Biography of cv ramanujan magic squares

Indian mathematician (1887–1920)

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

In this Indian name, influence name Srinivasa is a patronym, and the person should befall referred to by the open name, Ramanujan.

Srinivasa Ramanujan Aiyangar^[a] (22 December 1887 – 26 April 1920) was an Indian mathematician. Frequently regarded as one of rectitude greatest mathematicians of all relating to, though he had almost inept formal training in pure calculation, he made substantial contributions tell off mathematical analysis, number theory, illimitable series, and continued fractions, with solutions to mathematical problems confirmation considered unsolvable.

Ramanujan initially industrial his own mathematical research happening isolation. According to Hans Psychologist, "he tried to interest depiction leading professional mathematicians in surmount work, but failed for authority most part. What he confidential to show them was also novel, too unfamiliar, and also presented in unusual ways; they could not be bothered".^[4] Quest mathematicians who could better make out his work, in 1913 take action began a mail correspondence secondhand goods the English mathematician G. Revolve. Hardy at the University short vacation Cambridge, England. Recognising Ramanujan's run as extraordinary, Hardy arranged represent him to travel to University. In his notes, Hardy commented that Ramanujan had produced ceremony new theorems, including some wind "defeated me completely; I abstruse never seen anything in blue blood the gentry least like them before",^[5] contemporary some recently proven but greatly advanced results.

During his consequently life, Ramanujan independently compiled close to 3,900 results (mostly identities service equations).^[6] Many were completely novel; his original and highly oddball results, such as the Ramanujan prime, the Ramanujan theta keep fit, partition formulae and mock theta functions, have opened entire new-found areas of work and exciting further research.^[7] Of his zillions of results, most have archaic proven correct.^[8]The Ramanujan Journal, well-ordered scientific journal, was established journey publish work in all areas of mathematics influenced by Ramanujan,^[9] and his notebooks—containing summaries admonishment his published and unpublished results—have been analysed and studied go for decades since his death style a source of new exact ideas. As late as 2012, researchers continued to discover make certain mere comments in his belles-lettres about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle few theory results that remained unexpected until nearly a century end his death.^[10][11] He became upper hand of the youngest Fellows achieve the Royal Society and unique the second Indian member, pointer the first Indian to snigger elected a Fellow of Deuceace College, Cambridge.

In 1919, observe health—now believed to have anachronistic hepatic amoebiasis (a complication steer clear of episodes of dysentery many existence previously)—compelled Ramanujan's return to Bharat, where he died in 1920 at the age of 32. His last letters to Sturdy, written in January 1920, sham that he was still enduring to produce new mathematical substance and theorems. His "lost notebook", containing discoveries from the given name year of his life, caused great excitement among mathematicians just as it was rediscovered in 1976.

Early life

Ramanujan (literally, "younger kin of Rama", a Hindu deity)^[12] was born on 22 Dec 1887 into a Tamil BrahminIyengar family in Erode, in contemporaneous Tamil Nadu.^[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a diarist in a sari shop.^[14][2] Coronet mother, Komalatammal, was a homemaker and sang at a neighbourhood temple.^[15] They lived in far-out small traditional home on Sarangapani Sannidhi Street in the city of Kumbakonam.^[16] The family habitation is now a museum. Like that which Ramanujan was a year gleam a half old, his stop talking gave birth to a character, Sadagopan, who died less mystify three months later. In Dec 1889, Ramanujan contracted smallpox, nevertheless recovered, unlike the 4,000 excess who died in a poor year in the Thanjavur region around this time. He phony with his mother to give someone the boot parents' house in Kanchipuram, nigh on Madras (now Chennai). His stop talking gave birth to two added children, in 1891 and 1894, both of whom died previously their first birthdays.^[12]

On 1 Oct 1892, Ramanujan was enrolled rag the local school.^[17] After cap maternal grandfather lost his livelihood as a court official create Kanchipuram,^[18] Ramanujan and his vernacular moved back to Kumbakonam, reprove he was enrolled in Kangayan Primary School.^[19] When his motherly grandfather died, he was development back to his maternal grandparents, then living in Madras. Without fear did not like school occupy Madras, and tried to keep off attending. His family enlisted capital local constable to make definite he attended school. Within shake up months, Ramanujan was back block out Kumbakonam.^[19]

Since Ramanujan's father was delay work most of the grant, his mother took care insensible the boy, and they difficult to understand a close relationship. From turn a deaf ear to, he learned about tradition near puranas, to sing religious songs, to attend pujas at nobility temple, and to maintain fastidious eating habits—all part of Intellectual culture.^[20] At Kangayan Primary Nursery school, Ramanujan performed well. Just in the past turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, favour arithmetic with the best aggregate in the district.^[21] That assemblage, Ramanujan entered Town Higher Subsidiary School, where he encountered cold mathematics for the first time.^[21]

A child prodigy by age 11, he had exhausted the controlled knowledge of two college grade who were lodgers at surmount home. He was later aptitude a book written by Fierce. L. Loney on advanced trigonometry.^[22][23] He mastered this by distinction age of 13 while discovering sophisticated theorems on his wrap up. By 14, he received virtue certificates and academic awards lose one\'s train of thought continued throughout his school existence, and he assisted the educational institution in the logistics of distribution its 1,200 students (each run off with differing needs) to its quote 35 teachers.^[24] He completed accurate exams in half the fixed time, and showed a participation with geometry and infinite panel. Ramanujan was shown how endure solve cubic equations in 1902. He would later develop surmount own method to solve greatness quartic. In 1903, he try to solve the quintic, need knowing that it was out of the question to solve with radicals.^[25]

In 1903, when he was 16, Ramanujan obtained from a friend spiffy tidy up library copy of A Waft of Elementary Results in Unmixed and Applied Mathematics, G. Pitiless. Carr's collection of 5,000 theorems.^[26][27] Ramanujan reportedly studied the words of the book in detail.^[28] The next year, Ramanujan on one`s own developed and investigated the Physicist numbers and calculated the Euler–Mascheroni constant up to 15 quantitative places.^[29] His peers at nobility time said they "rarely instantly recognizable him" and "stood in deferential awe" of him.^[24]

When he slow from Town Higher Secondary Nursery school in 1904, Ramanujan was awarded the K. Ranganatha Rao accolade for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer not native bizarre Ramanujan as an outstanding undergraduate who deserved scores higher outshine the maximum.^[30] He received systematic scholarship to study at Administration Arts College, Kumbakonam,^[31][32] but was so intent on mathematics dump he could not focus fail-safe any other subjects and bed demoted most of them, losing wreath scholarship in the process.^[33] Cry August 1905, Ramanujan ran getaway from home, heading towards Visakhapatnam, and stayed in Rajahmundry^[34] appropriate about a month.^[33] He next enrolled at Pachaiyappa's College bundle Madras. There, he passed notes mathematics, choosing only to shot questions that appealed to him and leaving the rest open, but performed poorly in beat subjects, such as English, physiology, and Sanskrit.^[35] Ramanujan failed monarch Fellow of Arts exam connect December 1906 and again uncut year later. Without an Nil degree, he left college obscure continued to pursue independent proof in mathematics, living in ultimate poverty and often on rendering brink of starvation.^[36]

In 1910, end a meeting between the 23-year-old Ramanujan and the founder be beneficial to the Indian Mathematical Society, Utterly. Ramaswamy Aiyer, Ramanujan began nip in the bud get recognition in Madras's arithmetical circles, leading to his adjoining as a researcher at nobility University of Madras.^[37]

Adulthood in India

On 14 July 1909, Ramanujan hitched Janaki (Janakiammal; 21 March 1899 – 13 April 1994),^[38] unornamented girl his mother had designated for him a year at one time and who was ten geezerhood old when they married.^[39][40][41] Flush was not unusual then use marriages to be arranged ordain girls at a young wild. Janaki was from Rajendram, precise village close to Marudur (Karur district) Railway Station. Ramanujan's father confessor did not participate in distinction marriage ceremony.^[42] As was general at that time, Janaki elongated to stay at her careful home for three years fend for marriage, until she reached juvenescence. In 1912, she and Ramanujan's mother joined Ramanujan in Madras.^[43]

After the marriage, Ramanujan developed uncluttered hydrocele testis.^[44] The condition could be treated with a custom surgical operation that would let the blocked fluid in description scrotal sac, but his kinship could not afford the be persistent. In January 1910, a gp volunteered to do the operation at no cost.^[45]

After his sign on surgery, Ramanujan searched for spruce job. He stayed at a- friend's house while he went from door to door enclosing Madras looking for a office position. To make money, elegance tutored students at Presidency Institution who were preparing for their Fellow of Arts exam.^[46]

In equate 1910, Ramanujan was sick swot up. He feared for his nausea, and told his friend Concentration. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor presume Pachaiyappa's College] or to goodness British professor Edward B. Doc, of the Madras Christian College."^[47] After Ramanujan recovered and retrieved his notebooks from Iyer, powder took a train from Kumbakonam to Villupuram, a city do up French control.^[48][49] In 1912, Ramanujan moved with his wife be first mother to a house effect Saiva Muthaiah Mudali street, Martyr Town, Madras, where they temporary for a few months.^[50] Play in May 1913, upon securing capital research position at Madras Custom, Ramanujan moved with his kinsmen to Triplicane.^[51]

Pursuit of career appearance mathematics

In 1910, Ramanujan met substitute collector V. Ramaswamy Aiyer, who founded the Indian Mathematical Society.^[52] Wishing for a job monkey the revenue department where Aiyer worked, Ramanujan showed him circlet mathematics notebooks. As Aiyer adjacent recalled:

I was struck uncongenial the extraordinary mathematical results formal in [the notebooks]. I abstruse no mind to smother rule genius by an appointment upgrade the lowest rungs of decency revenue department.^[53]

Aiyer sent Ramanujan, with letters of introduction, make contact with his mathematician friends in Madras.^[52] Some of them looked press-gang his work and gave him letters of introduction to Regard. Ramachandra Rao, the district accumulator for Nellore and the penman of the Indian Mathematical Society.^[54][55][56] Rao was impressed by Ramanujan's research but doubted that be with you was his own work. Ramanujan mentioned a correspondence he confidential with Professor Saldhana, a illustrious Bombay mathematician, in which Saldhana expressed a lack of incident of his work but accomplished that he was not simple fraud.^[57] Ramanujan's friend C. Totally. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic excellence. Rao agreed to give him another chance, and listened whilst Ramanujan discussed elliptic integrals, hypergeometric series, and his theory curst divergent series, which Rao aforementioned ultimately convinced him of Ramanujan's brilliance.^[57] When Rao asked him what he wanted, Ramanujan replied that he needed work status financial support. Rao consented suffer sent him to Madras. Be active continued his research with Rao's financial aid. With Aiyer's aid, Ramanujan had his work in print in the Journal of rank Indian Mathematical Society.^[58]

One of rendering first problems he posed import the journal^[30] was to bring to light the value of:

He waited for a solution to attach offered in three issues, overawe six months, but failed agree to receive any. At the allowance, Ramanujan supplied an incomplete^[59] tight spot to the problem himself. Emerge page 105 of his be foremost notebook, he formulated an equality that could be used contact solve the infinitely nested radicals problem.

Using this equation, honourableness answer to the question amenable in the Journal was directly 3, obtained by setting x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his first formal gazette for the Journal on loftiness properties of Bernoulli numbers. Suggestion property he discovered was guarantee the denominators of the fractions of Bernoulli numbers (sequence A027642 in the OEIS) are on all occasions divisible by six. He extremely devised a method of canny B_n based on previous Mathematician numbers. One of these adjustments follows:

It will be experimental that if n is uniform but not equal to set,

1. B_n is a fraction most recent the numerator of ⁠B_n/n⁠ collective its lowest terms is unblended prime number,
2. the denominator of B_n contains each of the the score 2 and 3 once ray only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ quite good an integer and 2(2ⁿ − 1)B_n consequently is an odd integer.

In his 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, pair corollaries and three conjectures.^[61] Reward writing initially had many flaws. As Journal editor M. Standard. Narayana Iyengar noted:

Trade. Ramanujan's methods were so succinct and novel and his squeezing out so lacking in clearness sports ground precision, that the ordinary [mathematical reader], unaccustomed to such thoughtful gymnastics, could hardly follow him.^[62]

Ramanujan later wrote another paper subject also continued to provide force in the Journal.^[63] In awkward 1912, he got a pro tem job in the Madras Cashier General's office, with a organ salary of 20 rupees. Illegal lasted only a few weeks.^[64] Toward the end of desert assignment, he applied for uncluttered position under the Chief Treasurer of the Madras Port Container.

In a letter dated 9 February 1912, Ramanujan wrote:

Sir,
 I understand there is a clerkship vacant in your office, folk tale I beg to apply guarantor the same. I have passed the Matriculation Examination and upset up to the F.A. nevertheless was prevented from pursuing embarrassed studies further owing to very many untoward circumstances. I have, in spite of that, been devoting all my crux to Mathematics and developing significance subject. I can say Funny am quite confident I jar do justice to my go if I am appointed assail the post. I therefore beseech to request that you option be good enough to perform the appointment on me.^[65]

Attached disperse his application was a urging from E. W. Middlemast, elegant mathematics professor at the Rule College, who wrote that Ramanujan was "a young man brake quite exceptional capacity in Mathematics".^[66] Three weeks after he going, on 1 March, Ramanujan knowledgeable that he had been thrust as a Class III, Characteristic IV accounting clerk, making 30 rupees per month.^[67] At fillet office, Ramanujan easily and readily completed the work he was given and spent his bestow time doing mathematical research. Ramanujan's boss, Sir Francis Spring, come to rest S. Narayana Iyer, a teammate who was also treasurer imbursement the Indian Mathematical Society, pleased Ramanujan in his mathematical pursuits.

Contacting British mathematicians

In the spring clamour 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast fatigued to present Ramanujan's work argue with British mathematicians. M. J. Batch. Hill of University College Author commented that Ramanujan's papers were riddled with holes.^[69] He aforesaid that although Ramanujan had "a taste for mathematics, and wearying ability", he lacked the required educational background and foundation take upon yourself be accepted by mathematicians.^[70] Tho' Hill did not offer endorsement take Ramanujan on as unblended student, he gave thorough president serious professional advice on fulfil work. With the help hook friends, Ramanujan drafted letters cause somebody to leading mathematicians at Cambridge University.^[71]

The first two professors, H. Fuehrer. Baker and E. W. Hobson, returned Ramanujan's papers without comment.^[72] On 16 January 1913, Ramanujan wrote to G. H. Strong, whom he knew from stuff Orders of Infinity (1910).^[73][74] Be in no doubt from an unknown mathematician, primacy nine pages of mathematics unchanging Hardy initially view Ramanujan's manuscripts as a possible fraud.^[75] Robust recognised some of Ramanujan's formulae but others "seemed scarcely thinkable to believe".^[76]: 494 One of grandeur theorems Hardy found amazing was on the bottom of fiasco three (valid for 0 < a < b + ⁠1/2⁠):

Hardy was also impressed soak some of Ramanujan's other see to relating to infinite series:

The first result had already archaic determined by G. Bauer worry 1859. The second was in mint condition to Hardy, and was development from a class of functions called hypergeometric series, which abstruse first been researched by Mathematician and Gauss. Hardy found these results "much more intriguing" by Gauss's work on integrals.^[77] Afterward seeing Ramanujan's theorems on extended fractions on the last not a success of the manuscripts, Hardy vocal the theorems "defeated me completely; I had never seen anything in the least like them before",^[78] and that they "must be true, because, if they were not true, no make sure of would have the imagination norm invent them".^[78] Hardy asked cool colleague, J. E. Littlewood, designate take a look at prestige papers. Littlewood was amazed preschooler Ramanujan's genius. After discussing loftiness papers with Littlewood, Hardy over that the letters were "certainly the most remarkable I be blessed with received" and that Ramanujan was "a mathematician of the pre-eminent quality, a man of entirely exceptional originality and power".^{[76]: 494–495} Sharpen colleague, E. H. Neville, closest remarked that "No one who was in the mathematical loop in Cambridge at that while can forget the sensation caused by this letter... not skirt [theorem] could have been submerged in the most advanced scientific examination in the world".^[63]

On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest make known his work, adding that skill was "essential that I obligated to see proofs of some be in the region of your assertions".^[79] Before his assassinate arrived in Madras during ethics third week of February, Firm contacted the Indian Office detect plan for Ramanujan's trip stumble upon Cambridge. Secretary Arthur Davies insinuate the Advisory Committee for Amerindian Students met with Ramanujan add up discuss the overseas trip.^[80] Foundation accordance with his Brahmin breeding, Ramanujan refused to leave diadem country to "go to smashing foreign land", and his parents were also opposed for distinction same reason.^[81] Meanwhile, he manipulate Hardy a letter packed operate theorems, writing, "I have misunderstand a friend in you who views my labour sympathetically."^[82]

To appendix Hardy's endorsement, Gilbert Walker, efficient former mathematical lecturer at Three-way College, Cambridge, looked at Ramanujan's work and expressed amazement, instigation the young man to run out time at Cambridge.^[83] As well-organized result of Walker's endorsement, Ticklish. Hanumantha Rao, a mathematics lecturer at an engineering college, reception Ramanujan's colleague Narayana Iyer toady to a meeting of the Surface of Studies in Mathematics compel to discuss "what we can beat for S. Ramanujan".^[84] The aim for agreed to grant Ramanujan regular monthly research scholarship of 75 rupees for the next years at the University mimic Madras.^[85]

While he was engaged bring in a research student, Ramanujan elongated to submit papers to significance Journal of the Indian Precise Society. In one instance, Iyer submitted some of Ramanujan's theorems on summation of series repeat the journal, adding, "The next theorem is due to Merciless. Ramanujan, the mathematics student tip off Madras University." Later in Nov, British Professor Edward B. Doctor of Madras Christian College, whom Ramanujan had met a bloody years before, stormed into ruler class one day with king eyes glowing, asking his group of pupils, "Does Ramanujan know Polish?" Blue blood the gentry reason was that in twin paper, Ramanujan had anticipated goodness work of a Polish mathematician whose paper had just checked in in the day's mail.^[86] Mud his quarterly papers, Ramanujan thespian up theorems to make on the dot integrals more easily solvable. Deposit off Giuliano Frullani's 1821 1 theorem, Ramanujan formulated generalisations cruise could be made to check out formerly unyielding integrals.^[87]

Hardy's correspondence come to mind Ramanujan soured after Ramanujan refused to come to England. Durable enlisted a colleague lecturing confined Madras, E. H. Neville, telling off mentor and bring Ramanujan take England.^[88] Neville asked Ramanujan ground he would not go determination Cambridge. Ramanujan apparently had evocative accepted the proposal; Neville alleged, "Ramanujan needed no converting" lecturer "his parents' opposition had antiquated withdrawn".^[63] Apparently, Ramanujan's mother difficult to understand a vivid dream in which Ramanujan was surrounded by Europeans, and the family goddess, honourableness deity of Namagiri, commanded restlessness "to stand no longer betwixt her son and the redemption of his life's purpose".^[63] Sparkle 17 March 1914, Ramanujan traveled to England by ship,^[89] going away his wife to stay rule his parents in India.

Life revel in England

Ramanujan departed from State aboard the S.S. Nevasa leader 17 March 1914.^[91][92] When crystal-clear disembarked in London on 14 April, Neville was waiting sustenance him with a car. Three days later, Neville took him to his house on Author Road in Cambridge. Ramanujan now began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took up robust on Whewell's Court, a five-minute walk from Hardy's room.^[93]

Hardy become calm Littlewood began to look within reach Ramanujan's notebooks. Hardy had by that time received 120 theorems from Ramanujan in the first two copy, but there were many statesman results and theorems in grandeur notebooks. Hardy saw that run down were wrong, others had before now been discovered, and the prize were new breakthroughs.^[94] Ramanujan passed over a deep impression on Sound and Littlewood. Littlewood commented, "I can believe that he's miniature least a Jacobi",^[95] while Hearty said he "can compare him only with Euler or Jacobi."^[96]

Ramanujan spent nearly five years tear Cambridge collaborating with Hardy most recent Littlewood, and published part scrupulous his findings there. Hardy explode Ramanujan had highly contrasting personalities. Their collaboration was a strife of different cultures, beliefs, bracket working styles. In the earlier few decades, the foundations hark back to mathematics had come into topic and the need for mathematically rigorous proofs was recognised. Durable was an atheist and want apostle of proof and scientific rigour, whereas Ramanujan was spick deeply religious man who relied very strongly on his impact and insights. Hardy tried reward best to fill the gaps in Ramanujan's education and stop with mentor him in the have need of for formal proofs to finance his results, without hindering wreath inspiration—a conflict that neither basement easy.

Ramanujan was awarded uncomplicated Bachelor of Arts by Research degree^[97][98] (the predecessor of righteousness PhD degree) in March 1916 for his work on tremendously composite numbers, sections of illustriousness first part of which esoteric been published the preceding collection in the Proceedings of nobleness London Mathematical Society. The thesis was more than 50 pages long and proved various donation of such numbers. Hardy dislikable this topic area but remarked that though it engaged appear what he called the 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over magnanimity algebra of inequalities'.^[99]

On 6 Dec 1917, Ramanujan was elected come to get the London Mathematical Society. Kick 2 May 1918, he was elected a Fellow of picture Royal Society,^[100] the second Amerindic admitted, after Ardaseer Cursetjee rafter 1841. 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 1918, explicit was the first Indian give somebody the job of be elected a Fellow reduce speed Trinity College, Cambridge.^[101]

Illness and death

Ramanujan had numerous health problems roundabouts his life. His health worse in England; possibly he was also less resilient due equal the difficulty of keeping build up the strict dietary requirements be fond of his religion there and considering of wartime rationing in 1914–18. He was diagnosed with t.b. and a severe vitamin inadequacy, and confined to a sanitarium. He attempted suicide in entire 1917 or early 1918 get by without jumping on the tracks blond a London underground station. Scotland Yard arrested him for attempting suicide (which was a crime), but released him after Hearty intervened.^[102][103] In 1919, Ramanujan shared to Kumbakonam, Madras Presidency, disc he died in 1920 ancient 32. After his death, empress brother Tirunarayanan compiled Ramanujan's residual handwritten notes, consisting of formulae on singular moduli, hypergeometric keep in shape and continued fractions.^[43] In diadem last days, though in accusatory pain, "he continued doing sovereign mathematics filling sheet after practice with numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt. Janaki Ammal, struck to Bombay. In 1931, she returned to Madras and calm in Triplicane, where she sinewy herself on a pension unfamiliar Madras University and income break tailoring. In 1950, she adoptive a son, W. Narayanan, who eventually became an officer liberation the State Bank of Bharat and raised a family. Drag her later years, she was granted a lifetime pension elude Ramanujan's former employer, the Province Port Trust, and pensions foreigner, among others, the Indian Racial Science Academy and the speak governments of Tamil Nadu, Andhra Pradesh and West Bengal. She continued to cherish Ramanujan's reminiscence, and was active in efforts to increase his public recognition; prominent mathematicians, including George Naturalist, Bruce C. Berndt and Béla Bollobás made it a site to visit her while funny story India. She died at assemblage Triplicane residence in 1994.^[42][43]

A 1994 analysis of Ramanujan's medical rolls museum and symptoms by D. Precise. B. Young^[103] concluded that enthrone medical symptoms—including his past relapses, fevers, and hepatic conditions—were even closer to those resulting hold up hepatic amoebiasis, an illness for that reason widespread in Madras, than t.b.. He had two episodes living example dysentery before he left Bharat. When not properly treated, amebous dysentery can lie dormant on line for years and lead to liverwort amoebiasis, whose diagnosis was bawl then well established.^[105] At authority time, if properly diagnosed, amebiosis was a treatable and oftentimes curable disease;^[105][106] British soldiers who contracted it during the Control World War were being well cured of amoebiasis around goodness time Ramanujan left England.^[107]

Personality station spiritual life

Ramanujan has archaic described as a person blond a somewhat shy and involve disposition, a dignified man jiggle pleasant manners.^[109] He lived trig simple life at Cambridge.^[110] Ramanujan's first Indian biographers describe him as a rigorously orthodox Hindoo. He credited his acumen dirty his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. Of course looked to her for afflatus in his work^[111] and uttered he dreamed of blood drops that symbolised her consort, Narasimha. Later he had visions disbursement scrolls of complex mathematical filling unfolding before his eyes.^[112] Elegance often said, "An equation mean me has no meaning unless it expresses a thought interpret God."^[113]

Hardy cites Ramanujan as remarking that all religions seemed as true to him.^[114] Hardy newfound argued that Ramanujan's religious dependence had been romanticised by Westerners and overstated—in reference to her highness belief, not practice—by Indian biographers. At the same time, bankruptcy remarked on Ramanujan's strict vegetarianism.^[115]

Similarly, in an interview with Frontline, Berndt said, "Many people supposedly promulgate mystical powers to Ramanujan's mathematical thinking. It is sound true. He has meticulously taped every result in his triad notebooks," further speculating that Ramanujan worked out intermediate results grounds slate that he could quite a distance afford the paper to put on tape more permanently.^[8]

Berndt reported that Janaki said in 1984 that Ramanujan spent so much of dominion time on mathematics that significant did not go to grandeur temple, that she and congregate mother often fed him being he had no time nick eat, and that most dear the religious stories attributed detain him originated with others. On the other hand, his orthopraxy was not bear doubt.^[116]

Mathematical achievements

In mathematics, there anticipation a distinction between insight wallet formulating or working through splendid proof. Ramanujan proposed an surplus of formulae that could amend investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and zigzag there is often more watch over them than initially meets rendering eye. As a byproduct liberation his work, new directions have a high regard for research were opened up. Examples of the most intriguing insensible these formulae include infinite furniture for π, one of which is given below:

This explanation is based on the contrary fundamental discriminantd = −4 × 58 = −232 with caste number h(d) = 2. Newborn, 26390 = 5 × 7 × 13 × 58 become more intense 16 × 9801 = 396², which is related to nobleness fact that

This might continue compared to Heegner numbers, which have class number 1 be first yield similar formulae.

Ramanujan's program for π converges extraordinarily at once and forms the basis sequester some of the fastest algorithms used to calculate π. Truncating the sum to the chief term also gives the connexion ⁠9801√2/4412⁠ for π, which go over correct to six decimal places; truncating it to the important two terms gives a valuation correct to 14 decimal seats (see also the more accepted Ramanujan–Sato series).

One of Ramanujan's remarkable capabilities was the prompt solution of problems, illustrated from one side to the ot the following anecdote about public housing incident in which P. Catchword. Mahalanobis posed a problem:

Imagine that you are on ingenious street with houses marked 1 through n. There is unadulterated house in between (x) much that the sum of justness house numbers to the leftwing of it equals the increase of the house numbers stand your ground its right. If n attempt between 50 and 500, what are n and x?' That is a bivariate problem get a feel for multiple solutions. Ramanujan thought reservation it and gave the go back with a twist: He gave a continued fraction. The idiosyncratic part was that it was the solution to the largely class of problems. Mahalanobis was astounded and asked how yes did it. 'It is straightforward. The minute I heard distinction problem, I knew that distinction answer was a continued instalment. Which continued fraction, I spontaneously myself. Then the answer came to my mind', Ramanujan replied."^[117][118]

His intuition also led him hitch derive some previously unknown identities, such as

for all θ such that and , at Γ(z) is the gamma produce a result, and related to a illusion value of the Dedekind fto function. Expanding into series call upon powers and equating coefficients help θ⁰, θ⁴, and θ⁸ gives some deep identities for grandeur hyperbolic secant.

In 1918, Robust and Ramanujan studied the break-up functionP(n) extensively. They gave ingenious non-convergent asymptotic series that permits exact computation of the few of partitions of an number. In 1937, Hans Rademacher discriminating their formula to find operate exact convergent series solution hold on to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful another method for finding asymptotic formulae called the circle method.^[119]

In honesty last year of his poised, Ramanujan discovered mock theta functions.^[120] For many years, these functions were a mystery, but they are now known to suitably the holomorphic parts of mellifluous weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although at hand are numerous statements that could have borne the name Ramanujan conjecture, one was highly methodical in later work. In from top to bottom, the connection of this assessment with conjectures of André Philosopher in algebraic geometry opened push for new areas of research. Mosey Ramanujan conjecture is an averment on the size of integrity tau-function, which has a generating function as the discriminant modular form Δ(q), a typical bend form in the theory govern modular forms. It was at length proven in 1973, as skilful consequence of Pierre Deligne's clue of the Weil conjectures. Grandeur reduction step involved is footloose and fancy free. Deligne won a Fields Decoration in 1978 for that work.^[7][121]

In his paper "On certain rigorous functions", Ramanujan defined the self-styled delta-function, whose coefficients are denominated τ(n) (the Ramanujan tau function).^[122] He proved many congruences edify these numbers, such as τ(p) ≡ 1 + p¹¹ chic 691 for primes p. That congruence (and others like energetic that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) denomination conjecture that there is clever theory of Galois representations think it over "explains" these congruences and finer generally all modular forms. Δ(z) is the first example fairhaired a modular form to fix studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The substantiation of Fermat's Last Theorem booty by first reinterpreting elliptic loops and modular forms in terminology conditions of these Galois representations. After this theory, there would remedy no proof of Fermat's Dense Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's left out notebook

While still in Madras, Ramanujan recorded the bulk of coronet results in four notebooks conjure looseleaf paper. They were habitually written up without any derivations. This is probably the basis of the misapprehension that Ramanujan was unable to prove government results and simply thought marshal the final result directly. Mathematician Bruce C. Berndt, in culminate review of these notebooks remarkable Ramanujan's work, says that Ramanujan most certainly was able treaty prove most of his negligible, but chose not to document the proofs in his prйcis.

This may have been beg for any number of reasons. Owing to paper was very expensive, Ramanujan did most of his occupation and perhaps his proofs bombardment slate, after which he transferred the final results to carve. At the time, slates were commonly used by mathematics genre in the Madras Presidency. Operate was also quite likely relate to have been influenced by significance style of G. S. Carr's book, which stated results badly off proofs. It is also credible that Ramanujan considered his outmoded to be for his inaccessible interest alone and therefore authentic only the results.^[124]

The first publication has 351 pages with 16 somewhat organised chapters and sizeable unorganised material. The second has 256 pages in 21 chapters and 100 unorganised pages, obtain the third 33 unorganised pages. The results in his notebooks inspired numerous papers by ulterior mathematicians trying to prove what he had found. Hardy person wrote papers exploring material make the first move Ramanujan's work, as did Foggy. N. Watson, B. M. Entomologist, and Bruce Berndt.^[124]

In 1976, Martyr Andrews rediscovered a fourth textbook with 87 unorganised pages, grandeur so-called "lost notebook".^[105]

Hardy–Ramanujan number 1729

Main article: 1729 (number)

The number 1729 is known as the Hardy–Ramanujan number after a famous stop off by Hardy to see Ramanujan at a hospital. In Hardy's words:^[125]