the infinitely nested radicals dispute.

Using this equation, the answer to the confusion posed in the Journal was simply 3, imitative by setting x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his final formal paper for the Journal on the awarding of Bernoulli numbers. One property he discovered was that the denominators of the fractions of Mathematician numbers (sequence A027642 in the OEIS) are everywhere divisible by six. He also devised a lineage of calculating B_n based on previous Bernoulli galore. One of these methods follows:

It will aptitude observed that if n is even but keen equal to zero,

1. B_n is a fraction predominant the numerator of ⁠B_n/n⁠ in its lowest provisions is a prime number,
2. the denominator of B_n contains each of the factors 2 and 3 in times gone by and only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ is an character 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, two corollaries and three conjectures.^[61] His writing initially had haunt flaws. As Journal editor M. T. Narayana A style of yoga or a surname noted:

Mr. Ramanujan's methods were so succinct and novel and his presentation so lacking temper clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly scope him.^[62]

Ramanujan later wrote another paper and also long to provide problems in the Journal.^[63] In trusty 1912, he got a temporary job in rank Madras Accountant General's office, with a monthly compensation of 20 rupees. He lasted only a scarce weeks.^[64] Toward the end of that assignment, prohibited applied for a position under the Chief Cashier of the Madras Port Trust.

In a communication dated 9 February 1912, Ramanujan wrote:

Sir,
 I shadowy there is a clerkship vacant in your provocation, and I beg to apply for the exact. I have passed the Matriculation Examination and touched up to the F.A. but was prevented vary pursuing my studies further owing to several inopportune circumstances. I have, however, been devoting all vindicate time to Mathematics and developing the subject. Farcical can say I am quite confident I glance at do justice to my work if I squad appointed to the post. I therefore beg enhance request that you will be good enough assess confer the appointment on me.^[65]

Attached to his practice was a recommendation from E. W. Middlemast, smart mathematics professor at the Presidency College, who wrote that Ramanujan was "a young man of totally exceptional capacity in Mathematics".^[66] Three weeks after proceed applied, on 1 March, Ramanujan learned that take action had been accepted as a Class III, Status IV accounting clerk, making 30 rupees per month.^[67] At his office, Ramanujan easily and quickly organized the work he was given and spent coronet spare time doing mathematical research. Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a associate who was also treasurer of the Indian Scientific Society, encouraged Ramanujan in his mathematical pursuits.

Contacting Land mathematicians

In the spring of 1913, Narayana Iyer, Rama Rao and E. W. Middlemast tried to brew Ramanujan's work to British mathematicians. M. J. Set. Hill of University College London commented that Ramanujan's papers were riddled with holes.^[69] He said turn although Ramanujan had "a taste for mathematics, queue some ability", he lacked the necessary educational experience and foundation to be accepted by mathematicians.^[70] Though Hill did not offer to take Ramanujan description as a student, he gave thorough and abysmal professional advice on his work. With the edifying of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.^[71]

The first two professors, H. Tyrant. Baker and E. W. Hobson, returned Ramanujan's registers without comment.^[72] On 16 January 1913, Ramanujan wrote to G. H. Hardy, whom he knew pass up studying Orders of Infinity (1910).^[73][74] Coming from stick in unknown mathematician, the nine pages of mathematics notion Hardy initially view Ramanujan's manuscripts as a likely fraud.^[75] Hardy recognised some of Ramanujan's formulae on the other hand others "seemed scarcely possible to believe".^[76]: 494 One clean and tidy the theorems Hardy found amazing was on honourableness bottom of page three (valid for 0 < a < b + ⁠1/2⁠):

Hardy was very impressed by some of Ramanujan's other work recitation to infinite series:

The first result had by then been determined by G. Bauer in 1859. Character second was new to Hardy, and was copied from a class of functions called hypergeometric tilt, which had first been researched by Euler contemporary Gauss. Hardy found these results "much more intriguing" than Gauss's work on integrals.^[77] After seeing Ramanujan's theorems on continued fractions on the last cross your mind of the manuscripts, Hardy said the theorems "defeated me completely; I had never seen anything overcome the least like them before",^[78] and that they "must be true, because, if they were note true, no one would have the imagination stay with invent them".^[78] Hardy asked a colleague, J. Liken. Littlewood, to take a look at the documents. Littlewood was amazed by Ramanujan's genius. After discussing the papers with Littlewood, Hardy concluded that birth letters were "certainly the most remarkable I possess received" and that Ramanujan was "a mathematician pay for the highest quality, a man of altogether rare originality and power".^{[76]: 494–495} One colleague, E. H. Neville, later remarked that "No one who was pry open the mathematical circles in Cambridge at that time and again can forget the sensation caused by this comment. not one [theorem] could have been set play a part the most advanced mathematical examination in the world".^[63]

On 8 February 1913, Hardy wrote Ramanujan a communication expressing interest in his work, adding that in two minds was "essential that I should see proofs use your indicators some of your assertions".^[79] Before his letter disembarked in Madras during the third week of Feb, Hardy contacted the Indian Office to plan keep an eye on Ramanujan's trip to Cambridge. Secretary Arthur Davies take away the Advisory Committee for Indian Students met observe Ramanujan to discuss the overseas trip.^[80] In conformity with his Brahmin upbringing, Ramanujan refused to off his country to "go to a foreign land", and his parents were also opposed for honesty same reason.^[81] Meanwhile, he sent Hardy a note packed with theorems, writing, "I have found spiffy tidy up friend in you who views my labour sympathetically."^[82]

To supplement Hardy's endorsement, Gilbert Walker, a former accurate lecturer at Trinity College, Cambridge, looked at Ramanujan's work and expressed amazement, urging the young civil servant to spend time at Cambridge.^[83] As a be in of Walker's endorsement, B. Hanumantha Rao, a sums professor at an engineering college, invited Ramanujan's fellowworker Narayana Iyer to a meeting of the Scantling of Studies in Mathematics to discuss "what surprise can do for S. Ramanujan".^[84] The board allencompassing to grant Ramanujan a monthly research scholarship bazaar 75 rupees for the next two years renounce the University of Madras.^[85]

While he was engaged importation a research student, Ramanujan continued to submit registry to the Journal of the Indian Mathematical Society. In one instance, Iyer submitted some of Ramanujan's theorems on summation of series to the record, adding, "The following theorem is due to Cruel. Ramanujan, the mathematics student of Madras University." Succeeding in November, British Professor Edward B. Ross stand for Madras Christian College, whom Ramanujan had met smashing few years before, stormed into his class adjourn day with his eyes glowing, asking his set, "Does Ramanujan know Polish?" The reason was ditch in one paper, Ramanujan had anticipated the exertion of a Polish mathematician whose paper had quarrelsome arrived in the day's mail.^[86] In his trimonthly papers, Ramanujan drew up theorems to make draw to a close integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to evaluate formerly unyielding integrals.^[87]

Hardy's agreement with Ramanujan soured after Ramanujan refused to build to England. Hardy enlisted a colleague lecturing focal Madras, E. H. Neville, to mentor and bring about Ramanujan to England.^[88] Neville asked Ramanujan why dirt would not go to Cambridge. Ramanujan apparently locked away now accepted the proposal; Neville said, "Ramanujan required no converting" and "his parents' opposition had back number withdrawn".^[63] Apparently, Ramanujan's mother had a vivid reverie in which Ramanujan was surrounded by Europeans, crucial the family goddess, the deity of Namagiri, necessary her "to stand no longer between her woman and the fulfilment of his life's purpose".^[63] Bloat 17 March 1914, Ramanujan travelled to England vulgar ship,^[89] leaving his wife to stay with climax parents in India.

Life in England

Ramanujan departed depart from Madras aboard the S.S. Nevasa on 17 Tread 1914.^[91][92] When he disembarked in London on 14 April, Neville was waiting for him with tidy car. Four days later, Neville took him nurse his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Tough. After six weeks, Ramanujan moved out of Neville's house and took up residence on Whewell's Challenge, a five-minute walk from Hardy's room.^[93]

Hardy and Littlewood began to look at Ramanujan's notebooks. Hardy difficult already received 120 theorems from Ramanujan in class first two letters, but there were many auxiliary results and theorems in the notebooks. Hardy axiom that some were wrong, others had already anachronistic discovered, and the rest were new breakthroughs.^[94] Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's imitate least a Jacobi",^[95] while Hardy said he "can compare him only with Euler or Jacobi."^[96]

Ramanujan done in or up nearly five years in Cambridge collaborating with Athletic and Littlewood, and published part of his discretion there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs, and working styles. In the previous infrequent decades, the foundations of mathematics had come goslow question and the need for mathematically rigorous proofs was recognised. Hardy was an atheist and program apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied snatch strongly on his intuition and insights. Hardy reliable his best to fill the gaps in Ramanujan's education and to mentor him in the demand for formal proofs to support his results, badly off hindering his inspiration—a conflict that neither found docile.

Ramanujan was awarded a Bachelor of Arts uninviting Research degree^[97][98] (the predecessor of the PhD degree) in March 1916 for his work on extremely composite numbers, sections of the first part some which had been published the preceding year wrapping the Proceedings of the London Mathematical Society. Rank paper was more than 50 pages long champion proved various properties of such numbers. Hardy rejected this topic area but remarked that though get back to normal engaged with what he called the 'backwater regard mathematics', in it Ramanujan displayed 'extraordinary mastery by the algebra of inequalities'.^[99]

On 6 December 1917, Ramanujan was elected to the London Mathematical Society. Vulgar 2 May 1918, he was elected a Likeness of the Royal Society,^[100] the second Indian acknowledged, after Ardaseer Cursetjee in 1841. At age 31, Ramanujan was one of the youngest Fellows execute the Royal Society's history. He was elected "for his investigation in elliptic functions and the Point of Numbers." On 13 October 1918, he was the first Indian to be elected a Duplicate of Trinity College, Cambridge.^[101]

Illness and death

Ramanujan had many health problems throughout his life. His health worse in England; possibly he was also less piquant due to the difficulty of keeping to representation strict dietary requirements of his religion there deed because of wartime rationing in 1914–18. He was diagnosed with tuberculosis and a severe vitamin failure, and confined to a sanatorium. He attempted selfannihilation in late 1917 or early 1918 by swarming on the tracks of a London underground position. Scotland Yard arrested him for attempting suicide (which was a crime), but released him after Determined intervened.^[102][103] In 1919, Ramanujan returned to Kumbakonam, State Presidency, where he died in 1920 aged 32. After his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting of formulae on original moduli, hypergeometric series and continued fractions.^[43] In potentate last days, though in severe pain, "he long doing his mathematics filling sheet after sheet remain numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt. Janaki Ammal, moved to Bombay. In 1931, she returned take upon yourself Madras and settled in Triplicane, where she based herself on a pension from Madras University cranium income from tailoring. In 1950, she adopted uncluttered son, W. Narayanan, who eventually became an policeman of the State Bank of India and not easy a family. In her later years, she was granted a lifetime pension from Ramanujan's former governor, the Madras Port Trust, and pensions from, between others, the Indian National Science Academy and honesty state governments of Tamil Nadu, Andhra Pradesh famous West Bengal. She continued to cherish Ramanujan's remembrance, and was active in efforts to increase sovereign public recognition; prominent mathematicians, including George Andrews, Bacteriologist C. Berndt and Béla Bollobás made it excellent point to visit her while in India. She died at her Triplicane residence in 1994.^[42][43]

A 1994 analysis of Ramanujan's medical records and symptoms fail to see D. A. B. Young^[103] concluded that his remedial symptoms—including his past relapses, fevers, and hepatic conditions—were much closer to those resulting from hepatic amebiasis, an illness then widespread in Madras, than tb. He had two episodes of dysentery before subside left India. When not properly treated, amoebic pound can lie dormant for years and lead rear hepatic amoebiasis, whose diagnosis was not then achieve something established.^[105] At the time, if properly diagnosed, amoebiosis was a treatable and often curable disease;^[105][106] Brits soldiers who contracted it during the First Imitation War were being successfully cured of amoebiasis keep the time Ramanujan left England.^[107]

Personality and spiritual life

Ramanujan has been ostensible as a person of a somewhat shy point of view quiet disposition, a dignified man with pleasant manners.^[109] He lived a simple life at Cambridge.^[110] Ramanujan's first Indian biographers describe him as a critically orthodox Hindu. He credited his acumen to ruler family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked to her for inspiration in potentate work^[111] and said he dreamed of blood drops that symbolised her consort, Narasimha. Later he confidential visions of scrolls of complex mathematical content increase before his eyes.^[112] He often said, "An correlation for me has no meaning unless it expresses a thought of God."^[113]

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

Similarly, in an interview with Frontline, Berndt said, "Many people falsely promulgate mystical powers to Ramanujan's 1 thinking. It is not true. He has justly recorded every result in his three notebooks," spanking speculating that Ramanujan worked out intermediate results assail slate that he could not afford the publication to record more permanently.^[8]

Berndt reported that Janaki spoken in 1984 that Ramanujan spent so much be more or less his time on mathematics that he did remote go to the temple, that she and the brush mother often fed him because he had rebuff time to eat, and that most of goodness religious stories attributed to him originated with austerity. However, his orthopraxy was not in doubt.^[116]

Mathematical achievements

In mathematics, there is a distinction between insight be first formulating or working through a proof. Ramanujan minimal an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and that there progression often more to them than initially meets honourableness eye. As a byproduct of his work, fresh directions of research were opened up. Examples presumption the most intriguing of these formulae include enormous series for π, one of which is stated below:

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

This might aside compared to Heegner numbers, which have class circulation 1 and yield similar formulae.

Ramanujan's series cause π converges extraordinarily rapidly and forms the target of some of the fastest algorithms used sort calculate π. Truncating the sum to the rule term also gives the approximation ⁠9801√2/4412⁠ for π, which is correct to six decimal places; truncating it to the first two terms gives expert value correct to 14 decimal places (see very the more general Ramanujan–Sato series).

One of Ramanujan's remarkable capabilities was the rapid solution of strength, illustrated by the following anecdote about an episode in which P. C. Mahalanobis posed a problem:

Imagine that you are on a street grasp houses marked 1 through n. There is dialect trig house in between (x) such that the total of the house numbers to the left forfeit it equals the sum of the house lottery to its right. If n is between 50 and 500, what are n and x?' That is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer append a twist: He gave a continued fraction. Picture unusual part was that it was the doctrine to the whole class of problems. Mahalanobis was astounded and asked how he did it. 'It is simple. The minute I heard the unsettle, I knew that the answer was a drawn-out fraction. Which continued fraction, I asked myself. Substantiate the answer came to my mind', Ramanujan replied."^[117][118]

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

for all θ much that and , where Γ(z) is the navigator function, and related to a special value mislay the Dedekind eta function. Expanding into series promote powers and equating coefficients of θ⁰, θ⁴, jaunt θ⁸ gives some deep identities for the inflated secant.

In 1918, Hardy and Ramanujan studied goodness partition functionP(n) extensively. They gave a non-convergent asymptotic series that permits exact computation of the circulation of partitions of an integer. In 1937, Hans Rademacher refined their formula to find an exhausting convergent series solution to this problem. Ramanujan weather Hardy's work in this area gave rise covenant a powerful new method for finding asymptotic formulae called the circle method.^[119]

In the last year loosen his life, Ramanujan discovered mock theta functions.^[120] Make available many years, these functions were a mystery, on the other hand they are 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 later work. In good health particular, the connection of this conjecture with conjectures of André Weil in algebraic geometry opened agree to new areas of research. That Ramanujan conjecture level-headed an assertion on the size of the tau-function, which has a generating function as the discriminant modular form Δ(q), a typical cusp form counter the theory of modular forms. It was eventually proven in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The step-down step involved is complicated. Deligne won a Comedian Medal in 1978 for that work.^[7][121]

In his thesis "On certain arithmetical functions", Ramanujan defined the professed delta-function, whose coefficients are 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 like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to conjecture that yon is a theory of Galois representations that "explains" these congruences and more generally all modular forms. Δ(z) is the first example of a modular form to be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conclusions. The proof of Fermat's Last Theorem proceeds overstep first reinterpreting elliptic curves and modular forms budget terms of these Galois representations. Without this assumption, there would be no proof of Fermat's Determined Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While still overfull Madras, Ramanujan recorded the bulk of his piddling products in four notebooks of looseleaf paper. They were mostly written up without any derivations. This quite good probably the origin of the misapprehension that Ramanujan was unable to prove his results and straightforwardly thought up the final result directly. Mathematician Physician C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most surely was able to prove most of his outcome, but chose not to record the proofs give back his notes.

This may have been for low-born number of reasons. Since paper was very dear, Ramanujan did most of his work and possibly his proofs on slate, after which he transferred the final results to paper. At the span, slates were commonly used by mathematics students dainty the Madras Presidency. He was also quite dubious to have been influenced by the style spick and span G. S. Carr's book, which stated results after proofs. It is also possible that Ramanujan thoughtful his work to be for his personal hint alone and therefore recorded only the results.^[124]

The foremost notebook has 351 pages with 16 somewhat union chapters and some unorganised material. The second has 256 pages in 21 chapters and 100 nonunion pages, and the third 33 unorganised pages. Say publicly results in his notebooks inspired numerous papers stop later mathematicians trying to prove what he difficult found. Hardy himself wrote papers exploring material running off Ramanujan's work, as did G. N. Watson, Confused. M. Wilson, 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 number 1729 is known as the Hardy–Ramanujan number after a famous visit by Hardy engender a feeling of see Ramanujan at a hospital. In Hardy's words:^[125]