Srinivasa Ramanujan

Srinivasa Ramanujan 1

Srinivasa Ramanujan

Srinivasa Ramanujan

Born 22 December 1887Erode, British India

Died 26 April 1920 (aged 32)Chetput, (Madras), British India

Residence Tamil Nadu, India

Nationality Indian

Fields Mathematics

Alma mater Government Arts CollegePachaiyappa's College

Academic advisors G. H. HardyJ. E. Littlewood

Known for Landau–Ramanujan constantMock theta functionsRamanujan conjectureRamanujan primeRamanujan–Soldner constantRamanujan theta functionRamanujan's sumRogers–Ramanujan identities

Srīnivāsa Aiyangār Rāmānujam FRS, better known as Srinivasa Iyengar Ramanujan pronunciation (Tamil:சீனிவாச இராமானுஜன் or ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (22 December 1887 – 26 April 1920)was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, madesubstantial contributions to mathematical analysis, number theory, infinite series and continued fractions.Ramanujan's talent was said, by the prominent English mathematician G.H. Hardy, to be in the same league aslegendary mathematicians such as Euler, Gauss, Newton and Archimedes [1] .Born and raised in Erode, Tamil Nadu, India, Ramanujan first encountered formal mathematics at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney.[2] He had mastered them by age 12, and even discovered theorems of his own. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant. He received a scholarship to study at Government College in Kumbakonam, but lost it when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself.[3] In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge. Only G. H. Hardy recognized the brilliance of his work, subsequently inviting Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge,

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dying of illness, malnutrition and possibly liver infection in 1920 at the age of 32.During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identities andequations).[4] Although a small number of these results were actually false and some were already known, most ofhis claims have now been proven correct.[5] He stated results that were both original and highly unconventional, suchas the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of furtherresearch.[6] However, some of his major discoveries have been rather slow to enter the mathematical mainstream.Recently, Ramanujan's formulae have found applications in crystallography and string theory. The RamanujanJournal, an international publication, was launched to publish work in all areas of mathematics influenced by hiswork.[7]

Early life

Ramanujan's home on Sarangapani Street,Kumbakonam.

Ramanujan was born on 22 December 1887 in the city Erode, TamilNadu, India, at the residence of his maternal grandparents.[8] Hisfather, K. Srinivasa Iyengar worked as a clerk in a sari shop and hailedfrom the district of Thanjavur.[9] His mother, Komalatammal or KomalAmmal was a housewife and also sang at a local temple.[10] They livedin Sarangapani Street in a traditional home in the town ofKumbakonam. The family home is now a museum. When Ramanujanwas a year and a half old, his mother gave birth to a son namedSadagopan, who died less than three months later. In December 1889,Ramanujan had smallpox and recovered, unlike thousands in theThanjavur district who succumbed to the disease that year.[11] Hemoved with his mother to her parents' house in Kanchipuram, nearMadras (now Chennai). In November 1891, and again in 1894, hismother gave birth, but both children died in infancy.

On 1 October 1892, Ramanujan was enrolled at the local school.[12] InMarch 1894, he was moved to a Telugu medium school. After hismaternal grandfather lost his job as a court official in Kanchipuram,[13]

Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School.[14]

After his paternal grandfather died, he was sent back to his maternal grandparents, who were now living in Madras.He did not like school in Madras, and he tried to avoid going to school. His family enlisted a local constable to makesure he attended school. Within six months, Ramanujan was back in Kumbakonam.[14]

Since Ramanujan's father was at work most of the day, his mother took care of him as a child. He had a closerelationship with her. From her, he learned about tradition and puranas. He learned to sing religious songs, to attendpujas at the temple and particular eating habits – all of which are part of Brahmin culture.[15] At the KangayanPrimary School, Ramanujan performed well. Just before the age of 10, in November 1897, he passed his primaryexaminations in English, Tamil, geography and arithmetic. With his scores, he finished first in the district.[16] Thatyear, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the firsttime.[16]

By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book on advanced trigonometry written by S. L. Loney.[17] [18] He completely mastered this book by the age of 13 and discovered sophisticated theorems on his own. By 14, he was receiving merit certificates and academic awards which continued throughout his school career and also assisted the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers.[19] He completed mathematical exams in half the allotted time, and showed a familiarity with infinite series. When he was 16, Ramanujan came across the

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book A Synopsis of Elementary Results in Pure and Applied Mathematics by George S. Carr.[20] This book was acollection of 5000 theorems, and it introduced Ramanujan to the world of mathematics. The next year, he hadindependently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15decimal places.[21] His peers of the time commented that they "rarely understood him" and "stood in respectful awe"of him.[19]

When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Raoprize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstandingstudent who deserved scores higher than the maximum possible marks.[19] He received a scholarship to study atGovernment College in Kumbakonam,[22] [23] However, Ramanujan was so intent on studying mathematics that hecould not focus on any other subjects and failed most of them, losing his scholarship in the process.[24] In August1905, he ran away from home, heading towards Visakhapatnam.[25] He later enrolled at Pachaiyappa's College inMadras. He again excelled in mathematics but performed poorly in other subjects such as physiology. Ramanujanfailed his Fine Arts degree exam in December 1906 and again a year later. Without a degree, he left college andcontinued to pursue independent research in mathematics. At this point in his life, he lived in extreme poverty andwas often on the brink of starvation.[26]

Adulthood in IndiaOn 14 July 1909, Ramanujan was married to a nine-year old bride, Janaki Ammal.[27] – in the branch of Hinduism towhich Ramanujan belonged, marriage was a formal engagement that was consummated only after the bride turned17 or 18, as per the traditional calendar. After the marriage, Ramanujan developed a hydrocele testis, an abnormalswelling of the tunica vaginalis, an internal membrane in the testicle.[28] The condition could be treated with aroutine surgical operation that would release the blocked fluid in the scrotal sac. His family did not have the moneyfor the operation, but in January 1910, a doctor volunteered to do the surgery for free.[29] After his successfulsurgery, Ramanujan searched for a job. He stayed at friends' houses while he went door to door around the city ofMadras (now Chennai) looking for a clerical position. To make some money, he tutored some students at PresidencyCollege who were preparing for their F.A. exam.[30] In late 1910, Ramanujan was sick again, possibly as a result ofthe surgery earlier in the year. He feared for his health, and even told his friend, R. Radakrishna Iyer, to "hand these[my mathematical notebooks] over to Professor Singaravelu Mudaliar [mathematics professor at Pachaiyappa'sCollege] or to the British professor Edward B. Ross, of the Madras Christian College."[31] After Ramanujanrecovered and got back his notebooks from Iyer, he took a northbound train from Kumbakonam to Villupuram, acoastal city under French control.[32] [33]

Attention from mathematiciansHe met deputy collector V. Ramaswamy Aiyer, who had recently founded the Indian Mathematical Society.[34]

Ramanujan, wishing for a job at the revenue department where Aiyer worked, showed him his mathematicsnotebooks. As Aiyer later recalled:

I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had no mind tosmother his genius by an appointment in the lowest rungs of the revenue department.[35]

Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras.[34] Some of these friends looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Mathematical Society.[36] [37] [38] Ramachandra Rao was impressed by Ramanujan's research but doubted that it was actually his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understanding for his work but concluded that he was not a phony.[39] Ramanujan's friend, C. V. Rajagopalachari, persisted with Ramachandra Rao and tried to quell any doubts over Ramanujan's academic integrity. Rao agreed to give him another chance, and he listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of

Srinivasa Ramanujan 4

divergent series, which Rao said ultimately "converted" him to a belief in Ramanujan's mathematical brilliance.[39]

When Rao asked him what he wanted, Ramanujan replied that he needed some work and financial support. Raoconsented and sent him to Madras. He continued his mathematical research with Rao's financial aid taking care of hisdaily needs. Ramanujan, with the help of V. Ramaswamy Aiyer, had his work published in the Journal of IndianMathematical Society.[40]

One of the first problems he posed in the journal was:

He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end,Ramanujan supplied the solution to the problem himself. On page 105 of his first notebook, he formulated anequation that could be used to solve the infinitely nested radicals problem.Using this equation, the answer to the question posed in the Journal was simply 3.[41] Ramanujan wrote his firstformal paper for the Journal on the properties of Bernoulli numbers. One property he discovered was that thedenominators (sequence A027642 [42] in OEIS) of the fractions of Bernoulli numbers were always divisible by six.He also devised a method of calculating Bn based on previous Bernoulli numbers. One of these methods went asfollows:It will be observed that if n is even but not equal to zero,(i) Bn is a fraction and the numerator of in its lowest terms is a prime number,

(ii) the denominator of Bn contains each of the factors 2 and 3 once and only once,(iii) is an integer and consequently is an odd integer.

In his 17–page paper, "Some Properties of Bernoulli's Numbers", Ramanujan gave three proofs, two corollaries andthree conjectures.[43] Ramanujan's writing initially had many flaws. As Journal editor M. T. Narayana Iyengar noted:

Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness andprecision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, couldhardly follow him.[44]

Ramanujan later wrote another paper and also continued to provide problems in the Journal.[45] In early 1912, he gota temporary job in the Madras Accountant General's office, with a 20-rupee-a-month salary. He lasted for only a fewweeks.[46] Toward the end of that assignment he applied for a position under the Chief Accountant of the MadrasPort Trust. In a letter dated 9 February 1912, Ramanujan wrote:

Sir,I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed theMatriculation Examination and studied up to the F.A. but was prevented from pursuing my studiesfurther owing to several untoward circumstances. I have, however, been devoting all my time toMathematics and developing the subject. I can say I am quite confident I can do justice to my work if Iam appointed to the post. I therefore beg to request that you will be good enough to confer theappointment on me.[47]

Attached to his application was a recommendation from E. W. Middlemast, a mathematics professor at thePresidency College, who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics".[48]

Three weeks after he had applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade IVaccounting clerk, making 30 rupees per month.[49] At his office, Ramanujan easily and quickly completed the workhe was given, so he spent his spare time doing mathematical research. Ramanujan's boss, Sir Francis Spring, and S.Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in hismathematical pursuits.

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Contacting English mathematiciansSpring, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to Britishmathematicians. One mathematician, M. J. M. Hill of University College London, commented that Ramanujan'spapers were riddled with holes.[50] He said that although Ramanujan had "a taste for mathematics, and some ability",he lacked the educational background and foundation needed to be accepted by mathematicians.[51] Although Hilldid not offer to take Ramanujan on as a student, he did give thorough and serious professional advice on his work.With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.[52]

The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without comment.[53] On 16January 1913, Ramanujan wrote to G. H. Hardy. Coming from an unknown mathematician, the nine pages ofmathematical wonder made Hardy originally view Ramanujan's manuscripts as a possible "fraud".[54] Hardyrecognized some of Ramanujan's formulae but others "seemed scarcely possible to believe."[55] One of the theoremsHardy found so incredible was found on the bottom of page three (valid for 0 < a < b + 1/2):Hardy was also impressed by some of Ramanujan's other work relating to infinite series:

The first result had already been determined by a mathematician named Bauer. The second one was new to Hardy. Itwas derived from a class of functions called a hypergeometric series which had first been researched by LeonhardEuler and Carl Friedrich Gauss. Compared to Ramanujan's work on integrals, Hardy found these results "much moreintriguing".[56] After he saw Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardycommented that the "[theorems] defeated me completely; I had never seen anything in the least like them before."[57]

He figured that Ramanujan's theorems "must be true, because, if they were not true, no one would have theimagination to invent them."[57] Hardy asked a colleague, J. E. Littlewood, to take a look at the papers. Littlewoodwas amazed by the mathematical genius of Ramanujan. After discussing the papers with Littlewood, Hardyconcluded that the letters were "certainly the most remarkable I have received" and commented that Ramanujan was"a mathematician of the highest quality, a man of altogether exceptional originality and power."[58] One colleague, E.H. Neville, later commented that "not one [theorem] could have been set in the most advanced mathematicalexamination in the world."[59]

On 8 February 1913, Hardy wrote a letter to Ramanujan, expressing his interest for his work. Hardy also added thatit was "essential that I should see proofs of some of your assertions."[60] Before his letter arrived in Madras duringthe third week of February, Hardy contacted the Indian Office to plan for Ramanujan's trip to Cambridge. SecretaryArthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip.[61]

In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land."[62]

Meanwhile, Ramanujan sent a letter packed with theorems to Hardy, writing, "I have found a friend in you whoviews my labour sympathetically."[63]

To supplement Hardy's endorsement, a former mathematical lecturer at Trinity College, Cambridge, Gilbert Walker, looked at Ramanujan's work and expressed amazement, urging him to spend time at Cambridge.[64] As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan."[65] The board agreed to grant Ramanujan a research scholarship of 75 rupees per month for the next two years at the University of Madras.[66] While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one instance, Narayana Iyer submitted some theorems of Ramanujan on summation of series to the above mathematical journal adding “The following theorem is due to S. Ramanujan, the mathematics student of Madras University” Later in November, British Professor Edward.B.Ross of Madras Christian College, whom Ramanujan had met few years ago, stormed into his class one day with his eyes glowing, asking his students, “Does Ramanujan know Polish?” The reason was that in one paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had just arrived by the day’s mail. [67]

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In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working offGiuliano Frullani's 1821 integral theorem, Ramanujan formulated generalizations that could be made to evaluateformerly unyielding integrals.[68]

Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted acolleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England.[69] Neville askedRamanujan why he would not go to Cambridge. Ramanujan apparently had now accepted the proposal; as Nevilleput it, "Ramanujan needed no converting and that his parents' opposition had been withdrawn."[59] Apparently,Ramanujan's mother had a vivid dream in which the family Goddess Namagiri commanded her "to stand no longerbetween her son and the fulfillment of his life's purpose."[59]

Life in EnglandRamanujan boarded the S.S. Nevasa on 17 March 1914, and at 10 o'clock in the morning, the ship departed fromMadras.[70] He arrived in London on 14 April, with E. H. Neville waiting for him with a car. Four days later, Nevilletook him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewoodand Hardy. After six weeks, Ramanujan moved out of Neville's house and took up residence on Whewell's Court,just a five-minute walk from Hardy's room.[71] Hardy and Ramanujan began to take a look at Ramanujan'snotebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were manymore results and theorems to be found in the notebooks. Hardy saw that some were wrong, some had already beendiscovered, while the rest were new breakthroughs.[72] Ramanujan left a deep impression on Hardy and Littlewood.Littlewood commented, "I can believe that he's at least a Jacobi",[73] while Hardy said he "can compare him onlywith [Leonhard] Euler or Jacobi."[74]

Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood and published a part ofhis findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash ofdifferent cultures, beliefs and working styles. Hardy was an atheist and an apostle of proof and mathematical rigour,whereas Ramanujan was a deeply religious man and relied very strongly on his intuition. While in England, Hardytried his best to fill the gaps in Ramanujan's education without interrupting his spell of inspiration.Ramanujan was awarded a B.A. degree by research (this degree was later renamed PhD) in March 1916 for his workon highly composite numbers, which was published as a paper in the Journal of the London Mathematical Society.The paper was over 50 pages with different properties of such numbers proven. Hardy remarked that this was one ofthe most unusual papers seen in mathematical research at that time and that Ramanujan showed extraordinaryingenuity in handling it. On 6 December 1917, he was elected to the London Mathematical Society. He became aFellow of the Royal Society in 1918, becoming the second Indian to do so, following Ardaseer Cursetjee in 1841,and he was the youngest Fellow in the entire history of the Royal Society.[75] He was elected "for his investigation inElliptic functions and the Theory of Numbers." On 13 October 1918, he became the first Indian to be elected aFellow of Trinity College, Cambridge.[76]

Illness and return to IndiaPlagued by health problems throughout his life, living in a country far away from home, and obsessively involvedwith his mathematics, Ramanujan's health worsened in England, perhaps exacerbated by stress and by the scarcity ofvegetarian food during the First World War. He was diagnosed with tuberculosis and a severe vitamin deficiency andwas confined to a sanatorium.Ramanujan returned to Kumbakonam, India in 1919 and died soon thereafter at the age of 32. His widow, S. JanakiAmmal, lived in Chennai (formerly Madras) until her death in 1994.[77]

A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D.A.B. Young concluded that it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver widespread in Madras, where Ramanujan had

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spent time. He had had two episodes of dysentery before he left India. When not properly treated dysentery can liedormant for years and lead to hepatic amoebiasis,[3] a difficult disease to diagnose, but once diagnosed readilycured.[3]

Personality and spiritual lifeRamanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man with pleasantmanners.[78] He lived a rather Spartan life while at Cambridge. Ramanujan's first Indian biographers describe him asrigorously orthodox. Ramanujan credited his acumen to his family Goddess, Namagiri of Namakkal, and looked toher for inspiration in his work.[79] He often said, "An equation for me has no meaning, unless it represents a thoughtof God."[80] [81]

Hardy cites Ramanujan as remarking that all religions seemed equally true to him.[82] Hardy further argued thatRamanujan's religiousness had been romanticized by Westerners and overstated—in reference to his belief, notpractice—by Indian biographers. At the same time, he remarked on Ramanujan's strict observance of vegetarianism.

Mathematical achievementsIn mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested aplethora of formulae that could then be investigated in depth later. It is said that Ramanujan's discoveries areunusually rich and that there is often more in it than what initially meets the eye. As a by-product, new directions ofresearch were opened up. Examples of the most interesting of these formulae include the intriguing infinite series forπ, one of which is given below

This result is based on the negative fundamental discriminant d = −4×58 with class number h(d) = 2 (note that5×7×13×58 = 26390 and that 9801=99×99; 396=4×99) and is related to the fact that

Compare to Heegner numbers, which have class number 1 and yield similar formulae. Ramanujan's series for πconverges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently usedto calculate π. Truncating the sum to the first term also gives the approximation for π, which iscorrect to six decimal places.One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P. C.Mahalanobis who had a problem, "Imagine that you are on a street with houses marked 1 through n. There is a housein between (x) such that the sum of the house numbers to left of it equals the sum of the house numbers to its right. Ifn is between 50 and 500, what are n and x." This is a bivariate problem with multiple solutions. Ramanujan thoughtabout it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solutionto the whole class of problems. Mahalanobis was astounded and asked how he did it. "It is simple. The minute Iheard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Thenthe answer came to my mind", Ramanujan replied.[83] [84]

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

for all , where is the gamma function. Equating coefficients of , , and gives some deep identitiesfor the hyperbolic secant.In 1918, Hardy and Ramanujan studied the partition function P(n) extensively and gave a non-convergent asymptoticseries that permits exact computation of the number of partitions of an integer. Hans Rademacher, in 1937, was ableto refine their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work inthis area gave rise to a powerful new method for finding asymptotic formulae, called the circle method.[85]

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He discovered mock theta functions in the last year of his life. For many years these functions were a mystery, butthey are now known to be the holomorphic parts of harmonic weak Maass forms.

The Ramanujan conjectureAlthough there are numerous statements that could bear the name Ramanujan conjecture, there is one statement thatwas very influential on later work. In particular, the connection of this conjecture with conjectures of André Weil inalgebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the size of the taufunction, which has as generating function the discriminant modular form Δ(q), a typical cusp form in the theory ofmodular forms. It was finally proven in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures.The reduction step involved is complicated. Deligne won a Fields Medal in 1978 for his work on Weilconjectures.[86]

Ramanujan's notebooksWhile still in India, Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper. These resultswere mostly written up without any derivations. This is probably the origin of the misperception that Ramanujan wasunable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in hisreview of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to make the proofsof most of his results, but chose not to.This style of working may have been for several reasons. Since paper was very expensive, Ramanujan would domost of his work and perhaps his proofs on slate, and then transfer just the results to paper. Using a slate wascommon for mathematics students in India at the time. He was also quite likely to have been influenced by the styleof G. S. Carr's book, which stated results without proofs. Finally, it is possible that Ramanujan considered hisworkings to be for his personal interest alone; and therefore only recorded the results.[87]

The first notebook has 351 pages with 16 somewhat organized chapters and some unorganized material. The secondnotebook has 256 pages in 21 chapters and 100 unorganized pages, with the third notebook containing 33unorganized pages. The results in his notebooks inspired numerous papers by later mathematicians trying to provewhat he had found. Hardy himself created papers exploring material from Ramanujan's work as did G. N. Watson, B.M. Wilson, and Bruce Berndt.[87] A fourth notebook with 87 unorganized pages, the so-called "lost notebook", wasrediscovered in 1976 by George Andrews.[3]

Hardy–Ramanujan number 1729A common anecdote about Ramanujan relates to the number 1729. Hardy arrived at Ramanujan's residence in a cabnumbered 1729. Hardy commented that the number 1729 seemed to be uninteresting. Ramanujan is said to havestated on the spot that it was actually a very interesting number mathematically, being the smallest natural numberrepresentable in two different ways as a sum of two cubes:

Generalizations of this idea have spawned the notion of "taxicab numbers".

Other mathematicians' views of RamanujanRamanujan is generally hailed as an all-time great like Leonhard Euler, Carl Friedrich Gauss, and Carl Gustav Jacob Jacobi, for his natural mathematical genius.[88] Hardy quotes: "The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed

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but the vaguest idea of what a function of a complex variable was...".[89] Hardy went on to claim that his greatestcontribution to mathematics was discovering Ramanujan.Quoting K. Srinivasa Rao,[90] "As for his place in the world of Mathematics, we quote Bruce C. Berndt: 'Paul Erdőshas passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis ofpure talent on a scale from 0 to 100, Hardy gave himself a score of 25, J.E. Littlewood 30, David Hilbert 80 andRamanujan 100.'"In his book Scientific Edge, noted physicist Jayant Narlikar stated that "Srinivasa Ramanujan, discovered by theCambridge mathematician Hardy, whose great mathematical findings were beginning to be appreciated from 1915 to1919. His achievements were to be fully understood much later, well after his untimely death in 1920. For example,his work on the highly composite numbers (numbers with a large number of factors) started a whole new line ofinvestigations in the theory of such numbers."During his lifelong mission in educating and propagating mathematics among the school children in India, Nigeriaand elsewhere, P.K. Srinivasan has continually introduced Ramanujan's mathematical works.

RecognitionRamanujan's home state of Tamil Nadu celebrates 22 December (Ramanujan's birthday) as 'State IT Day',memorializing both the man and his achievements, as a native of Tamil Nadu. A stamp picturing Ramanujan wasreleased by the Government of India in 1962 – the 75th anniversary of Ramanujan's birth – commemorating hisachievements in the field of number theory.A prize for young mathematicians from developing countries has been created in the name of Ramanujan by theInternational Centre for Theoretical Physics (ICTP), in cooperation with the International Mathematical Union, whonominate members of the prize committee. The Shanmugha Arts, Science, Technology, Research Academy(SASTRA), based in the state of Tamil Nadu in South India, has instituted the SASTRA Ramanujan Prize of $10,000to be given annually to a mathematician not exceeding the age of 32 for outstanding contributions in an area ofmathematics influenced by Ramanujan. The age limit refers to the years Ramanujan lived, having nevertheless stillachieved many accomplishments. This prize has been awarded annually since 2005, at an international conferenceconducted by SASTRA in Kumbakonam, Ramanujan's hometown, around Ramanujan's birthday, December 22.

In popular culture• An international feature film on Ramanujan's life was announced in 2006 as due to begin shooting in 2007. It was

to be shot in Tamil Nadu state and Cambridge and be produced by an Indo-British collaboration and co-directedby Stephen Fry and Dev Benegal.[91] A play, First Class Man by Alter Ego Productions,[92] was based on DavidFreeman's First Class Man. The play is centered around Ramanujan and his complex and dysfunctionalrelationship with Hardy.

• Another film, based on the book The Man Who Knew Infinity: A Life of the Genius Ramanujan by RobertKanigel, is being made by Edward Pressman and Matthew Brown.[93]

• In the film Good Will Hunting, the titular character is compared to Ramanujan.• "Gomez", a short story by Cyril Kornbluth, describes the conflicted life of an untutored mathematical genius,

clearly based on Ramanujan.• A Disappearing Number is a recent British production that explores the relationship between Hardy and

Ramanujan.• The character Amita Ramanujan on the television show Numb3rs is named after Ramanujan.• The novel The Indian Clerk by David Leavitt explores in fiction the events following Ramanujan's letter to

Hardy.

Srinivasa Ramanujan 10

See also• List of amateur mathematicians• List of topics named after Srinivasa Ramanujan• Ramanujan–Petersson conjecture• 1729 (number)• Landau–Ramanujan constant• Ramanujan–Soldner constant• Ramanujan summation• Ramanujan theta function• Ramanujan graph• Ramanujan's tau function• Rogers–Ramanujan identities• Ramanujan prime• Ramanujan's constant• Ramanujan's sum• Ramanujan modular functions

Notes[1] C.P. Snow Foreword to "A Mathematician's Apology" by G.H. Hardy[2] Berndt, Bruce C. (2001). Ramanujan: Essays and Surveys. Providence, Rhode Island: American Mathematical Society. pp. 9.

ISBN 0-8218-2624-7.[3] Peterson, Doug. "Raiders of the Lost Notebook" (http:/ / web. archive. org/ web/ 20070517174549/ http:/ / www. las. uiuc. edu/ alumni/

news/ fall2006/ 06fall_lostnotebook. html). UIUC College of Liberal Arts and Sciences. Archived from the original (http:/ / www. las. uiuc.edu/ alumni/ news/ fall2006/ 06fall_lostnotebook. html) on May 17, 2007. . Retrieved 2007-06-22.

[4] Berndt, Bruce C. (2005). Ramanujan's Notebooks Part V. SpringerLink. pp. 4. ISBN 0-387-94941-0.[5] "Rediscovering Ramanujan" (http:/ / www. hinduonnet. com/ fline/ fl1617/ 16170810. htm). Frontline 16 (17): 650. August 1999. . Retrieved

2007-06-23.[6] Ono, Ken; Rankin, Robert A. (June–July 2006). "Honoring a Gift from Kumbakonam" (http:/ / www. ams. org/ notices/ 200606/ fea-ono.

pdf) (PDF). Notices of the American Mathematical Society (Mathematical Association of America) 53 (6): 650. doi:10.2307/2589114. .Retrieved 2007-06-23.

[7] Alladi, Krishnaswami (1998). Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdös. Norwell,Massachusetts: Kluwer Academic Publishers. pp. 6. ISBN 0-7923-8273-0.

[8] Kanigel, Robert (1991). The Man Who Knew Infinity: A Life of the Genius Ramanujan. New York: Charles Scribner's Sons. pp. 11.ISBN 0-684-19259-4.

[9] Kanigel (1991), p. 17–18.[10] Bruce C. Berndt; Robert Alexander Rankin (2001). Ramanujan: essays and surveys. AMS Bookstore. pp. 89. ISBN 0821826247, ISBN

978-0-8218-2624-9.[11] Kanigel (1991), p12.[12] Kanigel (1991), p13.[13] Kanigel (1991), p19.[14] Kanigel (1991), p14.[15] Kanigel (1991), p20.[16] Kanigel (1991), p25.[17] Hardy, G. H. (1999). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Providence, Rhode Island: American

Mathematical Society. pp. 2. ISBN 0-8218-2023-0.[18] Berndt, Bruce C.; Robert A. Rankin (2001). Ramanujan: Essays and Surveys. Providence, Rhode Island: American Mathematical Society.

pp. 9. ISBN 0-8218-2624-7.[19] Kanigel (1991), p27.[20] Kanigel (1991), p39.[21] Kanigel (1991), p90.[22] Kanigel (1991), p28.[23] Kanigel (1991), p45.[24] Kanigel (1991), p47.

Srinivasa Ramanujan 11

[25] Kanigel (1991), pp48–49.[26] Kanigel (1991), pp55–56.[27] Kanigel (1991), p71.[28] Kanigel (1991), p72.[29] Ramanujan, Srinivasa (1968). P. K. Srinivasan. ed. Ramanujan Memorial Number: Letters and Reminiscences. Madras: Muthialpet High

School. Vol. 1, p100.[30] Kanigel (1991), p73.[31] Kanigel (1991), pp74–75.[32] Ranganathan, S. R. (1967). Ramanujan: The Man and the Mathematician. Bombay: Asia Publishing House. pp. 23.[33] Srinivasan (1968), Vol. 1, p99.[34] Kanigel (1991), p77.[35] Srinivasan (1968), Vol. 1, p129.[36] Srinivasan (1968), Vol. 1, p86.[37] Neville, Eric Harold (January 1921). "The Late Srinivasa Ramanujan". Nature 106 (2673): 661–662. doi:10.1038/106661b0.[38] Ranganathan (1967), p24.[39] Kanigel (1991), p80.[40] Kanigel (1991), p86.[41] Kanigel (1991), p87.[42] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa027642[43] Kanigel (1991), p91.[44] Seshu Iyer, P. V. (June 1920). "The Late Mr. S. Ramanujan, B.A., F.R.S.". Journal of the Indian Mathematical Society 12 (3): 83.[45] Neville (March 1942), p292.[46] Srinivasan (1968), p176.[47] Srinivasan (1968), p31.[48] Srinivasan (1968), p49.[49] Kanigel (1991), p96.[50] Kanigel (1991), p105.[51] Letter from M. J. M. Hill to a C. L. T. Griffith (a former student who sent the request to Hill on Ramanujan's behalf), 28 November 1912.[52] Kanigel (1991), p106.[53] Kanigel (1991), pp170–171.[54] Snow, C. P. (1966). Variety of Men. New York: Charles Scribner's Sons. pp. 30–31.[55] Hardy, G. H.; Rankin, Robert A. (June 1920). "Obituary, S. Ramanujan" (http:/ / jstor. org/ stable/ 2589114). Nature (Mathematical

Association of America) 105 (7): 494. doi:10.2307/2589114. . Retrieved 2007-06-30.[56] Kanigel (1991), p167.[57] Kanigel (1991), p168.[58] Hardy (June 1920), pp494–495.[59] Neville, Eric Harold (March 1942). "Srinivasa Ramanujan". Nature 149 (3776): 293. doi:10.1038/149292a0.[60] Letter, Hardy to Ramanujan, 8 February 1913.[61] Letter, Ramanujan to Hardy, 22 January 1914.[62] Kanigel (1991), p185.[63] Letter, Ramanujan to Hardy, 27 February 1913, Cambridge University Library.[64] Kanigel (1991), p175.[65] Ram, Suresh (1972). Srinivasa Ramanujan. New Delhi: National Book Trust. pp. 29.[66] Ranganathan (1967), pp30–31.[67] Ranganathan (1967), p12.[68] Kanigel (1991), p183.[69] Kanigel (1991), p184.[70] Kanigel (1991), p196.[71] Kanigel (1991), p202.[72] Hardy, G. H. (1940). Ramanujan. Cambridge: Cambridge University Press. pp. 10.[73] Letter, Littlewood to Hardy, early March 1913.[74] Hardy, G. H. (1979). Collected Papers of G. H. Hardy. Oxford, England: Clarendon Press. Vol. 7, p720.[75] Kanigel (1991), p295.[76] Kanigel (1991), pp299–300.[77] "Ramanujan’s wife: Janakiammal (Janaki)" (http:/ / www. imsc. res. in/ ~rao/ ramanujan/ newnow/ janaki. pdf) (PDF). .[78] "Ramanujan's Personality" (http:/ / www. imsc. res. in/ ~rao/ ramanujan/ newnow/ pcm5. htm). .[79] Kanigel (1991), p36.[80] "Quote by Srinivasa Ramanujan Iyengar" (http:/ / lagrange. math. trinity. edu/ aholder/ misc/ quotes. shtml). .

Srinivasa Ramanujan 12

[81] Chaitin, Gregory; Rankin, Robert A. (2007-07-28). "Less Proof, More Truth" (http:/ / jstor. org/ stable/ 2589114). NewScientist(Mathematical Association of America) 107 (2614): 49. doi:10.2307/2589114. .

[82] Kanigel (1991), p283.[83] Ranganathan, Shiyali Ramamrita (1967). Ramanujan, the man and the mathematician (http:/ / books. google. com/

?id=OuTuAAAAMAAJ& q="Which+ continued+ fraction"). Asia Publishing House. p. 82. . Retrieved 2010-06-07.[84] Calyampudi Radhakrishna Rao (1997). Statistics and truth: putting chance to work (http:/ / books. google. com/ ?id=jqWd4oe3iwIC&

pg=PA185& dq="Which+ continued+ fraction"). World Scientific. p. 185. ISBN 9789810231118. . Retrieved 2010-06-07.[85] "Partition Formula" (http:/ / mathworld. wolfram. com/ PartitionFunctionP. html). .[86] Ono (June–July 2006), p649.[87] "Ramanujans Notebooks" (http:/ / www. amazon. com/ Ramanujans-Notebooks-Part-Bruce-Berndt/ dp/ 0387949410). .[88] K. Srinivasa Rao, "Srinivasa Ramanujan" (http:/ / www. imsc. res. in/ ~rao/ ramanujan. html). .[89] "Ramanujan quote" (http:/ / www-groups. dcs. st-and. ac. uk/ ~history/ Biographies/ Ramanujan. html). .[90] K Srinivasa Rao. "Srinivasa Ramanujan (22 December 1887 – 26 April 1920)" (http:/ / www. imsc. res. in/ ~rao/ ramanujan. html). .[91] "Film to celebrate maths genius" (http:/ / news. bbc. co. uk/ 2/ hi/ south_asia/ 4811920. stm). BBC News. 2006-03-16. . Retrieved

2008-08-08.[92] First Class Man (http:/ / www. alteregoproductions. org/ blog/ 2006/ 06/ alteregos_new_theater_season_b. htm)[93] Two Hollywood movies on Ramanujan (http:/ / sify. com/ news/ othernews/ fullstory. php?id=14173864)

Selected publications by Ramanujan• Srinivasa Ramanujan, G. H. Hardy, P. V. Seshu Aiyar, B. M. Wilson, Bruce C. Berndt (2000). Collected Papers

of Srinivasa Ramanujan. AMS. ISBN 0-8218-2076-1.This book was originally published in 1927 after Ramanujan's death. It contains the 37 papers published inprofessional journals by Ramanujan during his lifetime. The third re-print contains additional commentary byBruce C. Berndt.

• S. Ramanujan (1957). Notebooks (2 Volumes). Bombay: Tata Institute of Fundamental Research.These books contain photo copies of the original notebooks as written by Ramanujan.

• S. Ramanujan (1988). The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa.This book contains photo copies of the pages of the "Lost Notebook".

• Problems posed by Ramanujan (http:/ / www. imsc. res. in/ ~rao/ ramanujan/ collectedpapers/ question/ qJIMS.htm), Journal of the Indian Mathematical Society.

Selected publications about Ramanujan and his work• Berndt, Bruce C. " An Overview of Ramanujan's Notebooks (http:/ / www. math. uiuc. edu/ ~berndt/ articles/

aachen. pdf)." Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe. Ed. P. L.Butzer, W. Oberschelp, and H. Th. Jongen. Turnhout, Belgium: Brepols, 1998. 119–146.

• Berndt, Bruce C., and George E. Andrews. Ramanujan's Lost Notebook, Part I. New York: Springer, 2005. ISBN0-387-25529-X.

• Berndt, Bruce C., and George E. Andrews. Ramanujan's Lost Notebook, Part II. New York: Springer, 2008.ISBN 978-0-387-77765-8

• Berndt, Bruce C., and Robert A. Rankin. Ramanujan: Letters and Commentary. Vol. 9. Providence, Rhode Island:American Mathematical Society, 1995. ISBN 0-8218-0287-9.

• Berndt, Bruce C., and Robert A. Rankin. Ramanujan: Essays and Surveys. Vol. 22. Providence, Rhode Island:American Mathematical Society, 2001. ISBN 0-8218-2624-7.

• Berndt, Bruce C. Number Theory in the Spirit of Ramanujan. Providence, Rhode Island: American MathematicalSociety, 2006. ISBN 0-8218-4178-5.

• Berndt, Bruce C. Ramanujan's Notebooks, Part I. New York: Springer, 1985. ISBN 0-387-96110-0.• Berndt, Bruce C. Ramanujan's Notebooks, Part II. New York: Springer, 1999. ISBN 0-387-96794-X.• Berndt, Bruce C. Ramanujan's Notebooks, Part III. New York: Springer, 2004. ISBN 0-387-97503-9.

Srinivasa Ramanujan 13

• Berndt, Bruce C. Ramanujan's Notebooks, Part IV. New York: Springer, 1993. ISBN 0-387-94109-6.• Berndt, Bruce C. Ramanujan's Notebooks, Part V. New York: Springer, 2005. ISBN 0-387-94941-0.• Hardy, G. H. Ramanujan. New York, Chelsea Pub. Co., 1978. ISBN 0-8284-0136-5• Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Providence, Rhode

Island: American Mathematical Society, 1999. ISBN 0-8218-2023-0.• Henderson, Harry. Modern Mathematicians. New York: Facts on File Inc., 1995. ISBN 0-8160-3235-1.• Kanigel, Robert. The Man Who Knew Infinity: a Life of the Genius Ramanujan. New York: Charles Scribner's

Sons, 1991. ISBN 0-684-19259-4.• Kolata, Gina. "Remembering a 'Magical Genius'", Science, New Series, Vol. 236, No. 4808 (Jun. 19, 1987),

pp. 1519–1521, American Association for the Advancement of Science.• Leavitt, David. The Indian Clerk. London: Bloomsbury, 2007. ISBN 978-0-7475-9370-6 (paperback).• Narlikar, Jayant V. Scientific Edge: the Indian Scientist From Vedic to Modern Times. New Delhi, India: Penguin

Books, 2003. ISBN 0-14-303028-0.• T.M.Sankaran. "Srinivasa Ramanujan- Ganitha lokathile Mahaprathibha", (in Malayalam), 2005, Kerala Sastra

Sahithya Parishath, Kochi.

External links

Media links• "Film to celebrate mathematics genius" (http:/ / news. bbc. co. uk/ 2/ hi/ south_asia/ 4811920. stm). BBC. 16

March 2006. Retrieved 24 August 2006.• Feature Film on Mathematics Genius Ramanujan by Dev Benegal and Stephen Fry (http:/ / devbenegal. com/

2006/ 03/ 15/ feature-film-on-math-genius-ramanujan/ )• BBC radio programme about Ramanujan – episode 5 (http:/ / www. bbc. co. uk/ radio4/ science/ further5. shtml)• A biographical song about Ramanujan's life (http:/ / www. archive. org/ details/ Ramanujan)

Biographical links• O'Connor, John J.; Robertson, Edmund F., "Srinivasa Ramanujan" (http:/ / www-history. mcs. st-andrews. ac. uk/

Biographies/ Ramanujan. html), MacTutor History of Mathematics archive, University of St Andrews.• Weisstein, Eric W., Ramanujan, Srinivasa (1887–1920) (http:/ / scienceworld. wolfram. com/ biography/

Ramanujan. html) from ScienceWorld.• Biography of this mathematical genius at World of Biography (http:/ / www. worldofbiography. com/ 9094-S.

Ramanujan/ )• Srinivasan Ramanujan in One Hundred Tamils of 20th Century (http:/ / www. tamilnation. org/ hundredtamils/

ramanujan. htm)• Srinivasa Aiyangar Ramanujan (http:/ / intranet. woodvillehs. sa. edu. au/ pages/ resources/ maths/ History/

Rmnjn. htm)• A short biography of Ramanujan (http:/ / www. usna. edu/ Users/ math/ meh/ ramanujan. html)• "A passion for numbers" (http:/ / www. hinduonnet. com/ folio/ fo0102/ 01020480. htm)

Srinivasa Ramanujan 14

Other links• A Study Group For Mathematics: Srinivasa Ramanujan Iyengar (http:/ / groups. yahoo. com/ group/

srinivasaramanujan/ )• The Ramanujan Journal (http:/ / www. math. ufl. edu/ ~frank/ ramanujan. html) – An international journal

devoted to Ramanujan• International Math Union Prizes (http:/ / www. mathunion. org/ General/ Prizes/ ), including a Ramanujan Prize.• Complicite Production of "A Disappearing Number" (http:/ / www. complicite. org/ productions/ detail.

html?id=43) – a play about Ramanujan's work• Hindu.com: Norwegian and Indian mathematical geniuses (http:/ / www. hindu. com/ mag/ 2004/ 12/ 26/ stories/

2004122600610400. htm), RAMANUJAN — Essays and Surveys (http:/ / www. hindu. com/ thehindu/ br/ 2003/08/ 26/ stories/ 2003082600120300. htm), Ramanujan's growing influence (http:/ / www. hindu. com/ 2003/ 12/22/ stories/ 2003122204061100. htm), Ramanujan's mentor (http:/ / www. hindu. com/ thehindu/ mag/ 2002/ 12/22/ stories/ 2002122200040400. htm)

• Hindu.com: The sponsor of Ramanujan (http:/ / www. thehindu. com/ life-and-style/ metroplus/ article884010.ece)

• Bruce C. Berndt; Robert A. Rankin (2000). "The Books Studied by Ramanujan in India" (http:/ / links. jstor. org/sici?sici=0002-9890(200008/ 09)107:7<595:TBSBRI>2. 0. CO;2-6). American Mathematical Monthly(Mathematical Association of America) 107 (7): 595–601. doi:10.2307/2589114. MR1786233.

• "Ramanujan's mock theta function puzzle solved" (http:/ / www. maa. org/ news/ 030807puzzlesolved. html)• Ramanujan's papers and notebooks (http:/ / www. imsc. res. in/ ~rao/ ramanujan/ contentindex. html)• Sample page from the second notebook (http:/ / www. cecm. sfu. ca/ organics/ papers/ borwein/ paper/ html/

local/ ramnotebook. html)

Article Sources and Contributors 15

Article Sources and ContributorsSrinivasa Ramanujan  Source: http://en.wikipedia.org/w/index.php?oldid=399106116  Contributors: 100110100, 84user, Abdull, Acid2base, AdamRetchless, Adiswini, Adoniscik,Ahoerstemeier, Alansohn, Aldrich lucas, Allansteel, Alphachimp, Alren, Alsandro, AlvinMGO, Alwpoe, Ambarish, Ambi saba, An Justified Wikipedian, Ancheta Wis, Andrew Levine, Andy M.Wang, AnonMoos, Anonymous Dissident, Anthony, Antonio Lopez, Arch dude, ArglebargleIV, Arienh4, Ario, Ark pillai, Arthena, Arundhati bakshi, Asdofindia, Ash deb, Ashokmani,Ashwatham, Atif.t2, Autocracy, Aviad2001, Avmpsycho, Avnjay, AxelBoldt, Badripk, Bakasuprman, BalajiRamasubramanian, Balu.muthu, Banes, Banus, Barras, Baryonic Being, Bender235,Berasategui, Bhumiya, BigDunc, Bigmac31, BillFlis, Billlion, Black Falcon, Blakkandekka, Bobo192, Bsvprasad, Bucksburg, Bvssatish, CPWinter, Caltas, CanisRufus, Card, Causa sui,Chadhade, Charles Matthews, Chicheley, Chick Bowen, Chinju, Chocolateboy, Chuunen Baka, Clarityfiend, Code Wizard, CommonsDelinker, Conscious, Coyets, Cribananda, Curps, Cyphase,DARTH SIDIOUS 2, DESiegel, DHN, DRLB, DSachan, DVD R W, DYLAN LENNON, Da Stressor, DaGizza, DancingPenguin, Dankster, DavidCBryant, DeadEyeArrow, Defender of torch,Deltabeignet, DerHexer, Dexter73, Diddyeinstein, Disavian, Djkimmons, Djmdutch11, Dodiad, Dogah, DonSiano, Dr.K., Droll, Drummondjacob, Dycedarg, EamonnPKeane, Eiler7, Elite ferns,Enigmaman, Epbr123, Etaonish, Eugene van der Pijll, Everyking, Evolauxia, Exitr, Exploding Boy, Exteray, FT2, Fatal!ty, Favonian, Firstblossoms, Fish and karate, Francisco Tevez, Fredrik,Fuhghettaboutit, Furrykef, G716, GTBacchus, Gabbe, Gaius Cornelius, Gala.martin, Gamaliel, Gareth E Kegg, Garion96, Gauge00, Gaurav1146, Gershwinrb, Giftlite, Gilliam, Gloriamarie,Gnusbiz, Goalieprice13, Gogo Dodo, Gotyear, Gowthamjayapal, Gpohara, GrafZahl, Graham87, GregorB, Groovybill, Gyopi, Haemo, Haham hanuka, Hair Commodore, Hannes Eder, Haonhien,Harryboyles, Hasdrubal, Helix84, Herostratus, Hindduking, Hmrox, Husond, Hv, Hypergeometric2F1(a,b,c,x), Ian.homer, Ibroy00, Icairns, Iceager, Ijon, Ilmari Karonen, Indopug, Indosauros,Indythegeek, IntrepidWill, Ioannes Pragensis, Isnow, JaGa, Jackohare, Jagdeep, Jagged 85, Jak86, Jaredwf, Jason Carreiro, Jason Quinn, Jason Recliner, Esq., Jawed, Jeff G., Jengod, Jeronimo,JesseHogan, JidGom, Jitse Niesen, Joshua Issac, Jovianeye, Julia Rossi, Julien Tuerlinckx, Jumbuck, Justin W Smith, KJS77, Karl Dickman, Kaustubh, Keyan20, Kkm010, Koavf, Kobeisbeast,Kope, Krishnachandranvn, Krishnakoli, Kseferovic, Kseon, Kshieh, Kunalkishore55, Kungfuadam, La goutte de pluie, LachlanA, Lambiam, Larry V, LavosBacons, Ldrhcp, Lenin and McCarthy,Lenthe, Lestrade, Lethe, Libanar, Liface, Linas, Logan, LostLeviathan, LouScheffer, Lowe4091, Lowellian, Lquilter, M a s, Macslacker, Madman, Malapati, MarcelB612, Markizs,Masterpiece2000, Mathprofrockstar, Maurice Carbonaro, Maxal, Mbhagavan, Menchi, Merope, Mets501, Michael Angelkovich, Michael Hardy, Mik01aj, MikeCapone, Mikeblas, MilanJain,Mindmatrix, Misortie, Mohammed sultan, Mon4, Moral Simplex, Mouse Nightshirt, MoxJet, Mpatel, Mrholybrain, Ms2ger, Msrkiran, Mtommila, Mungomba, Myanw, Myasuda, N Shar,NERIC-Security, NSH001, NTK, Naa.ganesan, Nandakumarg, Nataraja, NawlinWiki, Nazra, Nbarth, Neilc, Nichalp, Nick C, NilsB, Nishkid64, Nonagonal Spider, Norm mit, Nv8200p, Omexis,Onathinwhiteline, Onorem, Opelio, Oracleofottawa, Outriggr, Oxymoron83, P.K.Niyogi, Paddu, Pandacomics, Parunach, Paul August, Paul venter, Paul1776, Pdelong, Peruvianllama, Pete.Hurd,Petello12000, Peter439, Pgk, Philip Trueman, Phys, Pichai Asokan, Pilotguy, Plasticspork, Plastikspork, Poddarrishabh, Pol098, Pomte, Poor Yorick, Prabaraj, Pranathi, Praveen pillay, Pred,Primes, Pseinstein, Pt, Puffin, Pyrop, QwertySama, R.e.b., R0uge, RNLion, RODERICKMOLASAR, Raguks, Rajamankkan, Rajivsundar, Rajprem, Rak3sh, Randhirreddy, Randomblue,Ranveig, Ravichandar84, Razorflame, Redtigerxyz, Relata refero, Resham Sivnarain, Resurgent insurgent, Riana, Rich Farmbrough, Richard L. 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Image Sources, Licenses and ContributorsImage:Ramanujan.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Ramanujan.jpg  License: Public Domain  Contributors: ABF, Aka, Angr, CBDunkerson, Gene.arboit, Kevyn,Kilom691, Mehmet Karatay, Phrood, Ranveig, Tarawneh, Yann, 竹麦魚(Searobin)Image:Ramanujanhome.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Ramanujanhome.jpg  License: Public Domain  Contributors: Adiswini, Nishkid64

LicenseCreative Commons Attribution-Share Alike 3.0 Unportedhttp:/ / creativecommons. org/ licenses/ by-sa/ 3. 0/