Shri ramanujan biography sample

Indian mathematician (–)

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

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

Srinivasa Ramanujan Aiyangar^[a] (22 December &#;&#; 26 April ) was an Amerind mathematician. Often regarded as one of the largest mathematicians of all time, though he had supposedly apparent no formal training in pure mathematics, he indebted substantial contributions to mathematical analysis, number theory, immense series, and continued fractions, including solutions to exact problems then considered unsolvable.

Ramanujan initially developed rule own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading outdated mathematicians in his work, but failed for high-mindedness most part. What he had to show them was too novel, too unfamiliar, and additionally nip in unusual ways; they could not be bothered".^[4] Seeking mathematicians who could better understand his outmoded, in he began a mail correspondence with prestige English mathematician G. H. Hardy at the Further education college of Cambridge, England. Recognising Ramanujan's work as incredible, Hardy arranged for him to travel to University. In his notes, Hardy commented that Ramanujan challenging produced groundbreaking new theorems, including some that "defeated me completely; I had never seen anything put over the least like them before",^[5] and some lately proven but highly advanced results.

During his subsequently life, Ramanujan independently compiled nearly 3, results (mostly identities and equations).^[6] Many were completely novel; monarch original and highly unconventional results, such as decency Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire unusual areas of work and inspired further research.^[7] Work his thousands of results, most have been recognized correct.^[8]The Ramanujan Journal, a scientific journal, was accustomed to publish work in all areas of calculation influenced by Ramanujan,^[9] and his notebooks—containing summaries unravel his published and unpublished results—have been analysed promote studied for decades since his death as clever source of new mathematical ideas. As late orangutan , researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound suggest subtle number theory results that remained unsuspected depending on nearly a century after his death.^[10][11] He became one of the youngest Fellows of the Kingly Society and only the second Indian member, professor the first Indian to be elected a Lookalike of Trinity College, Cambridge.

In , ill health—now believed to have been hepatic amoebiasis (a dispute from episodes of dysentery many years previously)—compelled Ramanujan's return to India, where he died in have an effect on the age of His last letters to Durable, written in January , show that he was still continuing to produce new mathematical ideas take up theorems. His "lost notebook", containing discoveries from grandeur last year of his life, caused great enjoyment among mathematicians when it was rediscovered in

Early life

Ramanujan (literally, "younger brother of Rama", a Asiatic deity)^[12] was born on 22 December into dinky Tamil BrahminIyengar family in Erode, in present-day Dravidian Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally running away Thanjavur district, worked as a clerk in uncut sari shop.^[14][2] His mother, Komalatammal, was a wife and sang at a local temple.^[15] They cursory in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam.^[16] The kinsmen home is now a museum. When Ramanujan was a year and a half old, his encircle gave birth to a son, Sadagopan, who grand mal less than three months later. In December , Ramanujan contracted smallpox, but recovered, unlike the 4, others who died in a bad year bay the Thanjavur district around this time. He non-natural with his mother to her parents' house pride Kanchipuram, near Madras (now Chennai). His mother gave birth to two more children, in and , both of whom died before their first birthdays.^[12]

On 1 October , Ramanujan was enrolled at blue blood the gentry local school.^[17] After his maternal grandfather lost dominion job as a court official in Kanchipuram,^[18] Ramanujan and his mother moved back to Kumbakonam, presentday he was enrolled in Kangayan Primary School.^[19] Conj at the time that his paternal grandfather died, he was sent curb to his maternal grandparents, then living in State. He did not like school in Madras, significant tried to avoid attending. His family enlisted undiluted local constable to make sure he attended nursery school. Within six months, Ramanujan was back in Kumbakonam.^[19]

Since Ramanujan's father was at work most of representation day, his mother took care of the fellow, and they had a close relationship. From squeeze up, he learned about tradition and puranas, to bring out religious songs, to attend pujas at the holy place, and to maintain particular eating habits—all part precision Brahmin culture.^[20] At Kangayan Primary School, Ramanujan executed well. Just before turning 10, in November , he passed his primary examinations in English, Dravidian, geography, and arithmetic with the best scores interject the district.^[21] That year, Ramanujan entered Town Superior Secondary School, where he encountered formal mathematics back the first time.^[21]

A child prodigy by age 11, he had exhausted the mathematical knowledge of four college students who were lodgers at his population. He was later lent a book written invitation S. L. Loney on advanced trigonometry.^[22][23] He down this by the age of 13 while discovering sophisticated theorems on his own. By 14, yes received merit certificates and academic awards that continuing throughout his school career, and he assisted righteousness school in the logistics of assigning its 1, students (each with differing needs) to its almost 35 teachers.^[24] He completed mathematical exams in portion the allotted time, and showed a familiarity be geometry and infinite series. Ramanujan was shown though to solve cubic equations in He would afterwards develop his own method to solve the biquadrate. In , he tried to solve the quintic, not knowing that it was impossible to settle with radicals.^[25]

In , when he was 16, Ramanujan obtained from a friend a library copy dear A Synopsis of Elementary Results in Pure survive Applied Mathematics, G. S. Carr's collection of 5, theorems.^[26][27] Ramanujan reportedly studied the contents of picture book in detail.^[28] The next year, Ramanujan in the flesh developed and investigated the Bernoulli numbers and canny the Euler–Mascheroni constant up to 15 decimal places.^[29] His peers at the time said they "rarely understood him" and "stood in respectful awe" supporting him.^[24]

When he graduated from Town Higher Secondary Kindergarten in , Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's superior, Krishnaswami Iyer. Iyer introduced Ramanujan as an famed student who deserved scores higher than the maximum.^[30] He received a scholarship to study at Administration Arts College, Kumbakonam,^[31][32] but was so intent separately mathematics that he could not focus on sizeable other subjects and failed most of them, deprivation his scholarship in the process.^[33] In August , Ramanujan ran away from home, heading towards Visakhapatnam, and stayed in Rajahmundry^[34] for about a month.^[33] He later enrolled at Pachaiyappa's College in Province. There, he passed in mathematics, choosing only get on the right side of attempt questions that appealed to him and dying the rest unanswered, but performed poorly in mother subjects, such as English, physiology, and Sanskrit.^[35] Ramanujan failed his Fellow of Arts exam in Dec and again a year later. Without an Not a bit degree, he left college and continued to court independent research in mathematics, living in extreme insufficiency and often on the brink of starvation.^[36]

In , after a meeting between the year-old Ramanujan nearby the founder of the Indian Mathematical Society, Completely. Ramaswamy Aiyer, Ramanujan began to get recognition lecture in Madras's mathematical circles, leading to his inclusion whereas a researcher at the University of Madras.^[37]

Adulthood increase India

On 14 July , Ramanujan married Janaki (Janakiammal; 21 March – 13 April ),^[38] a miss his mother had selected for him a period earlier and who was ten years old during the time that they married.^[39][40][41] It was not unusual then collaboration marriages to be arranged with girls at exceptional young age. Janaki was from Rajendram, a adjoining close to Marudur (Karur district) Railway Station. Ramanujan's father did not participate in the marriage ceremony.^[42] As was common at that time, Janaki spread to stay at her maternal home for triad years after marriage, until she reached puberty. Confine , she and Ramanujan's mother joined Ramanujan choose by ballot Madras.^[43]

After the marriage, Ramanujan developed a hydrocele testis.^[44] The condition could be treated with a monotonous surgical operation that would release the blocked damp in the scrotal sac, but his family could not afford the operation. In January , organized doctor volunteered to do the surgery at maladroit thumbs down d cost.^[45]

After his successful surgery, Ramanujan searched for trig job. He stayed at a friend's house measure he went from door to door around State looking for a clerical position. To make flat broke, he tutored students at Presidency College who were preparing for their Fellow of Arts exam.^[46]

In open out , Ramanujan was sick again. He feared merriment his health, and told his friend R. Radakrishna Iyer to "hand [his notebooks] over to Academic Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Squeeze out, of the Madras Christian College."^[47] After Ramanujan superior and retrieved his notebooks from Iyer, he took a train from Kumbakonam to Villupuram, a section under French control.^[48][49] In , Ramanujan moved make sense his wife and mother to a house in vogue Saiva Muthaiah Mudali street, George Town, Madras, wheel they lived for a few months.^[50] In Might , upon securing a research position at Province University, Ramanujan moved with his family to Triplicane.^[51]

Pursuit of career in mathematics

In , Ramanujan met second in com collector V. Ramaswamy Aiyer, who founded the Amerindic Mathematical Society.^[52] Wishing for a job at excellence revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled:

I was struck by the extraordinary mathematical results restrained in [the notebooks]. I had no mind find time for smother his genius by an appointment in excellence lowest rungs of the revenue department.^[53]

Aiyer deadlock Ramanujan, with letters of introduction, to his mathematician friends in Madras.^[52] Some of them looked go back his work and gave him letters of debut to R. Ramachandra Rao, the district collector agreeable Nellore and the secretary of the Indian Rigorous Society.^[54][55][56] Rao was impressed by Ramanujan's research however doubted that it was his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay mathematician, in which Saldhana verbalised a lack of understanding of his work on the contrary concluded that he was not a fraud.^[57] Ramanujan's friend C. V. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic integrity. Rao agreed resemble give him another chance, and listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his intention of divergent series, which Rao said ultimately decided him of Ramanujan's brilliance.^[57] When Rao asked him what he wanted, Ramanujan replied that he requisite work and financial support. Rao consented and hurl him to Madras. He continued his research stomach Rao's financial aid. With Aiyer's help, Ramanujan confidential his work published in the Journal of rank Indian Mathematical Society.^[58]

One of the first problems yes posed in the journal^[30] was to find description value of:

He waited for a solution contact be offered in three issues, over six months, but failed to receive any. At the stop, Ramanujan supplied an incomplete^[59] solution to the predicament himself. On page of his first notebook, agreed formulated an equation that could be used apropos solve the infinitely nested radicals problem.

Using that equation, the answer to the question posed directive the Journal was simply 3, obtained by location x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his first formal procedure for the Journal on the properties of Physicist numbers. One property he discovered was that grandeur denominators of the fractions of Bernoulli numbers (sequence A in the OEIS) are always divisible fail to notice six. He also devised a method of cunning B_n based on previous Bernoulli numbers. One commuter boat these methods follows:

It will be observed dump if n is even but not equal colloquium zero,

1. B_n is a fraction and the numerator of &#;B_n/n&#; in its lowest terms is organized prime number,
2. the denominator of B_n contains each party the factors 2 and 3 once and one once,
3. 2ⁿ(2ⁿ − 1)&#;B_n/n&#; is an integer and 2(2ⁿ − 1)B_n consequently is an odd integer.

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

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

Ramanujan next wrote another paper and also continued to fill problems in the Journal.^[63] In early , fair enough got a temporary job in the Madras Purser General's office, with a monthly salary of 20 rupees. He lasted only a few weeks.^[64] Advance the end of that assignment, he applied occupy a position under the Chief Accountant of greatness Madras Port Trust.

In a letter dated 9 February , Ramanujan wrote:

Sir,
&#;I understand there in your right mind a clerkship vacant in your office, and Frantic beg to apply for the same. I keep passed the Matriculation Examination and studied up be familiar with the F.A. but was prevented from pursuing adhesive studies further owing to several untoward circumstances. Hysterical have, however, been devoting all my time nip in the bud Mathematics and developing the subject. I can selfcontrol I am quite confident I can do ill-treat to my work if I am appointed inherit the post. I therefore beg to request delay you will be good enough to confer ethics appointment on me.^[65]

Attached to his application was clean up recommendation from E. W. Middlemast, a mathematics academic at the Presidency College, who wrote that Ramanujan was "a young man of quite exceptional influence in Mathematics".^[66] Three weeks after he applied, allusion 1 March, Ramanujan learned that he had antique accepted as a Class III, Grade IV merit clerk, making 30 rupees per month.^[67] At tiara office, Ramanujan easily and quickly completed the groove he was given and spent his spare date doing mathematical research. Ramanujan's boss, Sir Francis Pit, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, pleased Ramanujan in his mathematical pursuits.

Contacting British mathematicians

In prestige spring of , Narayana Iyer, Ramachandra Rao explode E. W. Middlemast tried to present Ramanujan's trench to British mathematicians. M. J. M. Hill sharing University College London commented that Ramanujan's papers were riddled with holes.^[69] He said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the necessary educational background and trigger to be accepted by mathematicians.^[70] Although Hill outspoken not offer to take Ramanujan on as clean student, he gave thorough and serious professional guidance on his work. With the help of flock, Ramanujan drafted letters to leading mathematicians at Metropolis University.^[71]

The first two professors, H. F. Baker near E. W. Hobson, returned Ramanujan's papers without comment.^[72] On 16 January , Ramanujan wrote to Misty. H. Hardy, whom he knew from studying Orders of Infinity ().^[73][74] Coming from an unknown mathematician, the nine pages of mathematics made Hardy primarily view Ramanujan's manuscripts as a possible fraud.^[75] Strong recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe".^[76]:&#;&#; One of the theorems Hardy found amazing was on the bottom line of attack page three (valid for 0 < a < b + &#;1/2&#;):

Hardy was also impressed mass some of Ramanujan's other work relating to unlimited series:

The first result had already been compress by G. Bauer in The second was newfound to Hardy, and was derived from a gigantic of functions called hypergeometric series, which had cardinal been researched by Euler and Gauss. Hardy originate these results "much more intriguing" than Gauss's operate on integrals.^[77] After seeing Ramanujan's theorems on extended fractions on the last page of the manuscripts, Hardy said the theorems "defeated me completely; Funny had never seen anything in the least need them before",^[78] and that they "must be genuine, because, if they were not true, no combine would have the imagination to invent them".^[78] Sturdy asked a colleague, J. E. Littlewood, to rigging a look at the papers. Littlewood was thunderstruck by Ramanujan's genius. After discussing the papers strike up a deal Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and lapse Ramanujan was "a mathematician of the highest noble, a man of altogether exceptional originality and power".^{[76]:&#;–&#;} One colleague, E. H. Neville, later remarked think it over "No one who was in the mathematical wind in Cambridge at that time can forget goodness sensation caused by this letter not one [theorem] could have been set in the most new mathematical examination in the world".^[63]

On 8 February , Hardy wrote Ramanujan a letter expressing interest reveal his work, adding that it was "essential ensure I should see proofs of some of your assertions".^[79] Before his letter arrived in Madras generous the third week of February, Hardy contacted character Indian Office to plan for Ramanujan's trip space Cambridge. Secretary Arthur Davies of the Advisory Chamber for Indian Students met with Ramanujan to review the overseas trip.^[80] In accordance with his Aristocrat upbringing, Ramanujan refused to leave his country satisfy "go to a foreign land", and his parents were also opposed for the same reason.^[81] Pause, he sent Hardy a letter packed with theorems, writing, "I have found a friend in order around who views my labour sympathetically."^[82]

To supplement Hardy's ratification, Gilbert Walker, a former mathematical lecturer at Trio College, Cambridge, looked at Ramanujan's work and verbalized amazement, urging the young man to spend repel at Cambridge.^[83] As a result of Walker's approbation, B. Hanumantha Rao, a mathematics professor at swindler engineering college, invited Ramanujan's colleague Narayana Iyer add up to a meeting of the Board of Studies pin down Mathematics to discuss "what we can do undertake S. Ramanujan".^[84] The board agreed to grant Ramanujan a monthly research scholarship of 75 rupees be the next two years at the University blond Madras.^[85]

While he was engaged as a research fan, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one occurrence, Iyer submitted some of Ramanujan's theorems on count of series to the journal, adding, "The adjacent theorem is due to S. Ramanujan, the math student of Madras University." Later in November, Country Professor Edward B. Ross of Madras Christian Institution, whom Ramanujan had met a few years in the past, stormed into his class one day with rulership eyes glowing, asking his students, "Does Ramanujan identify Polish?" The reason was that in one sighting, Ramanujan had anticipated the work of a Furbish mathematician whose paper had just arrived in leadership day's mail.^[86] In his quarterly papers, Ramanujan actor up theorems to make definite integrals more clearly solvable. Working off Giuliano Frullani's integral theorem, Ramanujan formulated generalisations that could be made to judge formerly unyielding integrals.^[87]

Hardy's correspondence with Ramanujan soured funds Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England.^[88] Neville asked Ramanujan why he would not go make somebody's day Cambridge. Ramanujan apparently had now accepted the proposal; Neville said, "Ramanujan needed no converting" and "his parents' opposition had been withdrawn".^[63] Apparently, Ramanujan's apathy had a vivid dream in which Ramanujan was surrounded by Europeans, and the family goddess, prestige deity of Namagiri, commanded her "to stand rebuff longer between her son and the fulfilment defer to his life's purpose".^[63] On 17 March , Ramanujan travelled to England by ship,^[89] leaving his spouse to stay with his parents in India.

Life uphold England

Ramanujan departed from Madras aboard the S.S. Nevasa on 17 March ^[91][92] When he disembarked in London on 14 April, Neville was vacillate for him with a car. Four days subsequent, Neville took him to his house on Writer Road in Cambridge. Ramanujan immediately began his have an effect with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took ask on somebody's behalf residence on Whewell's Court, a five-minute walk raid Hardy's room.^[93]

Hardy and Littlewood began to look equal finish Ramanujan's notebooks. Hardy had already received theorems alien Ramanujan in the first two letters, but near were many more results and theorems in position notebooks. Hardy saw that some were wrong, residuum had already been discovered, and the rest were new breakthroughs.^[94] Ramanujan left a deep impression succeed Hardy and Littlewood. Littlewood commented, "I can guess that he's at least a Jacobi",^[95] while Hearty said he "can compare him only with Mathematician or Jacobi."^[96]

Ramanujan spent nearly five years in Metropolis collaborating with Hardy and Littlewood, and published corrode of his findings there. Hardy and Ramanujan difficult to understand highly contrasting personalities. Their collaboration was a collision of different cultures, beliefs, and working styles. Sight the previous few decades, the foundations of reckoning had come into question and the need fit in mathematically rigorous proofs was recognised. Hardy was distinctive atheist and an apostle of proof and 1 rigour, whereas Ramanujan was a deeply religious person who relied very strongly on his intuition forward insights. Hardy tried his best to fill position gaps in Ramanujan's education and to mentor him in the need for formal proofs to point in time his results, without hindering his inspiration—a conflict saunter neither found easy.

Ramanujan was awarded a Bachelor of Arts by Research degree^[97][98] (the predecessor sell the PhD degree) in March for his run on highly composite numbers, sections of the have control over part of which had been published the aforementioned year in the Proceedings of the London Precise Society. The paper was more than 50 pages long and proved various properties of such information. Hardy disliked this topic area but remarked think it over though it engaged with what he called class 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra of inequalities'.^[99]

On 6 Dec , Ramanujan was elected to the London Scientific Society. On 2 May , he was elect a Fellow of the Royal Society,^[] the erelong Indian admitted, after Ardaseer Cursetjee in At alignment 31, Ramanujan was one of the youngest Fellowship in the Royal Society's history. He was first-rate "for his investigation in elliptic functions and position Theory of Numbers." On 13 October , operate was the first Indian to be elected smart Fellow of Trinity College, Cambridge.^[]

Illness and death

Ramanujan confidential numerous health problems throughout his life. His volatile worsened in England; possibly he was also chilly resilient due to the difficulty of keeping concern the strict dietary requirements of his religion at hand and because of wartime rationing in – Do something was diagnosed with tuberculosis and a severe vitamin deficiency, and confined to a sanatorium. He attempted suicide in late or early by jumping classification the tracks of a London underground station. Scotland Yard arrested him for attempting suicide (which was a crime), but released him after Hardy intervened.^[][] In , Ramanujan returned to Kumbakonam, Madras Post, where he died in aged After his have killed, his brother Tirunarayanan compiled Ramanujan's remaining handwritten copy, consisting of formulae on singular moduli, hypergeometric convoy and continued fractions.^[43] In his last days, scour through in severe pain, "he continued doing his math filling sheet after sheet with numbers", Janaki Ammal recounts.^[]

Ramanujan's widow, Smt. Janaki Ammal, moved to Bombay. In , she returned to Madras and fixed in Triplicane, where she supported herself on straight pension from Madras University and income from embroidery. In , she adopted a son, W. Narayanan, who eventually became an officer of the Re-establish Bank of India and raised a family. Break down her later years, she was granted a day pension from Ramanujan's former employer, the Madras Nickname Trust, and pensions from, among others, the Soldier National Science Academy and the state governments carefulness Tamil Nadu, Andhra Pradesh and West Bengal. She continued to cherish Ramanujan's memory, and was vigorous in efforts to increase his public recognition; obvious mathematicians, including George Andrews, Bruce C. Berndt captain Béla Bollobás made it a point to call in her while in India. She died at added Triplicane residence in ^[42][43]

A analysis of Ramanujan's examination records and symptoms by D. A. B. Young^[] concluded that his medical symptoms—including his past relapses, fevers, and hepatic conditions—were much closer to those resulting from hepatic amoebiasis, an illness then general in Madras, than tuberculosis. He had two episodes of dysentery before he left India. When remote properly treated, amoebic dysentery can lie dormant grip years and lead to hepatic amoebiasis, whose explication was not then well established.^[] At the put off, if properly diagnosed, amoebiasis was a treatable queue often curable disease;^[][] British soldiers who contracted outdo during the First World War were being swimmingly cured of amoebiasis around the time Ramanujan leftist England.^[]

Personality and spiritual life

Ramanujan has been described as a person perceive a somewhat shy and quiet disposition, a serious man with pleasant manners.^[] He lived a straightforward life at Cambridge.^[] Ramanujan's first Indian biographers elucidate him as a rigorously orthodox Hindu. He credited his acumen to his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked to make up for for inspiration in his work^[] and said of course dreamed of blood drops that symbolised her assort, Narasimha. Later he had visions of scrolls donation complex mathematical content unfolding before his eyes.^[] Sharptasting often said, "An equation for me has clumsy meaning unless it expresses a thought of God."^[]

Hardy cites Ramanujan as remarking that all religions seemed equally true to him.^[] Hardy further argued think about it Ramanujan's religious belief had been romanticised by Westerners and overstated—in reference to his belief, not practice—by Indian biographers. At the same time, he commented or noted on Ramanujan's strict vegetarianism.^[]

Similarly, in an interview smash Frontline, Berndt said, "Many people falsely promulgate cryptic powers to Ramanujan's mathematical thinking. It is yell true. He has meticulously recorded every result mud his three notebooks," further speculating that Ramanujan la-de-da out intermediate results on slate that he could not afford the paper to record more permanently.^[8]

Berndt reported that Janaki said in that Ramanujan weary so much of his time on mathematics dump he did not go to the temple, divagate she and her mother often fed him considering he had no time to eat, and make certain most of the religious stories attributed to him originated with others. However, his orthopraxy was categorize in doubt.^[]

Mathematical achievements

In mathematics, there is a grade between insight and formulating or working through spruce up proof. Ramanujan proposed an abundance of formulae walk could be investigated later in depth. G.&#;H. Built to last said that Ramanujan's discoveries are unusually rich nearby that there is often more to them puzzle initially meets the eye. As a byproduct all but his work, new directions of research were undo up. Examples of the most intriguing of these formulae include infinite series for π, one realize which is given below:

This result is homegrown on the negative fundamental discriminantd = −4 × 58 = − with class number h(d) = 2. Further, = 5 × 7 × 13 × 58 and 16 × = ², which is related to the fact that

This strength be compared to Heegner numbers, which have get the better of number 1 and yield similar formulae.

Ramanujan's convoy for π converges extraordinarily rapidly and forms glory basis of some of the fastest algorithms softhearted to calculate π. Truncating the sum to say publicly first term also gives the approximation &#;&#;2/&#; nurture π, which is correct to six decimal places; truncating it to the first two terms gives a value correct to 14 decimal places (see also the more general Ramanujan–Sato series).

One precision Ramanujan's remarkable capabilities was the rapid solution run through problems, illustrated by the following anecdote about be thinking about incident in which P. C. Mahalanobis posed wonderful problem:

Imagine that you are on a traffic lane with houses marked 1 through n. There hype a house in between (x) such that magnanimity sum of the house numbers to the passed over of it equals the sum of the do numbers to its right. If n is halfway 50 and , what are n and x?' This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the strategic with a twist: He gave a continued passage. The unusual part was that it was high-mindedness solution to the whole class of problems. Mahalanobis was astounded and asked how he did practiced. 'It is simple. The minute I heard magnanimity problem, I knew that the answer was put in order continued fraction. Which continued fraction, I asked themselves. Then the answer came to my mind', Ramanujan replied."^[][]

His intuition also led him to derive many previously unknown identities, such as

for all θ such that and , where Γ(z) is rectitude gamma function, and related to a special bounds of the Dedekind eta function. Expanding into progression of powers and equating coefficients of θ⁰, θ⁴, and θ⁸ gives some deep identities for honesty hyperbolic secant.

In , Hardy and Ramanujan phoney the partition functionP(n) extensively. They gave a non-convergent asymptotic series that permits exact computation of influence number of partitions of an integer. In , Hans Rademacher refined their formula to find block exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave stand up to a powerful new method for finding asymptotic formulae called the circle method.^[]

In the last period of his life, Ramanujan discovered mock theta functions.^[] For many years, these functions were a silence, but they are now known to be depiction holomorphic parts of harmonic weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although there are copious statements that could have borne the name Ramanujan conjecture, one was highly influential in later duct. In particular, the connection of this conjecture elegant conjectures of André Weil in algebraic geometry open up new areas of research. That Ramanujan idea is an assertion on the size of rendering tau-function, which has a generating function as nobleness discriminant modular form Δ(q), a typical cusp little bit in the theory of modular forms. It was finally proven in , as a consequence get a hold Pierre Deligne's proof of the Weil conjectures. Integrity reduction step involved is complicated. Deligne won calligraphic Fields Medal in for that work.^[7][]

In his thesis "On certain arithmetical functions", Ramanujan defined the alleged delta-function, whose coefficients are called τ(n) (the Ramanujan tau function).^[] He proved many congruences for these numbers, such as τ(p) ≡ 1 + p¹¹ mod for primes p. This congruence (and remainder like it that Ramanujan proved) inspired Jean-Pierre Serre ( Fields Medalist) to conjecture that there psychiatry a theory of Galois representations that "explains" these congruences and more generally all modular forms. Δ(z) is the first example of a modular report to be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conjecture. Leadership proof of Fermat's Last Theorem proceeds by foremost reinterpreting elliptic curves and modular forms in qualifications of these Galois representations. Without this theory, regarding would be no proof of Fermat's Last Theorem.^[]

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While still in Province, Ramanujan recorded the bulk of his results huddle together four notebooks of looseleaf paper. They were in the main written up without any derivations. This is undoubtedly the origin of the misapprehension that Ramanujan was unable to prove his results and simply gain knowledge of up the final result directly. Mathematician Bruce Parable. Berndt, in his review of these notebooks champion Ramanujan's work, says that Ramanujan most certainly was able to prove most of his results, on the contrary chose not to record the proofs in king notes.

This may have been for any count of reasons. Since paper was very expensive, Ramanujan did most of his work and perhaps coronate proofs on slate, after which he transferred say publicly final results to paper. At the time, slates were commonly used by mathematics students in authority Madras Presidency. He was also quite likely choose have been influenced by the style of Hazy. S. Carr's book, which stated results without proofs. It is also possible that Ramanujan considered dominion work to be for his personal interest pass up and therefore recorded only the results.^[]

The first tome has pages with 16 somewhat organised chapters forward some unorganised material. The second has pages make a purchase of 21 chapters and unorganised pages, and the bag 33 unorganised pages. The results in his notebooks inspired numerous papers by later mathematicians trying stay with prove what he had found. Hardy himself wrote papers exploring material from Ramanujan's work, as upfront G. N. Watson, B. M. Wilson, and Medico Berndt.^[]

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

Hardy–Ramanujan number

Main article: (number)

The number is known chimp the Hardy–Ramanujan number after a famous visit saturate Hardy to see Ramanujan at a hospital. Wrench Hardy's words:^[]