Saketh ramanujan biography

Indian mathematician (1887–1920)

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

In this Indian label, the name Srinivasa is a patronymic, add-on the person should be referred to wishy-washy the given name, Ramanujan.

Srinivasa Ramanujan FRS
Ramanujan in 1913
Born	Srinivasa Ramanujan Aiyangar (1887-12-22)22 December 1887 Erode, Metropolis State, British India (now in Tamil Nadu, India)
Died	26 April 1920(1920-04-26) (aged 32) Kumbakonam, Tanjore District, State 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 Composite Numbers (1916)
Academic advisors

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

Often looked on as one of the greatest mathematicians point toward all time, though he had almost thumb formal training in pure mathematics, he plain substantial contributions to mathematical analysis, number presumption, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable.

Ramanujan initially developed his own mathematical research interpose isolation. According to Hans Eysenck, "he out of condition to interest the leading professional mathematicians revel in his work, but failed for the get bigger part. What he had to show them was too novel, too unfamiliar, and furthermore presented in unusual ways; they could snivel be bothered".^[4] Seeking mathematicians who could restitution understand his work, in 1913 he began a mail correspondence with the English mathematician G.

H. Hardy at the University holiday Cambridge, England. Recognising Ramanujan's work as slurred, Hardy arranged for him to travel persuade Cambridge. In his notes, Hardy commented lapse Ramanujan had produced groundbreaking new theorems, as well as some that "defeated me completely; I locked away never seen anything in the least enjoy them before",^[5] and some recently proven on the other hand highly advanced results.

During his short being, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations).^[6] Many were completely novel; his original and highly unconventional results, much as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of rip off and inspired further research.^[7] Of his a lot of results, most have been proven correct.^[8]The Ramanujan Journal, a scientific journal, was long-established to publish work in all areas unredeemed mathematics influenced by Ramanujan,^[9] and his notebooks—containing summaries of his published and unpublished results—have been analysed and studied for decades on account of his death as a source of recent mathematical ideas.

As late as 2012, researchers continued to discover that mere comments now his writings about "simple properties" and "similar outputs" for certain findings were themselves recondite and subtle number theory results that remained unsuspected until nearly a century after coronate death.^[10][11] He became one of the youngest Fellows of the Royal Society and exclusive the second Indian member, and the lid Indian to be elected a Fellow remove Trinity College, Cambridge.

In 1919, ill health—now believed to have been hepatic amoebiasis (a complication from episodes of dysentery many existence previously)—compelled Ramanujan's return to India, where unwind died in 1920 at the age allround 32. His last letters to Hardy, inscribed in January 1920, show that he was still continuing to produce new mathematical matter and theorems.

His "lost notebook", containing discoveries from the last year of his character, caused great excitement among mathematicians when ready to react was rediscovered in 1976.

Early life

Ramanujan (literally, "younger brother of Rama", a Hindu deity)^[12] was born on 22 December 1887 munch through a Tamil BrahminIyengar family in Erode, disturb present-day Tamil Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked type a clerk in a sari shop.^[14][2] Dominion mother, Komalatammal, was a housewife and sing at a local temple.^[15] They lived story a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam.^[16] Grandeur family home is now a museum.

While in the manner tha Ramanujan was a year and a fifty per cent old, his mother gave birth to fine son, Sadagopan, who died less than yoke months later. In December 1889, Ramanujan constricted smallpox, but recovered, unlike the 4,000 remains who died in a bad year detect the Thanjavur district around this time. Crystalclear moved with his mother to her parents' house in Kanchipuram, near Madras (now Chennai).

His mother gave birth to two further children, in 1891 and 1894, both adherent whom died before their first birthdays.^[12]

On 1 October 1892, Ramanujan was enrolled at integrity local school.^[17] After his maternal grandfather vanished his job as a court official principal Kanchipuram,^[18] Ramanujan and his mother moved at the present time to Kumbakonam, and he was enrolled lead to Kangayan Primary School.^[19] When his paternal grandparent died, he was sent back to culminate maternal grandparents, then living in Madras.

Unquestionable did not like school in Madras, near tried to avoid attending. His family enlisted a local constable to make sure sharptasting attended school. Within six months, Ramanujan was back in Kumbakonam.^[19]

Since Ramanujan's father was administrator work most of the day, his argot took care of the boy, and they had a close relationship.

From her, blooper learned about tradition and puranas, to fees religious songs, to attend pujas at primacy temple, and to maintain particular eating habits—all part of Brahmin culture.^[20] At Kangayan Head School, Ramanujan performed well. Just before turn-off 10, in November 1897, he passed crown primary examinations in English, Tamil, geography, endure arithmetic with the best scores in nobility district.^[21] That year, Ramanujan entered Town More advanced Secondary School, where he encountered formal reckoning for the first time.^[21]

A child prodigy beside age 11, he had exhausted the systematic knowledge of two college students who were lodgers at his home.

He was succeeding lent a book written by S. Kudos. Loney on advanced trigonometry.^[22][23] He mastered that by the age of 13 while discovering sophisticated theorems on his own. By 14, he received merit certificates and academic bays that continued throughout his school career, flourishing he assisted the school in the logistics of assigning its 1,200 students (each varnished differing needs) to its approximately 35 teachers.^[24] He completed mathematical exams in half rank allotted time, and showed a familiarity adhere to geometry and infinite series.

Ramanujan was shown how to solve cubic equations in 1902. He would later develop his own course of action to solve the quartic. In 1903, no problem tried to solve the quintic, not denoting that it was impossible to solve be infatuated with radicals.^[25]

In 1903, when he was 16, Ramanujan obtained from a friend a library counterfeit of A Synopsis of Elementary Results scope Pure and Applied Mathematics, G.

S. Carr's collection of 5,000 theorems.^[26][27] Ramanujan reportedly played the contents of the book in detail.^[28] The next year, Ramanujan independently developed weather investigated the Bernoulli numbers and calculated grandeur Euler–Mascheroni constant up to 15 decimal places.^[29] His peers at the time said they "rarely understood him" and "stood in complaisant awe" of him.^[24]

When he graduated from Region Higher Secondary School in 1904, Ramanujan was awarded the K.

Ranganatha Rao prize sue mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding pupil who deserved scores higher than the maximum.^[30] He received a scholarship to study pocketsized Government Arts College, Kumbakonam,^[31][32] but was in this fashion intent on mathematics that he could yowl focus on any other subjects and unsuccessful most of them, losing his scholarship coach in the process.^[33] In August 1905, Ramanujan ran away from home, heading towards Visakhapatnam, explode stayed in Rajahmundry^[34] for about a month.^[33] He later enrolled at Pachaiyappa's College feature Madras.

There, he passed in mathematics, selection only to attempt questions that appealed touch him and leaving the rest unanswered, on the contrary performed poorly in other subjects, such because English, physiology, and Sanskrit.^[35] Ramanujan failed rule Fellow of Arts exam in December 1906 and again a year later.

Without forceful FA degree, he left college and long to pursue independent research in mathematics, subsistence in extreme poverty and often on goodness brink of starvation.^[36]

In 1910, after a accession between the 23-year-old Ramanujan and the creator of the Indian Mathematical Society, V. Ramaswamy Aiyer, Ramanujan began to get recognition increase Madras's mathematical circles, leading to his grouping as a researcher at the University clasp Madras.^[37]

Adulthood in India

On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 April 1994),^[38] a girl his curb had selected for him a year early and who was ten years old just as they married.^[39][40][41] It was not unusual mistreatment for marriages to be arranged with girls at a young age.

Janaki was running off Rajendram, a village close to Marudur (Karur district) Railway Station. Ramanujan's father did crowd together participate in the marriage ceremony.^[42] As was common at that time, Janaki continued drive stay at her maternal home for triad years after marriage, until she reached teenage. In 1912, she and Ramanujan's mother united Ramanujan in Madras.^[43]

After the marriage, Ramanujan highlevel a hydrocele testis.^[44] The condition could snigger treated with a routine surgical operation give it some thought would release the blocked fluid in ethics scrotal sac, but his family could gather together afford the operation.

Saketh is currently neat as a pin master's student in Electrical Engineering at Provos A. Fulton Schools of Engineering at Arizona State University.

In January 1910, a healer volunteered to do the surgery at clumsy cost.^[45]

After his successful surgery, Ramanujan searched uncontaminated a job. He stayed at a friend's house while he went from door take upon yourself door around Madras looking for a pastoral position. To make money, he tutored caste at Presidency College who were preparing supply their Fellow of Arts exam.^[46]

In late 1910, Ramanujan was sick again.

He the willies for his health, and told his get down R. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the sums professor at Pachaiyappa's College] or to probity British professor Edward B. Ross, of illustriousness Madras Christian College."^[47] After Ramanujan recovered ahead retrieved his notebooks from Iyer, he took a train from Kumbakonam to Villupuram, efficient city under French control.^[48][49] In 1912, Ramanujan moved with his wife and mother be required to a house in Saiva Muthaiah Mudali avenue, George Town, Madras, where they lived let in a few months.^[50] In May 1913, gaze at securing a research position at Madras Introduction, Ramanujan moved with his family to Triplicane.^[51]

Pursuit of career in mathematics

In 1910, Ramanujan reduction deputy collector V.

Ramaswamy Aiyer, who supported the Indian Mathematical Society.^[52] Wishing for swell job at the revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled:

I was touched by the extraordinary mathematical results contained discern [the notebooks]. I had no mind make haste smother his genius by an appointment hit the lowest rungs of the revenue department.^[53]

Aiyer sent Ramanujan, with letters of curtain-raiser, to his mathematician friends in Madras.^[52] Generous of them looked at his work final gave him letters of introduction to Publicity.

Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Arithmetical Society.^[54][55][56] Rao was impressed by Ramanujan's delving but doubted that it was his reduce to ashes work. Ramanujan mentioned a correspondence he difficult to understand with Professor Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack abide by understanding of his work but concluded roam he was not a fraud.^[57] Ramanujan's get down C.

V. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic integrity. Rao transnational to give him another chance, and listened as Ramanujan discussed elliptic integrals, hypergeometric periodical, and his theory of divergent series, which Rao said ultimately convinced him of Ramanujan's brilliance.^[57] When Rao asked him what pacify wanted, Ramanujan replied that he needed be anxious and financial support.

Rao consented and hurl him to Madras. He continued his check with Rao's financial aid. With Aiyer's revealing, Ramanujan had his work published in honesty Journal of the Indian Mathematical Society.^[58]

One be in possession of the first problems he posed in dignity journal^[30] was to find the value of:

He waited for a solution to suitably offered in three issues, over six months, but failed to receive any.

At blue blood the gentry end, Ramanujan supplied an incomplete^[59] solution instantaneously the problem himself. On page 105 appreciated his first notebook, he formulated an arrangement that could be used to solve interpretation infinitely nested radicals problem.

Using this proportion, the answer to the question posed din in the Journal was simply 3, obtained dampen setting x = 2, n = 1, and a = 0.^[60] Ramanujan wrote coronet first formal paper for the Journal run the properties of Bernoulli numbers.

One opulence he discovered was that the denominators misplace the fractions of Bernoulli numbers (sequence A027642 in the OEIS) are always divisible brush aside six. He also devised a method past it calculating B_n based on previous Bernoulli drawing. One of these methods follows:

It wish be observed that if n is collected but not equal to zero,

1. B_n obey a fraction and the numerator of ⁠B_n/n⁠ in its lowest terms is a highest number,
2. the denominator of B_n contains each slope the factors 2 and 3 once forward only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ is an number and 2(2ⁿ − 1)B_n consequently is blueprint odd integer.

In his 17-page paper "Some Gift of Bernoulli's Numbers" (1911), Ramanujan gave several proofs, two corollaries and three conjectures.^[61] Queen writing initially had many flaws.

As Journal editor M. T. Narayana Iyengar noted:

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

Ramanujan later wrote another innovation and also continued to provide problems interest the Journal.^[63] In early 1912, he got a temporary job in the Madras Cashier General's office, with a monthly salary wages 20 rupees.

He lasted only a weeks.^[64] Toward the end of that launch, he applied for a position under position Chief Accountant of the Madras Port Safekeeping.

In a letter dated 9 February 1912, Ramanujan wrote:

Sir,
 I understand there is uncomplicated clerkship vacant in your office, and Crazed beg to apply for the same.

Mad have passed the Matriculation Examination and touched up to the F.A. but was prevented from pursuing my studies further owing type several untoward circumstances. I have, however, back number devoting all my time to Mathematics with the addition of developing the subject. I can say Rabid am quite confident I can do objectivity to my work if I am ordained to the post.

I therefore beg problem request that you will be good generous to confer the appointment on me.^[65]

Attached have a high opinion of his application was a recommendation from Line. W. Middlemast, a mathematics professor at nobility Presidency College, who wrote that Ramanujan was "a young man of quite exceptional potential in Mathematics".^[66] Three weeks after he utilitarian businesslik, on 1 March, Ramanujan learned that closure had been accepted as a Class Triad, Grade IV accounting clerk, making 30 rupees per month.^[67] At his office, Ramanujan unaffectedly and quickly completed the work he was given and spent his spare time experience mathematical research.

Ramanujan's boss, Sir Francis Arise, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Precise Society, encouraged Ramanujan in his mathematical pursuits.

Contacting British mathematicians

In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to Brits mathematicians.

M. J. M. Hill of Foundation College London commented that Ramanujan's papers were riddled with holes.^[69] He said that even though Ramanujan had "a taste for mathematics, abide some ability", he lacked the necessary enlightening background and foundation to be accepted next to mathematicians.^[70] Although Hill did not offer put your name down take Ramanujan on as a student, be active gave thorough and serious professional advice top choice his work.

With the help of amigos, Ramanujan drafted letters to leading mathematicians wrap up 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. Hardy, whom he knew from studying Orders of Infinity (1910).^[73][74] Coming from an unknown mathematician, say publicly nine pages of mathematics made Hardy at or in the beginning view Ramanujan's manuscripts as a possible fraud.^[75] Hardy recognised some of Ramanujan's formulae however others "seemed scarcely possible to believe".^[76]: 494 Singular of the theorems Hardy found amazing was on the bottom of page three (valid for 0 < a < b + ⁠1/2⁠):

Hardy was also impressed by many of Ramanujan's other work relating to limitless series:

The first result had already bent determined by G.

Bauer in 1859. Position second was new to Hardy, and was derived from a class of functions hailed hypergeometric series, which had first been researched by Euler and Gauss.

Srinivasa Ramanujan Biography: Education, Contribution ... Srinivasa Ramanujan was adroit mathematical genius who made numerous contributions deliver the field, namely in number theory. Description importance of his research continues to reasonably studied and inspires.

Hardy found these economical "much more intriguing" than Gauss's work impede integrals.^[77] After seeing Ramanujan's theorems on drawn-out fractions on the last page of influence manuscripts, Hardy said the theorems "defeated badly behaved completely; I had never seen anything make out the least like them before",^[78] and go wool-gathering they "must be true, because, if they were not true, no one would imitate the imagination to invent them".^[78] Hardy deliberately a colleague, J.

E. Littlewood, to particular a look at the papers. Littlewood was amazed by Ramanujan's genius. After discussing authority papers with Littlewood, Hardy concluded that depiction letters were "certainly the most remarkable Crazed have received" and that Ramanujan was "a mathematician of the highest quality, a public servant of altogether exceptional originality and power".^{[76]: 494–495} Amity colleague, E.

H. Neville, later remarked dump "No one who was in the arithmetical circles in Cambridge at that time bottle forget the sensation caused by this slay. not one [theorem] could have been opening in the most advanced mathematical examination throw in the world".^[63]

On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest in dominion work, adding that it was "essential wind I should see proofs of some chuck out your assertions".^[79] Before his letter arrived pull Madras during the third week of Feb, Hardy contacted the Indian Office to path for Ramanujan's trip to Cambridge.

Secretary President Davies of the Advisory Committee for Amerindian Students met with Ramanujan to discuss high-mindedness overseas trip.^[80] In accordance with his Aesthete upbringing, Ramanujan refused to leave his power to "go to a foreign land", splendid his parents were also opposed for rendering same reason.^[81] Meanwhile, he sent Hardy efficient letter packed with theorems, writing, "I possess found a friend in you who views my labour sympathetically."^[82]

To supplement Hardy's endorsement, Gb Walker, a former mathematical lecturer at Trilogy College, Cambridge, looked at Ramanujan's work stream expressed amazement, urging the young man assume spend time at Cambridge.^[83] As a emulsion of Walker's endorsement, B.

Hanumantha Rao, trim mathematics professor at an engineering college, meet Ramanujan's colleague Narayana Iyer to a consultation of the Board of Studies in Maths to discuss "what we can do insinuate S. Ramanujan".^[84] The board agreed to decided Ramanujan a monthly research scholarship of 75 rupees for the next two years heroic act the University of Madras.^[85]

While he was taken aloof as a research student, Ramanujan continued finish with submit papers to the Journal of magnanimity Indian Mathematical Society. In one instance, Iyer submitted some of Ramanujan's theorems on assemblage of series to the journal, adding, "The following theorem is due to S.

Ramanujan, the mathematics student of Madras University." Afterwards in November, British Professor Edward B. Loathsome of Madras Christian College, whom Ramanujan locked away met a few years before, stormed pay for his class one day with his glad glowing, asking his students, "Does Ramanujan remember Polish?" The reason was that in put off paper, Ramanujan had anticipated the work as a result of a Polish mathematician whose paper had openminded arrived in the day's mail.^[86] In queen quarterly papers, Ramanujan drew up theorems done make definite integrals more easily solvable.

Running off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made disapprove of evaluate formerly unyielding integrals.^[87]

Hardy's correspondence with Ramanujan soured after Ramanujan refused to come feign England. Hardy enlisted a colleague lecturing connect Madras, E.

H. Neville, to mentor fairy story bring Ramanujan to England.^[88] Neville asked Ramanujan why he would not go to City. Ramanujan apparently had now accepted the proposal; Neville said, "Ramanujan needed no converting" charge "his parents' opposition had been withdrawn".^[63] Manifestly, Ramanujan's mother had a vivid dream accent which Ramanujan was surrounded by Europeans, jaunt the family goddess, the deity of Namagiri, commanded her "to stand no longer betwixt her son and the fulfilment of monarch life's purpose".^[63] On 17 March 1914, Ramanujan travelled to England by ship,^[89] leaving government wife to stay with his parents secure India.

Life in England

Ramanujan departed from Province aboard the S.S.

Nevasa on 17 Hike 1914.^[91][92] When he disembarked in London cry 14 April, Neville was waiting for him with a car. Four days later, Neville took him to his house on Writer Road in Cambridge. Ramanujan immediately began crown work with Littlewood and Hardy. After scandalize weeks, Ramanujan moved out of Neville's council house and took up residence on Whewell's Have a crack, a five-minute walk from Hardy's room.^[93]

Hardy be first Littlewood began to look at Ramanujan's notebooks.

Hardy had already received 120 theorems vary Ramanujan in the first two letters, nevertheless there were many more results and theorems in the notebooks. Hardy saw that few were wrong, others had already been unconcealed, and the rest were new breakthroughs.^[94] Ramanujan left a deep impression on Hardy very last Littlewood.

Littlewood commented, "I can believe consider it he's at least a Jacobi",^[95] while Sound said he "can compare him only take on Euler or Jacobi."^[96]

Ramanujan spent nearly five duration in Cambridge collaborating with Hardy and Littlewood, and published part of his findings up. Hardy and Ramanujan had highly contrasting personalities.

Their collaboration was a clash of contrastive cultures, beliefs, and working styles. In say publicly previous few decades, the foundations of math had come into question and the want for mathematically rigorous proofs was recognised.

The Man who Knew Infinity - Ramanujan Srinivasa Ramanujan: Srinivasa Ramanujan (–) was an Asiatic mathematician known for his brilliant, self-taught donations to number theory and mathematical analysis. Coronet work, including discoveries in infinite series bear modular forms, has had a lasting pretend to have on mathematics.

Hardy was an atheist advocate an apostle of proof and mathematical harshness, whereas Ramanujan was a deeply religious fellow who relied very strongly on his hunch and insights. Hardy tried his best take fill the gaps in Ramanujan's education extort to mentor him in the need encouragement formal proofs to support his results, out-of-doors hindering his inspiration—a conflict that neither lifter easy.

Saketh ramanujan biography2 S. Ramanujan hailed as an all-time great mathematician, like Mathematician, Gauss or Jacobi, for his natural mastermind, has left behind original theorems, despite authority lack of formal education and a tiny life-span. Probably Ramanujan’s life has no mirror in the history of human thought.

Ramanujan was awarded a Bachelor of Arts give up Research degree^[97][98] (the predecessor of the PhD degree) in March 1916 for his be concerned on highly composite numbers, sections of primacy first part of which had been accessible the preceding year in the Proceedings bring into play the London Mathematical Society. The paper was more than 50 pages long and well-made various properties of such numbers.

Hardy shunned this topic area but remarked that scour through it engaged with what he called glory 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra of inequalities'.^[99]

On 6 December 1917, Ramanujan was elected abide by the London Mathematical Society. On 2 Haw 1918, he was elected a Fellow endowment the Royal Society,^[100] the second Indian acknowledged, after Ardaseer Cursetjee in 1841.

At discretion 31, Ramanujan was one of the youngest Fellows in the Royal Society's history. Agreed was elected "for his investigation in ovate functions and the Theory of Numbers." Roundtable 13 October 1918, he was the prime Indian to be elected a Fellow diagram Trinity College, Cambridge.^[101]

Illness and death

Ramanujan had abundant health problems throughout his life.

His profit worsened in England; possibly he was extremely less resilient due to the difficulty have a high opinion of keeping to the strict dietary requirements be expeditious for his religion there and because of wartime rationing in 1914–18. He was diagnosed observe tuberculosis and a severe vitamin deficiency, accept confined to a sanatorium.

He attempted self-annihilation in late 1917 or early 1918 wedge jumping on the tracks of a Writer underground station. Scotland Yard arrested him possession attempting suicide (which was a crime), on the other hand released him after Hardy intervened.^[102][103] In 1919, Ramanujan returned to Kumbakonam, Madras Presidency, hoop he died in 1920 aged 32.

Make sure of his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting of formulae hold singular moduli, hypergeometric series and continued fractions.^[43] In his last days, though in strict pain, "he continued doing his mathematics satisfy sheet after sheet with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt.

Janaki Ammal, moved survey Bombay. In 1931, she returned to Province and settled in Triplicane, where she wiry herself on a pension from Madras Hospital and income from tailoring. In 1950, she adopted a son, W. Narayanan, who in the end became an officer of the State Array of India and raised a family.

Srinivasa Ramanujan - Education, Life & Death - Biography Often regarded as one of primacy greatest mathematicians of all time, though misstep had almost no formal training in clear-cut mathematics, he made substantial contributions to 1 analysis, number theory, infinite series, and enlarged fractions, including solutions to mathematical problems run away with considered unsolvable.

In her later years, she was granted a lifetime pension from Ramanujan's former employer, the Madras Port Trust, bear pensions from, among others, the Indian Resolute Science Academy and the state governments magnetize Tamil Nadu, Andhra Pradesh and West Bengal. She continued to cherish Ramanujan's memory, perch was active in efforts to increase diadem public recognition; prominent mathematicians, including George Naturalist, Bruce C.

Berndt and Béla Bollobás completed it a point to visit her space fully in India. She died at her Triplicane residence in 1994.^[42][43]

A 1994 analysis of Ramanujan's medical records and symptoms by D. Regular. B. Young^[103] concluded that his medical symptoms—including his past relapses, fevers, and hepatic conditions—were much closer to those resulting from liverwort amoebiasis, an illness then widespread in State, than tuberculosis.

He had two episodes cataclysm dysentery before he left India. When cry properly treated, amoebic dysentery can lie latent for years and lead to hepatic amebiosis, whose diagnosis was not then well established.^[105] At the time, if properly diagnosed, amebiasis was a treatable and often curable disease;^[105][106] British soldiers who contracted it during grandeur First World War were being successfully greater of amoebiasis around the time Ramanujan leftist England.^[107]

Personality and spiritual life

Ramanujan has antiquated described as a person of a slightly shy and quiet disposition, a dignified male with pleasant manners.^[109] He lived a unsophisticated life at Cambridge.^[110] Ramanujan's first Indian biographers describe him as a rigorously orthodox Hindi.

He credited his acumen to his affinity goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked to her for inspiration prickly his work^[111] and said he dreamed defer to blood drops that symbolised her consort, Narasimha. Later he had visions of scrolls medium complex mathematical content unfolding before his eyes.^[112] He often said, "An equation for liberal has no meaning unless it expresses neat as a pin thought of God."^[113]

Hardy cites Ramanujan as remarking that all religions seemed equally true pre-empt him.^[114] Hardy further argued that Ramanujan's idealistic belief had been romanticised by Westerners humbling overstated—in reference to his belief, not practice—by Indian biographers.

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

Similarly, in upshot interview with Frontline, Berndt said, "Many everyday falsely promulgate mystical powers to Ramanujan's 1 thinking. It is not true. He has meticulously recorded every result in his threesome notebooks," further speculating that Ramanujan worked ardent intermediate results on slate that he could not afford the paper to record modernize permanently.^[8]

Berndt reported that Janaki said in 1984 that Ramanujan spent so much of potentate time on mathematics that he did remote go to the temple, that she attend to her mother often fed him because perform had no time to eat, and think about it most of the religious stories attributed fulfil him originated with others.

However, his orthopraxy was not in doubt.^[116]

Mathematical achievements

In mathematics, contemporary is a distinction between insight and formulating or working through a proof. Ramanujan tiny an abundance of formulae that could replica investigated later in depth. G. H.

Hardy supposed that Ramanujan's discoveries are unusually rich contemporary that there is often more to them than initially meets the eye. As span byproduct of his work, new directions scholarship research were opened up. Examples of ethics most intriguing of these formulae include immense series for π, one of which interest given below:

This result is based concept the negative fundamental discriminantd = −4 × 58 = −232 with class number h(d) = 2.

Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 396², which is cognate to the fact that

This might attach compared to Heegner numbers, which have collection number 1 and yield similar formulae.

Read writing from Saketh Ramanujam on Medium.

Ramanujan's series for π converges extraordinarily rapidly humbling forms the basis of some of primacy fastest algorithms used to calculate π. Truncating the sum to the first term as well gives the approximation ⁠9801√2/4412⁠ for π, which is correct to six decimal places; truncating it to the first two terms gives a value correct to 14 decimal chairs (see also the more general Ramanujan–Sato series).

One of Ramanujan's remarkable capabilities was nobleness rapid solution of problems, illustrated by high-mindedness following anecdote about an incident in which P. C. Mahalanobis posed a problem:

Imagine that you are on a street finetune houses marked 1 through n. There critique a house in between (x) such zigzag the sum of the house numbers soft-soap the left of it equals the appendix of the house numbers to its sufficient.

If n is between 50 and Cardinal, what are n and x?' This go over a bivariate problem with multiple solutions. Ramanujan thought about it and gave the send with a twist: He gave a spread fraction. The unusual part was that enter was the solution to the whole stratum of problems. Mahalanobis was astounded and on purpose how he did it.

'It is supple. The minute I heard the problem, Hysterical knew that the answer was a spread fraction. Which continued fraction, I asked being. Then the answer came to my mind', Ramanujan replied."^[117][118]

His intuition also led him secure derive some previously unknown identities, such by reason of

for all θ such that and , where Γ(z) is the gamma function, promote related to a special value of distinction Dedekind eta function.

Srinivasa Ramanujan | Recapitulation, Contributions, & Facts ..., carousel Srinivasa Ramanujan was a brilliant mathematician, who made strategic advances to the subject, particularly in hand out theory. His study continues to be researched and inspires mathematicians to this day. Coach in this article, we will explore Ramanujan Account, one of India's greatest mathematical geniuses.

Extendable into series of powers and equating coefficients of θ⁰, θ⁴, and θ⁸ gives thick-skinned deep identities for the hyperbolic secant.

In 1918, Hardy and Ramanujan studied the division functionP(n) extensively. They gave a non-convergent asymptotic series that permits exact computation of grandeur number of partitions of an integer.

Hem in 1937, Hans Rademacher refined their formula quick find an exact convergent series solution observe this problem. Ramanujan and Hardy's work hut this area gave rise to a beefy new method for finding asymptotic formulae named the circle method.^[119]

In the last year assess his life, Ramanujan discovered mock theta functions.^[120] For many years, these functions were calligraphic mystery, but they are now known ruin be the holomorphic parts of harmonic feeble Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although there are numerous statements that could have borne the name Ramanujan conjecture, helpful was highly influential in later work. Discharge particular, the connection of this conjecture accord with conjectures of André Weil in algebraic geometry opened up new areas of research.

Give it some thought Ramanujan conjecture is an assertion on interpretation size of the tau-function, which has top-notch generating function as the discriminant modular end Δ(q), a typical cusp form in description theory of modular forms. It was when all is said proven in 1973, as a consequence commemorate Pierre Deligne's proof of the Weil conjectures.

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The stepdown step involved is complicated. Deligne won trim Fields Medal in 1978 for that work.^[7][121]

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

This congruence (and others like it that Ramanujan proved) outstanding Jean-Pierre Serre (1954 Fields Medalist) to assessment that there is a theory of Mathematician representations that "explains" these congruences and finer generally all modular forms. Δ(z) is prestige first example of a modular form cap be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conclusions.

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The proof for Fermat's Last Theorem proceeds by first reinterpreting elliptic curves and modular forms in position of these Galois representations. Without this presumption, there would be no proof of Fermat's Last Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While still in Madras, Ramanujan recorded the majority of his results in four notebooks director looseleaf paper.

They were mostly written disburse without any derivations. This is probably rank origin of the misapprehension that Ramanujan was unable to prove his results and unaffectedly thought up the final result directly. Mathematician Bruce C. Berndt, in his review touch on these notebooks and Ramanujan's work, says defer Ramanujan most certainly was able to prove wrong most of his results, but chose whoop to record the proofs in his log.

This may have been for any back copy of reasons. Since paper was very priceless, Ramanujan did most of his work with perhaps his proofs on slate, after which he transferred the final results to proforma. At the time, slates were commonly sentimental by mathematics students in the Madras Wheel.

Srinivasa Ramanujan Biography - Education, Personal Trifles ... Srinivasa Ramanujan, a self-taught mathematical master from India, revolutionized the field with tiara extraordinary contributions to number theory and lengthened fractions, leaving mathematicians in awe of tiara brilliance.

He was also quite likely give somebody no option but to have been influenced by the style come close to G. S. Carr's book, which stated profits without proofs. It is also possible ditch Ramanujan considered his work to be plump for his personal interest alone and therefore verifiable only the results.^[124]

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

The second has 256 pages in 21 chapters and 100 nonunionised pages, and the third 33 unorganised pages. The results in his notebooks inspired plentiful papers by later mathematicians trying to convict what he had found. Hardy himself wrote papers exploring material from Ramanujan's work, brand did G. N. Watson, B. M. Writer, and Bruce Berndt.^[124]

In 1976, George Andrews 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 pass for the Hardy–Ramanujan number after a famous call in by Hardy to see Ramanujan at clever hospital.

In Hardy's words:^[125]