Ramanujan mathematical biography and contributions

Indian mathematician (–)

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

In this Indian name, primacy name Srinivasa is a patronym, and the person should last referred to by the stated name, Ramanujan.

Srinivasa Ramanujan FRS
Ramanujan in
Born	Srinivasa Ramanujan Aiyangar ()22 Dec Erode, Mysore State, British Bharat (now in Tamil Nadu, India)
Died	26 April () (aged&#;32) Kumbakonam, Tanjore Division, Madras Presidency, British India (now Thanjavur district, Tamil Nadu, India)
Citizenship	British Indian
Education
Known&#;for
Awards	Fellow of the Royal Society ()
Scientific career
Fields	Mathematics
Institutions	University of Cambridge
Thesis	Highly Synthesis Numbers&#;()
Academic advisors

Srinivasa Ramanujan Aiyangar^[a] (22 December &#;&#; 26 April ) was an Indian mathematician.

Over and over again regarded as one of glory greatest mathematicians of all heart, though he had almost maladroit thumbs down d formal training in pure calculation, he made substantial contributions adjoin mathematical analysis, number theory, unbridled series, and continued fractions, containing solutions to mathematical problems hence considered unsolvable.

Ramanujan initially dash his own mathematical research affluent isolation. According to Hans Psychologist, "he tried to interest decency leading professional mathematicians in rulership work, but failed for honourableness most part. What he confidential to show them was moreover novel, too unfamiliar, and furthermore presented in unusual ways; they could not be bothered".^[4] Hunt mathematicians who could better fluffy his work, in he began a mail correspondence with distinction English mathematician G.

H. Sturdy at the University of University, England. Recognising Ramanujan's work tempt extraordinary, Hardy arranged for him to travel to Cambridge. Grasp his notes, Hardy commented dump Ramanujan had produced groundbreaking additional theorems, including some that "defeated me completely; I had not seen anything in the smallest amount like them before",^[5] and whatsoever recently proven but highly fresh results.

During his short courage, Ramanujan independently compiled nearly 3, results (mostly identities and equations).^[6] Many were completely novel; consummate original and highly unconventional conservative, such as the Ramanujan peak, the Ramanujan theta function, splitup formulae and mock theta functions, have opened entire new areas of work and inspired mint research.^[7] Of his thousands deduction results, most have been demonstrated correct.^[8]The Ramanujan Journal, a wellregulated journal, was established to advise work in all areas very last mathematics influenced by Ramanujan,^[9] stomach his notebooks—containing summaries of jurisdiction published and unpublished results—have antediluvian analysed and studied for decades since his death as uncut source of new mathematical gist.

As late as , researchers continued to discover that unmixed comments in his writings star as "simple properties" and "similar outputs" for certain findings were herself profound and subtle number hypothesis results that remained unsuspected depending on nearly a century after reward death.^[10][11] He became one bear witness the youngest Fellows of representation Royal Society and only decency second Indian member, and class first Indian to be choose a Fellow of Trinity School, Cambridge.

In , ill health—now believed to have been liverwort amoebiasis (a complication from episodes of dysentery many years previously)—compelled Ramanujan's return to India, place he died in at high-mindedness age of His last hand to Hardy, written in Jan , show that he was still continuing to produce another mathematical ideas and theorems.

Consummate "lost notebook", containing discoveries vary the last year of realm life, caused great excitement in the middle of mathematicians when it was rediscovered in

Early life

Ramanujan (literally, "younger brother of Rama", a Faith deity)^[12] was born on 22 December into a Tamil BrahminIyengar family in Erode, in parallel Tamil Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a recorder in a sari shop.^[14][2] Climax mother, Komalatammal, was a homemaker and sang at a go out of business temple.^[15] They lived in put in order small traditional home on Sarangapani Sannidhi Street in the oppidan of Kumbakonam.^[16] The family soupзon is now a museum.

Like that which Ramanujan was a year extremity a half old, his materfamilias gave birth to a jointly, Sadagopan, who died less prevail over three months later. In Dec , Ramanujan contracted smallpox, on the other hand recovered, unlike the 4, blankness who died in a awful year in the Thanjavur part around this time. He unnatural with his mother to congregate parents' house in Kanchipuram, away Madras (now Chennai).

His progenitrix gave birth to two bonus children, in and , both of whom died before their first birthdays.^[12]

On 1 October , Ramanujan was enrolled at justness local school.^[17] After his fatherly grandfather lost his job type a court official in Kanchipuram,^[18] Ramanujan and his mother troubled back to Kumbakonam, and smartness was enrolled in Kangayan Prime School.^[19] When his paternal elder died, he was sent in reply to his maternal grandparents, followed by living in Madras.

He exact not like school in State, and tried to avoid assembly. His family enlisted a provincial constable to make sure proceed attended school. Within six months, Ramanujan was back in Kumbakonam.^[19]

Since Ramanujan's father was at enquiry most of the day, ruler mother took care of probity boy, and they had wonderful close relationship.

From her, crystalclear learned about tradition and puranas, to sing religious songs, follow a line of investigation attend pujas at the church, and to maintain particular feeding habits—all part of Brahmin culture.^[20] At Kangayan Primary School, Ramanujan performed well. Just before unsettled 10, in November , unwind passed his primary examinations deduct English, Tamil, geography, and arithmetical with the best scores observe the district.^[21] That year, Ramanujan entered Town Higher Secondary High school, where he encountered formal science for the first time.^[21]

A youngster prodigy by age 11, smartness had exhausted the mathematical admit of two college students who were lodgers at his soupзon.

He was later lent undiluted book written by S. Renown. Loney on advanced trigonometry.^[22][23] Smartness mastered this by the winner of 13 while discovering polished theorems on his own. Emergency 14, he received merit certificates and academic awards that extended throughout his school career, famous he assisted the school well-off the logistics of assigning treason 1, students (each with various needs) to its approximately 35 teachers.^[24] He completed mathematical exams in half the allotted intention, and showed a familiarity vacate geometry and infinite series.

Ramanujan was shown how to unwavering cubic equations in He would later develop his own ploy to solve the quartic. Execute , he tried to get to the bottom of the quintic, not knowing deviate it was impossible to solution with radicals.^[25]

In , when crystal-clear was 16, Ramanujan obtained non-native a friend a library fake of A Synopsis of Lurking Results in Pure and Going Mathematics, G.

S. Carr's collecting of 5, theorems.^[26][27] Ramanujan reportedly studied the contents of leadership book in detail.^[28] The effort year, Ramanujan independently developed highest investigated the Bernoulli numbers enjoin calculated the Euler–Mascheroni constant become conscious to 15 decimal places.^[29] Diadem peers at the time alleged they "rarely understood him" boss "stood in respectful awe" be fitting of him.^[24]

When he graduated from Inner-city Higher Secondary School in , Ramanujan was awarded the Infant.

Ranganatha Rao prize for calculation by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan owing to an outstanding student who owed scores higher than the maximum.^[30] He received a scholarship proficient study at Government Arts Institution, Kumbakonam,^[31][32] but was so fishinging expedition on mathematics that he could not focus on any thought subjects and failed most extent them, losing his scholarship herbaceous border the process.^[33] In August , Ramanujan ran away from spiteful, heading towards Visakhapatnam, and stayed in Rajahmundry^[34] for about clean month.^[33] He later enrolled watch Pachaiyappa's College in Madras.

Present, he passed in mathematics, decision only to attempt questions dump appealed to him and going the rest unanswered, but ended poorly in other subjects, much as English, physiology, and Sanskrit.^[35] Ramanujan failed his Fellow castigate Arts exam in December jaunt again a year later. Deficient in an FA degree, he nautical port college and continued to marks independent research in mathematics, days in extreme poverty and generally on the brink of starvation.^[36]

In , after a meeting betwixt the year-old Ramanujan and say publicly founder of the Indian Exact Society, V.

Ramaswamy Aiyer, Ramanujan began to get recognition cede Madras's mathematical circles, leading disclose his inclusion as a scientist at the University of Madras.^[37]

Adulthood in India

On 14 July , Ramanujan married Janaki (Janakiammal; 21 March – 13 April ),^[38] a girl his mother confidential selected for him a crop earlier and who was necessity years old when they married.^[39][40][41] It was not unusual authenticate for marriages to be remain with girls at a callow age.

Janaki was from Rajendram, a village close to Marudur (Karur district) Railway Station. Ramanujan's father did not participate hem in the marriage ceremony.^[42] As was common at that time, Janaki continued to stay at torment maternal home for three age after marriage, until she reached puberty. In , she with the addition of Ramanujan's mother joined Ramanujan manifestation Madras.^[43]

After the marriage, Ramanujan matured a hydrocele testis.^[44] The requirement could be treated with natty routine surgical operation that would release the blocked fluid tabled the scrotal sac, but rulership family could not afford picture operation.

In January , adroit doctor volunteered to do integrity surgery at no cost.^[45]

After tiara successful surgery, Ramanujan searched funding a job. He stayed comic story a friend's house while take action went from door to dawn around Madras looking for spick clerical position. To make method, he tutored students at Saddle College who were preparing make public their Fellow of Arts exam.^[46]

In late , Ramanujan was ailing again.

He feared for realm health, and told his comrade R. Radakrishna Iyer to "hand [his notebooks] over to Lecturer Singaravelu Mudaliar [the mathematics senior lecturer at Pachaiyappa's College] or command somebody to the British professor Edward Inelegant. Ross, of the Madras Religionist College."^[47] After Ramanujan recovered splendid retrieved his notebooks from Iyer, he took a train pass up Kumbakonam to Villupuram, a skill under French control.^[48][49] In , Ramanujan moved with his helpmate and mother to a back-to-back in Saiva Muthaiah Mudali structure, George Town, Madras, where they lived for a few months.^[50] In May , upon obtaining a research position at Province University, Ramanujan moved with top family to Triplicane.^[51]

Pursuit of calling in mathematics

In , Ramanujan fall over deputy collector V.

Ramaswamy Aiyer, who founded the Indian Scientific Society.^[52] Wishing for a work at the revenue department site Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled:

I was hit by the extraordinary mathematical frugal contained in [the notebooks]. Wild had no mind to snuff out his genius by an post 2 in the lowest rungs catch the revenue department.^[53]

Aiyer tie Ramanujan, with letters of entry, to his mathematician friends insipid Madras.^[52] Some of them looked at his work and gave him letters of introduction on two legs R.

Ramachandra Rao, the division collector for Nellore and grandeur secretary of the Indian Arithmetical Society.^[54][55][56] Rao was impressed fail to notice Ramanujan's research but doubted desert it was his own enquiry. Ramanujan mentioned a correspondence fair enough had with Professor Saldhana, elegant notable Bombay mathematician, in which Saldhana expressed a lack learn understanding of his work on the contrary concluded that he was pule a fraud.^[57] Ramanujan's friend Catchword.

V. Rajagopalachari tried to compose Rao's doubts about Ramanujan's scholastic integrity. Rao agreed to bring in him another chance, and listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his timidly of divergent series, which Rao said ultimately convinced him very last Ramanujan's brilliance.^[57] When Rao gratuitously him what he wanted, Ramanujan replied that he needed make a hole and financial support.

Rao consented and sent him to State. He continued his research monitor Rao's financial aid. With Aiyer's help, Ramanujan had his bradawl published in the Journal appreciate the Indian Mathematical Society.^[58]

One time off the first problems he friendly in the journal^[30] was on touching find the value of:

He waited for a solution come close to be offered in three issues, over six months, but futile to receive any.

At rectitude end, Ramanujan supplied an incomplete^[59] solution to the problem person. On page of his gain victory notebook, he formulated an equality that could be used process solve the infinitely nested radicals problem.

Using this equation, nobility answer to the question amenable in the Journal was purely 3, obtained by setting x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his first formal system for the Journal on probity properties of Bernoulli numbers.

See to property he discovered was meander the denominators of the fractions of Bernoulli numbers (sequence Span in the OEIS) are at all times divisible by six. He further devised a method of sly B_n based on previous Mathematician numbers. One of these approachs follows:

It will be experiential that if n is much but not equal to digit,

1. B_n is a fraction final the numerator of &#;B_n/n&#; shrub border its lowest terms is splendid prime number,
2. the denominator of B_n contains each of the reality 2 and 3 once endure only once,
3. 2ⁿ(2ⁿ − 1)&#;B_n/n&#; silt an integer and 2(2ⁿ − 1)B_n consequently is an odd integer.

In his page paper "Some Properties of Bernoulli's Numbers" (), Ramanujan gave three proofs, couple corollaries and three conjectures.^[61] Government writing initially had many flaws.

As Journal editor M. Regular. Narayana Iyengar noted:

Worldwide. Ramanujan's methods were so compact and novel and his shape so lacking in clearness prosperous precision, that the ordinary [mathematical reader], unaccustomed to such academic gymnastics, could hardly follow him.^[62]

Ramanujan later wrote another paper unacceptable also continued to provide compression in the Journal.^[63] In anciently , he got a transcribe job in the Madras Teller General's office, with a journal salary of 20 rupees.

Forbidden lasted only a few weeks.^[64] Toward the end of defer assignment, he applied for unadulterated position under the Chief Break of the Madras Port Credit.

In a letter dated 9 February , Ramanujan wrote:

Sir,
&#;I understand there is a clerkship vacant in your office, good turn I beg to apply funds the same.

I have passed the Matriculation Examination and hollow up to the F.A. on the contrary was prevented from pursuing slump studies further owing to a sprinkling untoward circumstances. I have, regardless, been devoting all my offend to Mathematics and developing justness subject. I can say Funny am quite confident I glare at do justice to my research paper if I am appointed join the post.

I therefore press to request that you longing be good enough to converse the appointment on me.^[65]

Attached observe his application was a counsel from E. W. Middlemast, dialect trig mathematics professor at the Leadership College, who wrote that Ramanujan was "a young man closing stages quite exceptional capacity in Mathematics".^[66] Three weeks after he welldesigned, on 1 March, Ramanujan knowledgeable that he had been push as a Class III, Seminar IV accounting clerk, making 30 rupees per month.^[67] At tiara office, Ramanujan easily and flashy completed the work he was given and spent his additional time doing mathematical research.

Ramanujan's boss, Sir Francis Spring, opinion S. Narayana Iyer, a associate who was also treasurer get the message the Indian Mathematical Society, pleased Ramanujan in his mathematical pursuits.

Contacting British mathematicians

In the spring disturb , Narayana Iyer, Ramachandra Rao and E.

W. Middlemast well-tried to present Ramanujan's work make British mathematicians. M. J. Class. Hill of University College Writer commented that Ramanujan's papers were riddled with holes.^[69] He articulate that although Ramanujan had "a taste for mathematics, and harsh ability", he lacked the lawful educational background and foundation go down with be accepted by mathematicians.^[70] Though Hill did not offer distribute take Ramanujan on as clean student, he gave thorough gain serious professional advice on fulfil work.

With the help model friends, Ramanujan drafted letters dealings leading mathematicians at Cambridge University.^[71]

The first two professors, H. Oppressor. Baker and E. W. Hobson, returned Ramanujan's papers without comment.^[72] On 16 January , Ramanujan wrote to G. H.

Durable, whom he knew from lost in thought Orders of Infinity ().^[73][74] Time to come from an unknown mathematician, rectitude nine pages of mathematics obliged Hardy initially view Ramanujan's manuscripts as a possible fraud.^[75] Determined recognised some of Ramanujan's formulae but others "seemed scarcely imaginable to believe".^[76]:&#;&#; One of honourableness theorems Hardy found amazing was on the bottom of fence three (valid for 0 < a < b + &#;1/2&#;):

Hardy was also impressed coarse some of Ramanujan's other gratuitous relating to infinite series:

The first result had already antique determined by G.

Bauer change for the better The second was new assail Hardy, and was derived cheat a class of functions christened hypergeometric series, which had be in first place been researched by Euler stake Gauss. Hardy found these paltry "much more intriguing" than Gauss's work on integrals.^[77] After sight Ramanujan's theorems on continued fractions on the last page friendly the manuscripts, Hardy said interpretation theorems "defeated me completely; Rabid had never seen anything just right the least like them before",^[78] and that they "must mistrust true, because, if they were not true, no one would have the imagination to originate them".^[78] Hardy asked a ally, J.

E. Littlewood, to blur a look at the archives. Littlewood was amazed by Ramanujan's genius. After discussing the credentials with Littlewood, Hardy concluded put off the letters were "certainly distinction most remarkable I have received" and that Ramanujan was "a mathematician of the highest respectable, a man of altogether elite originality and power".^{[76]:&#;–&#;} One confrere, E.

H. Neville, later remarked that "No one who was in the mathematical circles note Cambridge at that time bottle forget the sensation caused fail to see this letter not one [theorem] could have been set unite the most advanced mathematical scrutiny in the world".^[63]

On 8 Feb , Hardy wrote Ramanujan put in order letter expressing interest in emperor work, adding that it was "essential that I should shroud proofs of some of your assertions".^[79] Before his letter attained in Madras during the tertiary week of February, Hardy contacted the Indian Office to scheme for Ramanujan's trip to University.

Secretary Arthur Davies of greatness Advisory Committee for Indian Rank met with Ramanujan to consult the overseas trip.^[80] In assent with his Brahmin upbringing, Ramanujan refused to leave his kingdom to "go to a non-native land", and his parents were also opposed for the be the same as reason.^[81] Meanwhile, he sent Sound a letter packed with theorems, writing, "I have found systematic friend in you who views my labour sympathetically."^[82]

To supplement Hardy's endorsement, Gilbert Walker, a nark mathematical lecturer at Trinity School, Cambridge, looked at Ramanujan's toil and expressed amazement, urging prestige young man to spend over and over again at Cambridge.^[83] As a upshot of Walker's endorsement, B.

Hanumantha Rao, a mathematics professor quandary an engineering college, invited Ramanujan's colleague Narayana Iyer to unblended meeting of the Board regard Studies in Mathematics to agree "what we can do represent S. Ramanujan".^[84] The board large-scale to grant Ramanujan a organ research scholarship of 75 rupees for the next two ripen at the University of Madras.^[85]

While he was engaged as unadorned research student, Ramanujan continued confront submit papers to the Journal of the Indian Mathematical Society. In one instance, Iyer submitted some of Ramanujan's theorems quick summation of series to representation journal, adding, "The following assumption is due to S.

Ramanujan, the mathematics student of Province University." Later in November, Nation Professor Edward B. Ross translate Madras Christian College, whom Ramanujan had met a few days before, stormed into his get the better of one day with his seeing glowing, asking his students, "Does Ramanujan know Polish?" The explanation was that in one weekly, Ramanujan had anticipated the trench of a Polish mathematician whose paper had just arrived hurt the day's mail.^[86] In crown quarterly papers, Ramanujan drew hurry theorems to make definite integrals more easily solvable.

Working slacken off Giuliano Frullani's integral theorem, Ramanujan formulated generalisations that could tweak made to evaluate formerly scenery integrals.^[87]

Hardy's correspondence with Ramanujan turned after Ramanujan refused to build to England.

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Hardy enlisted natty colleague lecturing in Madras, Family. H. Neville, to mentor at an earlier time bring Ramanujan to England.^[88] Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparently had now accepted position proposal; Neville said, "Ramanujan requisite no converting" and "his parents' opposition had been withdrawn".^[63] Ostensibly, Ramanujan's mother had a brilliant dream in which Ramanujan was surrounded by Europeans, and rectitude family goddess, the deity supplementary Namagiri, commanded her "to be subjected to no longer between her top soil and the fulfilment of monarch life's purpose".^[63] On 17 Hoof it , Ramanujan travelled to England by ship,^[89] leaving his helpmeet to stay with his parents in India.

Life in England

Ramanujan departed from Madras aboard loftiness S.S.

Nevasa on 17 Hike ^[91][92] When he disembarked etch London on 14 April, Neville was waiting for him form a junction with a car. Four days succeeding, Neville took him to cap house on Chesterton Road resource Cambridge. Ramanujan immediately began government work with Littlewood and Flourishing.

After six weeks, Ramanujan laid hold of out of Neville's house skull took up residence on Whewell's Court, a five-minute walk circumvent Hardy's room.^[93]

Hardy and Littlewood began to look at Ramanujan's notebooks. Hardy had already received theorems from Ramanujan in the primary two letters, but there were many more results and theorems in the notebooks.

Hardy gnome that some were wrong, bareness had already been discovered, bid the rest were new breakthroughs.^[94] Ramanujan left a deep notion on Hardy and Littlewood. Littlewood commented, "I can believe put off he's at least a Jacobi",^[95] while Hardy said he "can compare him only with Mathematician or Jacobi."^[96]

Ramanujan spent nearly cardinal years in Cambridge collaborating know Hardy and Littlewood, and publicized part of his findings nearby.

Hardy and Ramanujan had immensely contrasting personalities. Their collaboration was a clash of different cultures, beliefs, and working styles. Etch the previous few decades, magnanimity foundations of mathematics had evenly into question and the require for mathematically rigorous proofs was recognised. Hardy was an idel and an apostle of clue and mathematical rigour, whereas Ramanujan was a deeply religious person who relied very strongly accede his intuition and insights.

Tough tried his best to overindulgence the gaps in Ramanujan's teaching and to mentor him explain the need for formal proofs to support his results, on one\'s uppers hindering his inspiration—a conflict deviate neither found easy.

Ramanujan was awarded a Bachelor of School of dance by Research degree^[97][98] (the forefather of the PhD degree) concern March for his work extent highly composite numbers, sections bargain the first part of which had been published the earlier year in the Proceedings suffer defeat the London Mathematical Society. Distinction paper was more than 50 pages long and proved several properties of such numbers.

Rugged disliked this topic area nevertheless remarked that though it pledged with what he called interpretation 'backwater of mathematics', in give Ramanujan displayed 'extraordinary mastery go out with the algebra of inequalities'.^[99]

On 6 December , Ramanujan was pick to the London Mathematical Theatre company. On 2 May , dirt was elected a Fellow exempt the Royal Society,^[] the superfluous Indian admitted, after Ardaseer Cursetjee in At age 31, Ramanujan was one of the youngest Fellows in the Royal Society's history.

He was elected "for his investigation in elliptic functions and the Theory of Numbers." On 13 October , do something was the first Indian come to be elected a Fellow stand for Trinity College, Cambridge.^[]

Illness and death

Ramanujan had numerous health problems all over his life. His health degenerate in England; possibly he was also less resilient due although the difficulty of keeping withstand the strict dietary requirements considerate his religion there and considering of wartime rationing in – He was diagnosed with tb and a severe vitamin lack, and confined to a dispensary.

He attempted suicide in socialize or early by jumping mother the tracks of a Author underground station. Scotland Yard slow him for attempting suicide (which was a crime), but free him after Hardy intervened.^[][] Show , Ramanujan returned to Kumbakonam, Madras Presidency, where he dreary in aged After his surround, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting livestock formulae on singular moduli, hypergeometric series and continued fractions.^[43] Attach his last days, though rerouteing severe pain, "he continued knowledge his mathematics filling sheet make sure of sheet with numbers", Janaki Ammal recounts.^[]

Ramanujan's widow, Smt.

Janaki Ammal, moved to Bombay. In , she returned to Madras opinion settled in Triplicane, where she supported herself on a allowance from Madras University and means from tailoring. In , she adopted a son, W. Narayanan, who eventually became an government agent of the State Bank hook India and raised a affinity. In her later years, she was granted a lifetime allotment from Ramanujan's former employer, grandeur Madras Port Trust, and pensions from, among others, the Amerindian National Science Academy and blue blood the gentry state governments of Tamil Nadu, Andhra Pradesh and West Bengal.

She continued to cherish Ramanujan's memory, and was active imprison efforts to increase his general recognition; prominent mathematicians, including Martyr Andrews, Bruce C. Berndt direct Béla Bollobás made it simple point to visit her onetime in India. She died fatigued her Triplicane residence in ^[42][43]

A analysis of Ramanujan's medical rolls museum and symptoms by D.

Straight. B. Young^[] concluded that her majesty medical symptoms—including his past relapses, fevers, and hepatic conditions—were ostentatious closer to those resulting break hepatic amoebiasis, an illness thence widespread in Madras, than t.b.. He had two episodes appropriate dysentery before he left Bharat.

When not properly treated, amebic dysentery can lie dormant book years and lead to liverwort amoebiasis, whose diagnosis was categorize then well established.^[] At rectitude time, if properly diagnosed, amebiasis was a treatable and usually curable disease;^[][] British soldiers who contracted it during the Cheeriness World War were being in triumph cured of amoebiasis around honourableness time Ramanujan left England.^[]

Personality gift spiritual life

Ramanujan has bent described as a person manipulate a somewhat shy and face disposition, a dignified man narrow pleasant manners.^[] He lived elegant simple life at Cambridge.^[] Ramanujan's first Indian biographers describe him as a rigorously orthodox Asiatic.

He credited his acumen guideline his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. Recognized looked to her for impulse in his work^[] and thought he dreamed of blood drops that symbolised her consort, Narasimha. Later he had visions prime scrolls of complex mathematical volume unfolding before his eyes.^[] Why not?

often said, "An equation give reasons for me has no meaning unless it expresses a thought weekend away God."^[]

Hardy cites Ramanujan as remarking that all religions seemed like one another true to him.^[] Hardy extremely argued that Ramanujan's religious love had been romanticised by Westerners and overstated—in reference to her majesty belief, not practice—by Indian biographers.

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

Similarly, in an interview with Frontline, Berndt said, "Many people under the pretext of ad promulgate mystical powers to Ramanujan's mathematical thinking. It is party true. He has meticulously documented every result in his troika notebooks," further speculating that Ramanujan worked out intermediate results superior slate that he could groan afford the paper to copy more permanently.^[8]

Berndt reported that Janaki said in that Ramanujan clapped out so much of his age on mathematics that he frank not go to the place, that she and her indolence often fed him because forbidden had no time to unmistakable, and that most of loftiness religious stories attributed to him originated with others.

However, wreath orthopraxy was not in doubt.^[]

Mathematical achievements

In mathematics, there is clean distinction between insight and formulating or working through a sponsorship. Ramanujan proposed an abundance presumption formulae that could be investigated later in depth.

G.&#;H. Rugged said that Ramanujan's discoveries preparation unusually rich and that in attendance is often more to them than initially meets the vision. As a byproduct of emperor work, new directions of trial were opened up. Examples own up the most intriguing of these formulae include infinite series backer π, one of which quite good given below:

This result anticipation based on the negative key discriminantd = −4 × 58 = − with class integer h(d) = 2.

Further, = 5 × 7 × 13 × 58 and 16 × = ², which is affiliated to the fact that

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

Ramanujan's series for π converges extraordinarily rapidly and forms significance basis of some of integrity fastest algorithms used to number π.

Truncating the sum draw near the first term also gives the approximation &#;&#;2/&#; for π, which is correct to shake up decimal places; truncating it fail the first two terms gives a value correct to 14 decimal places (see also rendering more general Ramanujan–Sato series).

One of Ramanujan's remarkable capabilities was the rapid solution of straits, illustrated by the following account about an incident in which P.

C. Mahalanobis posed adroit problem:

Imagine that you idea on a street with buildings marked 1 through n. To is a house in in the middle of (x) such that the affixing of the house numbers on a par with the left of it equals the sum of the homestead numbers to its right. Allowing n is between 50 be proof against , what are n stomach x?' This is a bivariate problem with multiple solutions.

Ramanujan thought about it and gave the answer with a twist: He gave a continued division. The unusual part was lapse it was the solution concord the whole class of persuasion. Mahalanobis was astounded and gratuitously how he did it. 'It is simple. The minute Uncontrolled heard the problem, I knew that the answer was top-hole continued fraction.

Which continued cipher, I asked myself. Then honesty answer came to my mind', Ramanujan replied."^[][]

His intuition also anxious him to derive some heretofore unknown identities, such as

for all θ such that obscure , where Γ(z) is significance gamma function, and related be determined a special value of primacy Dedekind eta function.

Expanding pay for series of powers and agreement coefficients of θ⁰, θ⁴, deliver θ⁸ gives some deep identities for the hyperbolic secant.

In , Hardy and Ramanujan phoney the partition functionP(n) extensively. They gave a non-convergent asymptotic keep fit that permits exact computation oust the number of partitions funding an integer.

In , Hans Rademacher refined their formula familiar with find an exact convergent group solution to this problem. Ramanujan and Hardy's work in that area gave rise to efficient powerful new method for decree asymptotic formulae called the salvo method.^[]

In the last year very last his life, Ramanujan discovered ersatz theta functions.^[] For many time eon, these functions were a huggermugger, but they are now acknowledged to be the holomorphic endowments of harmonic weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although there are numerous statements that could have borne blue blood the gentry name Ramanujan conjecture, one was highly influential in later drain. In particular, the connection be successful this conjecture with conjectures present André Weil in algebraic geometry opened up new areas hold research.

That Ramanujan conjecture psychotherapy an assertion on the prove correct of the tau-function, which has a generating function as significance discriminant modular form Δ(q), unornamented typical cusp form in righteousness theory of modular forms. Colour up rinse was finally proven in , as a consequence of Pierre Deligne's proof of the Mathematician conjectures.

The reduction step active is complicated. Deligne won a-okay Fields Medal in for focus work.^[7][]

In his paper "On set arithmetical functions", Ramanujan defined illustriousness so-called delta-function, whose coefficients strategy called τ(n) (the Ramanujan tau function).^[] He proved many congruences for these numbers, such because τ(p) ≡ 1 + p¹¹ mod for primes p.

That congruence (and others like give rise to that Ramanujan proved) inspired Jean-Pierre Serre ( Fields Medalist) cause problems conjecture that there is unadorned theory of Galois representations lose one\'s train of thought "explains" these congruences and improved generally all modular forms. Δ(z) is the first example brake a modular form to print studied in this way.

Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The admonish of Fermat's Last Theorem return by first reinterpreting elliptic turnings and modular forms in price of these Galois representations. Outdoors this theory, there would verbal abuse no proof of Fermat's Surname Theorem.^[]

Ramanujan's notebooks

Further information: Ramanujan's left out notebook

While still in Madras, Ramanujan recorded the bulk of coronet results in four notebooks round looseleaf paper.

They were habitually written up without any derivations. This is probably the basis of the misapprehension that Ramanujan was unable to prove authority results and simply thought delineate the final result directly. Mathematician Bruce C. Berndt, in rule review of these notebooks dispatch Ramanujan's work, says that Ramanujan most certainly was able confront prove most of his skimpy, but chose not to write down the proofs in his carbon copy.

This may have been backer any number of reasons. In that paper was very expensive, Ramanujan did most of his travail and perhaps his proofs fraudulent slate, after which he transferred the final results to engrave. At the time, slates were commonly used by mathematics category in the Madras Presidency. Earth was also quite likely get to the bottom of have been influenced by illustriousness style of G.

S. Carr's book, which stated results insolvent proofs. It is also imaginable that Ramanujan considered his office to be for his lonely interest alone and therefore reliable only the results.^[]

The first publication has pages with 16 slightly organised chapters and some nonunionized material.

The second has pages in 21 chapters and nonunion pages, and the third 33 unorganised pages. The results tidy his notebooks inspired numerous credentials by later mathematicians trying swing by prove what he had speck. Hardy himself wrote papers interested material from Ramanujan's work, owing to did G. N. Watson, Uneasy. M. Wilson, and Bruce Berndt.^[]

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

Hardy–Ramanujan number

Main article: (number)

The consider is known as the Hardy–Ramanujan number after a famous send back by Hardy to see Ramanujan at a hospital.

In Hardy's words:^[]