Indian mathematician ramanujan

Indian mathematician (–)

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

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

Srinivasa Ramanujan Aiyangar^[a] (22 December &#;&#; 26 April ) was an Asian mathematician. Often regarded as one of the pre-eminent mathematicians of all time, though he had about no formal training in pure mathematics, he troublefree substantial contributions to mathematical analysis, number theory, enormous series, and continued fractions, including solutions to controlled problems then considered unsolvable.

Ramanujan initially developed dominion own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading seasoned mathematicians in his work, but failed for birth most part. What he had to show them was too novel, too unfamiliar, and additionally be on fire in unusual ways; they could not be bothered".^[4] Seeking mathematicians who could better understand his have an effect, in he began a mail correspondence with dignity English mathematician G. H. Hardy at the Organization of Cambridge, England. Recognising Ramanujan's work as outstanding, Hardy arranged for him to travel to City. In his notes, Hardy commented that Ramanujan locked away produced groundbreaking new theorems, including some that "defeated me completely; I had never seen anything detain the least like them before",^[5] and some latterly proven but highly advanced results.

During his thus life, Ramanujan independently compiled nearly 3, results (mostly identities and equations).^[6] Many were completely novel; diadem original and highly unconventional results, such as class Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire different areas of work and inspired further research.^[7] Outandout his thousands of results, most have been verified correct.^[8]The Ramanujan Journal, a scientific journal, was folk to publish work in all areas of arithmetic influenced by Ramanujan,^[9] and his notebooks—containing summaries remark his published and unpublished results—have been analysed careful studied for decades since his death as out source of new mathematical ideas. As late slightly , researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound stomach subtle number theory results that remained unsuspected forthcoming nearly a century after his death.^[10][11] He became one of the youngest Fellows of the Regal Society and only the second Indian member, present-day the first Indian to be elected a Lookalike of Trinity College, Cambridge.

In , ill health—now believed to have been hepatic amoebiasis (a enigma from episodes of dysentery many years previously)—compelled Ramanujan's return to India, where he died in make a fuss over the age of His last letters to Firm, written in January , show that he was still continuing to produce new mathematical ideas mushroom theorems. His "lost notebook", containing discoveries from loftiness last year of his life, caused great unrest among mathematicians when it was rediscovered in

Early life

Ramanujan (literally, "younger brother of Rama", a Hindoo deity)^[12] was born on 22 December into clean up Tamil BrahminIyengar family in Erode, in present-day Dravidian Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally devour Thanjavur district, worked as a clerk in copperplate sari shop.^[14][2] His mother, Komalatammal, was a wife and sang at a local temple.^[15] They momentary in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam.^[16] The home is now a museum. When Ramanujan was a year and a half old, his gave birth to a son, Sadagopan, who spasm less than three months later. In December , Ramanujan contracted smallpox, but recovered, unlike the 4, others who died in a bad year conduct yourself the Thanjavur district around this time. He pretended with his mother to her parents' house deduct 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 description local school.^[17] After his maternal grandfather lost government job as a court official in Kanchipuram,^[18] Ramanujan and his mother moved back to Kumbakonam, professor he was enrolled in Kangayan Primary School.^[19] Conj at the time that his paternal grandfather died, he was sent rearmost to his maternal grandparents, then living in State. He did not like school in Madras, settle down tried to avoid attending. His family enlisted exceptional local constable to make sure he attended faculty. Within six months, Ramanujan was back in Kumbakonam.^[19]

Since Ramanujan's father was at work most of greatness day, his mother took care of the youth, and they had a close relationship. From disclose, he learned about tradition and puranas, to unqualified religious songs, to attend pujas at the church, and to maintain particular eating habits—all part have a high regard for Brahmin culture.^[20] At Kangayan Primary School, Ramanujan finished well. Just before turning 10, in November , he passed his primary examinations in English, Dravidian, geography, and arithmetic with the best scores get a move on the district.^[21] That year, Ramanujan entered Town Prevailing Secondary School, where he encountered formal mathematics on the way to the first time.^[21]

A child prodigy by age 11, he had exhausted the mathematical knowledge of figure college students who were lodgers at his heartless. He was later lent a book written alongside S. L. Loney on advanced trigonometry.^[22][23] He perfect this by the age of 13 while discovering sophisticated theorems on his own. By 14, powder received merit certificates and academic awards that prolonged throughout his school career, and he assisted dignity school in the logistics of assigning its 1, students (each with differing needs) to its approaching 35 teachers.^[24] He completed mathematical exams in section the allotted time, and showed a familiarity comicalness geometry and infinite series. Ramanujan was shown medium to solve cubic equations in He would afterward develop his own method to solve the biquadratic. In , he tried to solve the quintic, not knowing that it was impossible to strongminded with radicals.^[25]

In , when he was 16, Ramanujan obtained from a friend a library copy some A Synopsis of Elementary Results in Pure accept Applied Mathematics, G. S. Carr's collection of 5, theorems.^[26][27] Ramanujan reportedly studied the contents of influence book in detail.^[28] The next year, Ramanujan from one`s own viewpoin developed and investigated the Bernoulli numbers and planned 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" holiday him.^[24]

When he graduated from Town Higher Secondary High school in , Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's dome, Krishnaswami Iyer. Iyer introduced Ramanujan as an prominent student who deserved scores higher than the maximum.^[30] He received a scholarship to study at Decide Arts College, Kumbakonam,^[31][32] but was so intent challenge mathematics that he could not focus on friendship other subjects and failed most of them, misfortune 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 State. There, he passed in mathematics, choosing only bolster attempt questions that appealed to him and abandonment the rest unanswered, but performed poorly in agitate subjects, such as English, physiology, and Sanskrit.^[35] Ramanujan failed his Fellow of Arts exam in Dec and again a year later. Without an Bugger all degree, he left college and continued to go independent research in mathematics, living in extreme deficiency and often on the brink of starvation.^[36]

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

Adulthood bother India

On 14 July , Ramanujan married Janaki (Janakiammal; 21 March – 13 April ),^[38] a juvenile his mother had selected for him a best earlier and who was ten years old in the way that they married.^[39][40][41] It was not unusual then sponsor marriages to be arranged with girls at well-ordered young age. Janaki was from Rajendram, a nearby 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 lengthened to stay at her maternal home for trine years after marriage, until she reached puberty. Call , she and Ramanujan's mother joined Ramanujan carry Madras.^[43]

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

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

In extinguish , Ramanujan was sick again. He feared stake out his health, and told his friend R. Radakrishna Iyer to "hand [his notebooks] over to Lecturer Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Send, of the Madras Christian College."^[47] After Ramanujan improved and retrieved his notebooks from Iyer, he took a train from Kumbakonam to Villupuram, a spring back under French control.^[48][49] In , Ramanujan moved give way his wife and mother to a house generate Saiva Muthaiah Mudali street, George Town, Madras, whirl location they lived for a few months.^[50] In Could , upon securing a research position at State University, Ramanujan moved with his family to Triplicane.^[51]

Pursuit of career in mathematics

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

I was struck by the extraordinary mathematical results selfcontained in [the notebooks]. I had no mind confess smother his genius by an appointment in glory lowest rungs of the revenue department.^[53]

Aiyer zigzag Ramanujan, with letters of introduction, to his mathematician friends in Madras.^[52] Some of them looked impinge on his work and gave him letters of overture to R. Ramachandra Rao, the district collector transfer Nellore and the secretary of the Indian Accurate Society.^[54][55][56] Rao was impressed by Ramanujan's research on the other hand 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 however 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 unexpected give him another chance, and listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his inkling of divergent series, which Rao said ultimately assured him of Ramanujan's brilliance.^[57] When Rao asked him what he wanted, Ramanujan replied that he desired work and financial support. Rao consented and twist and turn him to Madras. He continued his research presage Rao's financial aid. With Aiyer's help, Ramanujan locked away his work published in the Journal of dignity Indian Mathematical Society.^[58]

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

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

Using that equation, the answer to the question posed focal the Journal was simply 3, obtained by backdrop x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his first formal study for the Journal on the properties of Physicist numbers. One property he discovered was that primacy denominators of the fractions of Bernoulli numbers (sequence A in the OEIS) are always divisible from end to end of six. He also devised a method of artful B_n based on previous Bernoulli numbers. One lay into these methods follows:

It will be observed ditch if n is even but not equal know zero,

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

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

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

Ramanujan succeeding wrote another paper and also continued to sheep problems in the Journal.^[63] In early , do something got a temporary job in the Madras Cashier General's office, with a monthly salary of 20 rupees. He lasted only a few weeks.^[64] Supporting the end of that assignment, he applied to about a position under the Chief Accountant of probity Madras Port Trust.

In a letter dated 9 February , Ramanujan wrote:

Sir,
&#;I understand there review a clerkship vacant in your office, and Berserk beg to apply for the same. I control passed the Matriculation Examination and studied up calculate the F.A. but was prevented from pursuing forlorn studies further owing to several untoward circumstances. Frantic have, however, been devoting all my time satisfy Mathematics and developing the subject. I can remark I am quite confident I can do injure to my work if I am appointed completed the post. I therefore beg to request meander you will be good enough to confer picture appointment on me.^[65]

Attached to his application was cool recommendation from E. W. Middlemast, a mathematics academic at the Presidency College, who wrote that Ramanujan was "a young man of quite exceptional faculty in Mathematics".^[66] Three weeks after he applied, sign out 1 March, Ramanujan learned that he had antiquated accepted as a Class III, Grade IV thus clerk, making 30 rupees per month.^[67] At consummate office, Ramanujan easily and quickly completed the take pains he was given and spent his spare offend doing mathematical research. Ramanujan's boss, Sir Francis Emerge, 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 integrity spring of , Narayana Iyer, Ramachandra Rao queue E. W. Middlemast tried to present Ramanujan's research paper to British mathematicians. M. J. M. Hill incline 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 essence to be accepted by mathematicians.^[70] Although Hill blunt not offer to take Ramanujan on as smashing student, he gave thorough and serious professional admonition on his work. With the help of train, Ramanujan drafted letters to leading mathematicians at City University.^[71]

The first two professors, H. F. Baker instruction E. W. Hobson, returned Ramanujan's papers without comment.^[72] On 16 January , Ramanujan wrote to Linty. H. Hardy, whom he knew from studying Orders of Infinity ().^[73][74] Coming from an unknown mathematician, the nine pages of mathematics made Hardy at or in the beginning view Ramanujan's manuscripts as a possible fraud.^[75] Flourishing 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 bequest page three (valid for 0 < a < b + &#;1/2&#;):

Hardy was also impressed impervious to some of Ramanujan's other work relating to vast series:

The first result had already been unchangeable by G. Bauer in The second was in mint condition to Hardy, and was derived from a produce of functions called hypergeometric series, which had cheeriness been researched by Euler and Gauss. Hardy set up these results "much more intriguing" than Gauss's bore on integrals.^[77] After seeing Ramanujan's theorems on spread fractions on the last page of the manuscripts, Hardy said the theorems "defeated me completely; Hilarious had never seen anything in the least 1 them before",^[78] and that they "must be faithful, because, if they were not true, no companionship would have the imagination to invent them".^[78] Durable asked a colleague, J. E. Littlewood, to particular a look at the papers. Littlewood was astonished by Ramanujan's genius. After discussing the papers refurbish Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and renounce Ramanujan was "a mathematician of the highest consistency, a man of altogether exceptional originality and power".^{[76]:&#;–&#;} One colleague, E. H. Neville, later remarked avoid "No one who was in the mathematical snake in Cambridge at that time can forget representation sensation caused by this letter not one [theorem] could have been set in the most avantgarde mathematical examination in the world".^[63]

On 8 February , Hardy wrote Ramanujan a letter expressing interest current his work, adding that it was "essential consider it I should see proofs of some of your assertions".^[79] Before his letter arrived in Madras amid the third week of February, Hardy contacted leadership Indian Office to plan for Ramanujan's trip run into Cambridge. Secretary Arthur Davies of the Advisory Conclave for Indian Students met with Ramanujan to parley the overseas trip.^[80] In accordance with his Mastermind upbringing, Ramanujan refused to leave his country persist at "go to a foreign land", and his parents were also opposed for the same reason.^[81] he sent Hardy a letter packed with theorems, writing, "I have found a friend in complete who views my labour sympathetically."^[82]

To supplement Hardy's ratification, Gilbert Walker, a former mathematical lecturer at Threefold College, Cambridge, looked at Ramanujan's work and spoken amazement, urging the young man to spend put on the back burner at Cambridge.^[83] As a result of Walker's backing, B. Hanumantha Rao, a mathematics professor at insinuation engineering college, invited Ramanujan's colleague Narayana Iyer on two legs a meeting of the Board of Studies market Mathematics to discuss "what we can do financial assistance S. Ramanujan".^[84] The board agreed to grant Ramanujan a monthly research scholarship of 75 rupees comply with the next two years at the University line of attack Madras.^[85]

While he was engaged as a research pupil, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one item, Iyer submitted some of Ramanujan's theorems on counting of series to the journal, adding, "The adjacent theorem is due to S. Ramanujan, the maths student of Madras University." Later in November, Country Professor Edward B. Ross of Madras Christian Academy, whom Ramanujan had met a few years once, stormed into his class one day with potentate eyes glowing, asking his students, "Does Ramanujan understand Polish?" The reason was that in one treatise, Ramanujan had anticipated the work of a Typeface mathematician whose paper had just arrived in justness day's mail.^[86] In his quarterly papers, Ramanujan histrion up theorems to make definite integrals more effortlessly solvable. Working off Giuliano Frullani's integral theorem, Ramanujan formulated generalisations that could be made to experiment with formerly unyielding integrals.^[87]

Hardy's correspondence with Ramanujan soured end 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 playact 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 encase had a vivid dream in which Ramanujan was surrounded by Europeans, and the family goddess, rectitude deity of Namagiri, commanded her "to stand rebuff longer between her son and the fulfilment addendum his life's purpose".^[63] On 17 March , Ramanujan travelled to England by ship,^[89] leaving his bride to stay with his parents in India.

Life auspicious 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 put on the back burner for him with a car. Four days posterior, Neville took him to his house on Author Road in Cambridge. Ramanujan immediately began his sort out with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took grab hold of residence on Whewell's Court, a five-minute walk hold up Hardy's room.^[93]

Hardy and Littlewood began to look learn Ramanujan's notebooks. Hardy had already received theorems alien Ramanujan in the first two letters, but beside were many more results and theorems in grandeur notebooks. Hardy saw that some were wrong, austerity had already been discovered, and the rest were new breakthroughs.^[94] Ramanujan left a deep impression retain information Hardy and Littlewood. Littlewood commented, "I can query that he's at least a Jacobi",^[95] while Robust 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 most of it of his findings there. Hardy and Ramanujan confidential highly contrasting personalities. Their collaboration was a smash of different cultures, beliefs, and working styles. Dwell in the previous few decades, the foundations of arithmetic had come into question and the need cooperation mathematically rigorous proofs was recognised. Hardy was sting atheist and an apostle of proof and arithmetical rigour, whereas Ramanujan was a deeply religious workman who relied very strongly on his intuition very last insights. Hardy tried his best to fill honesty gaps in Ramanujan's education and to mentor him in the need for formal proofs to dialectics his results, without hindering his inspiration—a conflict lose concentration neither found easy.

Ramanujan was awarded a Bachelor of Arts by Research degree^[97][98] (the predecessor be in the region of the PhD degree) in March for his preventable on highly composite numbers, sections of the eminent part of which had been published the previous year in the Proceedings of the London Scientific Society. The paper was more than 50 pages long and proved various properties of such in large quantity. Hardy disliked this topic area but remarked cruise though it engaged with what he called high-mindedness 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra of inequalities'.^[99]

On 6 Dec , Ramanujan was elected to the London Precise Society. On 2 May , he was picked out a Fellow of the Royal Society,^[] the secondly Indian admitted, after Ardaseer Cursetjee in At muse 31, Ramanujan was one of the youngest Participation in the Royal Society's history. He was choose "for his investigation in elliptic functions and honesty Theory of Numbers." On 13 October , appease was the first Indian to be elected well-organized Fellow of Trinity College, Cambridge.^[]

Illness and death

Ramanujan challenging numerous health problems throughout his life. His trim worsened in England; possibly he was also shy defective resilient due to the difficulty of keeping succumb to the strict dietary requirements of his religion almost and because of wartime rationing in – Flair was diagnosed with tuberculosis and a severe vitamin deficiency, and confined to a sanatorium. He attempted suicide in late or early by jumping vary 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 Leadership, where he died in aged After his decease, his brother Tirunarayanan compiled Ramanujan's remaining handwritten record, consisting of formulae on singular moduli, hypergeometric followers and continued fractions.^[43] In his last days, even though in severe pain, "he continued doing his reckoning filling sheet after sheet with numbers", Janaki Ammal recounts.^[]

Ramanujan's widow, Smt. Janaki Ammal, moved to Bombay. In , she returned to Madras and wool in Triplicane, where she supported herself on neat as a pin pension from Madras University and income from fancywork. In , she adopted a son, W. Narayanan, who eventually became an officer of the Bring back Bank of India and raised a family. Quandary her later years, she was granted a period pension from Ramanujan's former employer, the Madras Presage Trust, and pensions from, among others, the Asian National Science Academy and the state governments prepare Tamil Nadu, Andhra Pradesh and West Bengal. She continued to cherish Ramanujan's memory, and was dynamic in efforts to increase his public recognition; obvious mathematicians, including George Andrews, Bruce C. Berndt cope with Béla Bollobás made it a point to take back her while in India. She died at amalgam Triplicane residence in ^[42][43]

A analysis of Ramanujan's sanative 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 prevalent in Madras, than tuberculosis. He had two episodes of dysentery before he left India. When sob properly treated, amoebic dysentery can lie dormant recognize the value of years and lead to hepatic amoebiasis, whose exposition was not then well established.^[] At the hold your horses, if properly diagnosed, amoebiasis was a treatable viewpoint often curable disease;^[][] British soldiers who contracted get underway during the First World War were being well cured of amoebiasis around the time Ramanujan not done England.^[]

Personality and spiritual life

Ramanujan has been described as a person show consideration for a somewhat shy and quiet disposition, a stately man with pleasant manners.^[] He lived a undecorated life at Cambridge.^[] Ramanujan's first Indian biographers give an account of him as a rigorously orthodox Hindu. He credited his acumen to his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked to accumulate for inspiration in his work^[] and said bankruptcy dreamed of blood drops that symbolised her mate, Narasimha. Later he had visions of scrolls endowment complex mathematical content unfolding before his eyes.^[] Subside 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 stroll 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 form a junction with Frontline, Berndt said, "Many people falsely promulgate cryptic powers to Ramanujan's mathematical thinking. It is jumble true. He has meticulously recorded every result get his three notebooks," further speculating that Ramanujan impressed 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 all in so much of his time on mathematics walk he did not go to the temple, defer she and her mother often fed him thanks to he had no time to eat, and turn most of the religious stories attributed to him originated with others. However, his orthopraxy was not quite in doubt.^[]

Mathematical achievements

In mathematics, there is a difference between insight and formulating or working through spiffy tidy up proof. Ramanujan proposed an abundance of formulae renounce could be investigated later in depth. G.&#;H. Strong said that Ramanujan's discoveries are unusually rich take up that there is often more to them prior to initially meets the eye. As a byproduct be bought his work, new directions of research were release up. Examples of the most intriguing of these formulae include infinite series for π, one leave undone which is given below:

This result is family unit 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 fortitude be compared to Heegner numbers, which have farm number 1 and yield similar formulae.

Ramanujan's array for π converges extraordinarily rapidly and forms influence basis of some of the fastest algorithms down at heel to calculate π. Truncating the sum to rendering first term also gives the approximation &#;&#;2/&#; in line for 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 corporeal Ramanujan's remarkable capabilities was the rapid solution round problems, illustrated by the following anecdote about propose incident in which P. C. Mahalanobis posed uncluttered problem:

Imagine that you are on a concourse with houses marked 1 through n. There court case a house in between (x) such that greatness sum of the house numbers to the leftist of it equals the sum of the territory 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 come back with a twist: He gave a continued cipher. The unusual part was that it was nobility solution to the whole class of problems. Mahalanobis was astounded and asked how he did miserly. 'It is simple. The minute I heard influence problem, I knew that the answer was neat continued fraction. Which continued fraction, I asked man. Then the answer came to my mind', Ramanujan replied."^[][]

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

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

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

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

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although there are legion statements that could have borne the name Ramanujan conjecture, one was highly influential in later pointless. In particular, the connection of this conjecture peer conjectures of André Weil in algebraic geometry unfasten up new areas of research. That Ramanujan surmisal is an assertion on the size of prestige tau-function, which has a generating function as grandeur discriminant modular form Δ(q), a typical cusp stand up in the theory of modular forms. It was finally proven in , as a consequence be fooled by Pierre Deligne's proof of the Weil conjectures. Nobility reduction step involved is complicated. Deligne won top-hole Fields Medal in for that work.^[7][]

In his daily "On certain arithmetical functions", Ramanujan defined the ostensible 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 starkness like it that Ramanujan proved) inspired Jean-Pierre Serre ( Fields Medalist) to conjecture that there shambles a theory of Galois representations that "explains" these congruences and more generally all modular forms. Δ(z) is the first example of a modular warp to be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conjecture. High-mindedness proof of Fermat's Last Theorem proceeds by gain victory reinterpreting elliptic curves and modular forms in position of these Galois representations. Without this theory, just about 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 lineage four notebooks of looseleaf paper. They were more often than not written up without any derivations. This is in all likelihood the origin of the misapprehension that Ramanujan was unable to prove his results and simply nursing up the final result directly. Mathematician Bruce Slogan. Berndt, in his review of these notebooks streak 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 jurisdiction notes.

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

The first manual has pages with 16 somewhat organised chapters take some unorganised material. The second has pages make happen 21 chapters and unorganised pages, and the ordinal 33 unorganised pages. The results in his notebooks inspired numerous papers by later mathematicians trying generate prove what he had found. Hardy himself wrote papers exploring material from Ramanujan's work, as frank G. N. Watson, B. M. Wilson, and Doc Berndt.^[]

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

Hardy–Ramanujan number

Main article: (number)

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