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| {{Indian name|Ramanujan|Srinivasa}}
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| {{Use British English|date=November 2011}}
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| {{Use dmy dates|date=November 2011}}
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| {{Infobox scientist
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| | name = Srinivasa Ramanujan
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| | image = Srinivasa Ramanujan - OPC - 1.jpg
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| | image_size = 220px
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| | birth_date = {{Birth date|1887|12|22|df=y}}
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| | birth_place = [[Erode]], [[Madras Presidency]] (now [[Tamil Nadu]])
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| | death_date = {{Death date and age|1920|4|26|1887|12|22|df=y}}
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| | death_place = [[Chetput]], [[Madras]], Madras Presidency (now Tamil Nadu)
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| | residence = [[Kumbakonam]], Tamil Nadu
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| | nationality = [[British Raj|Indian]]
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| | field = [[Mathematician|Mathematics]]
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| | religion = [[Hinduism]]
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| | alma_mater = {{nowrap|[[Government Arts College, Kumbakonam|Government Arts College]] <br /> [[Pachaiyappa's College]]}}
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| | academic_advisors = [[G. H. Hardy]] <br /> [[John Edensor Littlewood|J. E. Littlewood]]
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| | known_for = {{nowrap|[[Landau–Ramanujan constant]] <br /> [[Mock theta function]]s <br /> [[Ramanujan conjecture]] <br /> [[Ramanujan prime]] <br /> [[Ramanujan–Soldner constant]] <br /> [[Ramanujan theta function]] <br /> [[Ramanujan's sum]] <br /> [[Rogers–Ramanujan identities]] <br /> [[Ramanujan's master theorem]]}}
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| | awards =
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| | influences = [[G. H. Hardy]]
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| |signature=Srinivasa Ramanujan signature.gif
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| |signature alt=Srinivasa Ramanujan signature
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| }}
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| '''Srinivasa Ramanujan''' [[Fellow of the Royal Society|FRS]] ({{Audio|Srinivasa_ramanujan_wikipedia.ogg|pronunciation}}) (22 December 1887{{spaced ndash}}26 April 1920) was an Indian [[mathematician]] and [[autodidact]] who, with almost no formal training in [[pure mathematics]], made extraordinary contributions to [[mathematical analysis]], [[number theory]], [[infinite series]], and [[continued fraction]]s. Living in India with no access to the larger mathematical community, which was centred in Europe at the time, Ramanujan developed his own mathematical research in isolation. As a result, he rediscovered known theorems in addition to producing new work. Ramanujan was said to be a natural genius by the English mathematician [[G. H. Hardy]], in the same league as mathematicians such as [[Leonhard Euler|Euler]] and [[Carl Friedrich Gauss|Gauss]].<ref>[[C.P. Snow]] Foreword to "[[A Mathematician's Apology]]" by [[G.H. Hardy]]</ref>
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| Ramanujan was born at [[Erode]], [[Madras Presidency]] (now [[Tamil Nadu]]) in a [[Tamil people|Tamil]] [[Brahmin]] family of [[Thenkalai]] Iyengar sect.<ref>{{cite news|title=Ramanujan lost and found: a 1905 letter from The Hindu|url=http://www.thehindu.com/arts/history-and-culture/ramanujan-lost-and-found-a-1905-letter-from-the-hindu/article2745164.ece|date=December 25, 2011}}</ref><ref>http://www.3quarksdaily.com/3quarksdaily/2011/01/the-use-and-misuse-of-srinivasa-ramanujan.html</ref><ref>http://mathsc.tripod.com/ramanujan/sramanujan.htm</ref> His introduction to formal [[mathematics]] began at age 10. He demonstrated a natural ability, and was given books on advanced [[trigonometry]] written by [[S. L. Loney]] that he mastered by the age of 12; he even discovered [[theorem]]s of his own, and re-discovered [[Euler's identity]] independently.<ref name="berndt9">{{Harvnb|Berndt|Rankin|2001|p=9}}</ref> He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan had conducted his own mathematical research on [[Bernoulli number]]s and the [[Euler–Mascheroni constant]].
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| Ramanujan received a scholarship to study at Government College in [[Kumbakonam]], which was later rescinded 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.<ref name="lostnotebook">{{cite web
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| |url=http://www.las.illinois.edu/alumni/magazine/articles/2006/lostnotebook/|title=Raiders of the Lost Notebook|accessdate=11 Jan 2014|last=Peterson|first=Doug|publisher=[[UIUC College of Liberal Arts and Sciences]]}}</ref> In 1912–1913, he sent samples of his theorems to three academics at the [[University of Cambridge]]. [[G. H. Hardy]], recognizing the brilliance of his work, invited Ramanujan to visit and work with him at [[Cambridge]]. He became a [[Fellow of the Royal Society]] and a Fellow of [[Trinity College, Cambridge]]. Ramanujan died of illness, malnutrition, and possibly liver infection in 1920 at the age of 32.
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| During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly [[identity (mathematics)|identities]] and [[equation]]s).<ref>{{Cite book|last=Berndt |first=Bruce C. |title= Ramanujan's Notebooks Part V|year= 2005 |publisher= [[Springer Science+Business Media|SpringerLink]] |isbn= 0-387-94941-0 | page=4}}</ref> Nearly all his claims have now been proven correct, although a small number of these results were actually false and some were already known.<ref>{{Cite journal|year=1999 |month=August |title=Rediscovering Ramanujan |journal= [[Frontline (magazine)|Frontline]]|volume=16 |issue=17 |page=650|url=http://www.frontlineonnet.com/fl1617/16170810.htm |accessdate=20 December 2012|ref=harv }}</ref> He stated results that were both original and highly unconventional, such as the [[Ramanujan prime]] and the [[Ramanujan theta function]], and these have inspired a vast amount of further research.<ref>{{Cite journal|last=Ono |first= Ken|authorlink=Ken Ono |date=June–July 2006 |title=Honoring a Gift from Kumbakonam |journal= [[Notices of the American Mathematical Society]]|volume=53 |issue=6 |page=650|url=http://www.ams.org/notices/200606/fea-ono.pdf |format=PDF|accessdate=23 June 2007 |publisher=Mathematical Association of America|ref=harv }}</ref> However, the mathematical mainstream has been rather slow in absorbing some of his major discoveries.{{citation needed|date=June 2013}} The ''Ramanujan Journal'', an international publication, was launched to publish work in all areas of mathematics influenced by his work.<ref>{{Cite book| last=Alladi | first=Krishnaswami | title=Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdös| publisher=Kluwer Academic Publishers | location=Norwell, Massachusetts | year = 1998|isbn=0-7923-8273-0 | page=6 }}</ref>
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| In December 2011, in recognition of his contribution to mathematics, the Government of India declared that Ramanujan's birthday (22 December) should be celebrated every year as [[National Mathematics Day]], and also declared 2012 the [[National Mathematics Year]].<ref>{{cite web|author=C. Jaishankar |url=http://www.thehindu.com/news/cities/chennai/article2750402.ece |title=Ramanujan's birthday will be National Mathematics Day |publisher=Thehindu.com |date=27 December 2011 |accessdate=20 November 2012}}</ref><ref name=vigyanprasar>[http://www.vigyanprasar.gov.in/nmy2012/National_Mathematical_Year2012.htm National Mathematical Year 2012], Vigyan Prasar Science Portal, vigyanprasar.gov.in</ref>
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| ==Early life==
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| [[Image:Ramanujanhome.jpg|thumb|Ramanujan's home on Sarangapani Street, Kumbakonam]]
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| Ramanujan was born on 22 December 1887 in [[Erode]], [[Madras Presidency]] (now Pallipalayam Agraharam, Namakkal Disrict, [[Tamil Nadu]]), at the residence of his maternal grandparents.<ref>{{Harvnb|Kanigel|1991|p=11}}</ref> His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and hailed from the district of [[Thanjavur]].<ref>{{Harvnb|Kanigel|1991|pp=17–18}}</ref> His mother, Komalatammal, was a [[homemaker|housewife]] and also sang at a local temple.<ref name="ramanujan_essays_p89">{{Harvnb|Berndt|Rankin|2001|p=89}}</ref> They lived in Sarangapani Street in a traditional home in the town of Kumbakonam. The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son named Sadagopan, who died less than three months later. In December 1889, Ramanujan had [[smallpox]] and recovered, unlike thousands in the [[Thanjavur District]] who died from the disease that year.<ref name="p12">{{Harvnb|Kanigel|1991|p=12}}</ref> He moved with his mother to her parents' house in [[Kanchipuram]], near Madras (now [[Chennai]]). In November 1891, and again in 1894, his mother gave birth to two children, but both children died in infancy.
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| On 1 October 1892, Ramanujan was enrolled at the local school.<ref>{{Harvnb|Kanigel|1991|p=13}}</ref> In March 1894, he was moved to a [[Medium of instruction|Telugu medium]] school. After his maternal grandfather lost his job as a court official in Kanchipuram,<ref>{{Harvnb|Kanigel|1991|p=19}}</ref> Ramanujan and his mother moved back to [[Kumbakonam]] and he was enrolled in the Kangayan Primary School.<ref name="p14">{{Harvnb|Kanigel|1991|p=14}}</ref> When 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 attending. His family enlisted a local constable to make sure he attended school. Within six months, Ramanujan was back in Kumbakonam.<ref name="p14"/>
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| Since Ramanujan's father was at work most of the day, his mother took care of him as a child. He had a close relationship with her. From her, he learned about tradition and [[puranas]]. He learned to sing religious songs, to attend pujas at the temple and particular eating habits – all of which are part of [[Brahmin]] culture.<ref>{{Harvnb|Kanigel|1991|p=20}}</ref> At the Kangayan Primary School, Ramanujan performed well. Just before the age of 10, in November 1897, he passed his primary examinations in English, [[Tamil language|Tamil]], geography and arithmetic. With his scores, he stood first in the district.<ref name="Kanigel 1991, p25">{{Harvnb|Kanigel|1991|p=25}}</ref> That year, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the first time.<ref name="Kanigel 1991, p25"/>
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| 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]].<ref name="berndt9"/><ref>{{Cite book| last=Hardy | first=G. H.| title=Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work| publisher=[[American Mathematical Society]] | location=Providence, Rhode Island | year = 1999|isbn=0-8218-2023-0 | page=2 |authorlink4=[[G. H. Hardy]] }}</ref> 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.<ref name="p27">{{Harvnb|Kanigel|1991|p=27}}</ref> He completed mathematical exams in half the allotted time, and showed a familiarity with [[geometry]] and [[infinite series]]. Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.
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| In 1903 when he was 16, Ramanujan obtained from a friend a library-loaned copy of a book by [[G. S. Carr]].<ref>{{Harvnb|Kanigel|1991|p=39}}</ref><ref>''A to Z of mathematicians'' by Tucker McElroy 2005 ISBN 0-8160-5338-3 page 221</ref> The book was titled ''[[Synopsis of Pure Mathematics|A Synopsis of Elementary Results in Pure and Applied Mathematics]]'' and was a collection of 5000 theorems. Ramanujan reportedly studied the contents of the book in detail.<ref name=papers /> The book is generally acknowledged as a key element in awakening the genius of Ramanujan.<ref name=papers >''Collected papers of Srinivasa Ramanujan'' Srinivasa Ramanujan Aiyangar, Godfrey Harold Hardy, P. Veṅkatesvara Seshu Aiyar 2000 ISBN 0-8218-2076-1 page xii</ref> The next year, he had independently developed and investigated the [[Bernoulli number]]s and had calculated the [[Euler–Mascheroni constant]] up to 15 decimal places.<ref>{{Harvnb|Kanigel|1991|p=90}}</ref> His peers at the time commented that they "rarely understood him" and "stood in respectful awe" of him.<ref name="p27"/>
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| When he graduated from [[Town Higher Secondary School]] in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than the maximum possible marks.<ref name="p27"/> He received a scholarship to study at [[Government Arts College, Kumbakonam]],<ref>{{Harvnb|Kanigel|1991|p=28}}</ref><ref>{{Harvnb|Kanigel|1991|p=45}}</ref> However, Ramanujan was so intent on studying mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process.<ref>{{Harvnb|Kanigel|1991|p=47}}</ref> In August 1905, he ran away from home, heading towards [[Visakhapatnam]] and stayed in [[Rajahmundry]] <ref>http://www.thehindu.com/arts/history-and-culture/ramanujan-lost-and-found-a-1905-letter-from-the-hindu/article2745164.ece</ref> for about a month.<ref>{{Harvnb|Kanigel|1991|pp=48–49}}</ref> He later enrolled at [[Pachaiyappa's College]] in Madras. He again excelled in mathematics but performed poorly in other subjects such as physiology. Ramanujan failed his [[Fellow of Arts]] exam in December 1906 and again a year later. Without a degree, he left college and continued to pursue independent research in mathematics. At this point in his life, he lived in extreme poverty and was often on the brink of starvation.<ref>{{Harvnb|Kanigel|1991|pp=55–56}}</ref>
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| ==Adulthood in India==
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| On 14 July 1909, Ramanujan was married to a ten-year old bride, Janakiammal (21 March 1899 – 13 April 1994).<ref>{{Harvnb|Kanigel|1991|p=71}}</ref> She came from Rajendram, a village close to Marudur (Karur district) Railway Station. Ramanujan's father did not participate in the marriage ceremony.<ref name="Janaki">{{cite web|title=Ramanujan’s wife: Janakiammal (Janaki)|url=http://www.imsc.res.in/~rao/ramanujan/newnow/janaki.pdf|publisher=Institute of Mathematical Sciences, Chennai|accessdate=10 November 2012}}</ref>
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| After the marriage, Ramanujan developed a [[hydrocele testis]], an abnormal swelling of the [[tunica vaginalis]], an internal membrane in the testicle.<ref>{{Harvnb|Kanigel|1991|p=72}}</ref> The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac. His family did not have the money for the operation, but in January 1910, a doctor volunteered to do the surgery for free.<ref>{{Cite book|last=Ramanujan|first=Srinivasa|editor = P. K. Srinivasan |title= Ramanujan Memorial Number: Letters and Reminiscences |year= 1968|publisher= Muthialpet High School |location=Madras|isbn= | pages=Vol. 1, p100 |nopp=true}}</ref>
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| After his successful surgery, Ramanujan searched for a job. He stayed at friends' houses while he went door to door around the city of [[Chennai|Madras]] (now Chennai) looking for a clerical position. To make some money, he tutored some students at Presidency College who were preparing for their F.A. exam.<ref>{{Harvnb|Kanigel|1991|p=73}}</ref>
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| In late 1910, Ramanujan was sick again, possibly as a result of the surgery earlier in the year. He feared for his health, and even told his friend, R. Radakrishna Iyer, to "hand these [Ramanujan's mathematical notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the [[Madras Christian College]]."<ref>{{Harvnb|Kanigel|1991|pp=74–75}}</ref> After Ramanujan recovered and got back his notebooks from Iyer, he took a northbound train from Kumbakonam to [[Viluppuram (city)|Villupuram]], a coastal city under French control.<ref>{{Cite book |last=Ranganathan |first=Shiyali Ramamrita |authorlink=Shiyali Ramamrita Ranganathan |title=Ramanujan: The Man and the Mathematician |year=1967 |publisher=[[Asia Publishing House]] |location=Bombay |isbn= |ref=harv |url=http://books.google.com/?id=OuTuAAAAMAAJ }}, p. 23.</ref><ref>Srinivasan (1968), Vol. 1, p99.</ref>
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| ===Attention towards mathematics===
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| Ramanujan met deputy collector [[V. Ramaswamy Aiyer]], who had recently founded the Indian Mathematical Society.<ref name="p77">{{Harvnb|Kanigel|1991|p=77}}</ref> Ramanujan, wishing for a job at the revenue department where Ramaswamy Aiyer worked, showed him his mathematics notebooks. As Ramaswamy Aiyer later recalled:
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| <blockquote>
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| I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.<ref>Srinivasan (1968), Vol. 1, p129.</ref>
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| </blockquote>
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| Ramaswamy Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras.<ref name="p77"/> 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.<ref>Srinivasan (1968), Vol. 1, p86.</ref><ref>{{Cite journal|last=Neville |first=Eric Harold |date=January 1921 |title=The Late Srinivasa Ramanujan |journal=[[Nature (journal)|Nature]]|volume=106 |issue=2673 |pages=661–662 |doi=10.1038/106661b0 |bibcode = 1921Natur.106..661N|ref=harv }}</ref><ref>{{Harvnb|Ranganathan|1967|p=24}}</ref> 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 [[Mumbai|Bombay]] mathematician, in which Saldhana expressed a lack of understanding for his work but concluded that he was not a phony.<ref name="p80">{{Harvnb|Kanigel|1991|p=80}}</ref> 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 integral]]s, [[Gaussian hypergeometric series|hypergeometric series]], and his theory of [[divergent series]], which Rao said ultimately "converted" him to a belief in Ramanujan's mathematical brilliance.<ref name="p80"/> When Rao asked him what he wanted, Ramanujan replied that he needed some work and financial support. Rao consented and sent him to Madras. He continued his mathematical research with Rao's financial aid taking care of his daily needs. Ramanujan, with the help of Ramaswamy Aiyer, had his work published in the ''Journal of the Indian Mathematical Society''.<ref>{{Harvnb|Kanigel|1991|p=86}}</ref>
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| One of the first problems he posed in the journal was:
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| : <math>\sqrt{1+2\sqrt{1+3 \sqrt{1+\cdots}}}.</math>
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| 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 an equation that could be used to solve the infinitely [[nested radical]]s problem.
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| : <math>x+n+a = \sqrt{ax+(n+a)^2 +x\sqrt{a(x+n)+(n+a)^2+(x+n) \sqrt{\cdots}}}</math>
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| Using this equation, the answer to the question posed in the ''Journal'' was simply 3.<ref>{{Harvnb|Kanigel|1991|p=87}}</ref> Ramanujan wrote his first formal paper for the ''Journal'' on the properties of [[Bernoulli number]]s. One property he discovered was that the denominators {{OEIS|id=A027642}} of the fractions of Bernoulli numbers were always divisible by six. He also devised a method of calculating ''B<sub>n</sub>'' based on previous Bernoulli numbers. One of these methods went as follows:
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| It will be observed that if ''n'' is even but not equal to zero,<br>
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| (i) ''B<sub>n</sub>'' is a fraction and the numerator of <math>{B_n \over n}</math> in its lowest terms is a prime number,<br>
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| (ii) the denominator of ''B<sub>n</sub>'' contains each of the factors 2 and 3 once and only once,<br>
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| (iii) <math>2^n(2^n-1){b_n \over n}</math> is an integer and <math>2(2^n-1)B_n\,</math> consequently is an ''odd'' integer.
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| In his 17-page paper, "Some Properties of Bernoulli's Numbers", Ramanujan gave three proofs, two corollaries and three conjectures.<ref>{{Harvnb|Kanigel|1991|p=91}}</ref> Ramanujan's writing initially had many flaws. As ''Journal'' editor M. T. Narayana Iyengar noted:
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| <blockquote>
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| Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.<ref>{{Cite journal|last=Seshu Iyer |first=P. V. |date=June 1920 |title=The Late Mr. S. Ramanujan, B.A., F.R.S|journal=Journal of the Indian Mathematical Society|volume=12 |issue=3 |page=83|ref=harv }}</ref></blockquote>
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| Ramanujan later wrote another paper and also continued to provide problems in the ''Journal''.<ref>Neville (March 1942), p292.</ref> In early 1912, he got a temporary job in the Madras [[Accountant General]]'s office, with a salary of 20 rupees per month. He lasted for only a few weeks.<ref>Srinivasan (1968), p176.</ref> Toward the end of that assignment he applied for a position under the Chief Accountant of the Madras Port Trust. In a letter dated 9 February 1912, Ramanujan wrote:
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| <blockquote>
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| Sir,<br>
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| I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.<ref>Srinivasan (1968), p31.</ref>
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| </blockquote>
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| Attached to his application was a recommendation from [[E. W. Middlemast]], a mathematics professor at the [[Presidency College, Chennai|Presidency College]], who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics".<ref>Srinivasan (1968), p49.</ref> Three weeks after he had applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making 30 rupees per month.<ref>{{Harvnb|Kanigel|1991|p=96}}</ref> At his office, Ramanujan easily and quickly completed the work he was given, so he spent his spare time doing mathematical research. Ramanujan's boss, [[Francis Spring|Sir Francis Spring]], and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits.
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| ===Contacting English mathematicians===
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| In the spring of 1913, Narayana Iyer, Ramachandra Rao and [[E. W. Middlemast]] tried to present Ramanujan's work to British mathematicians. One mathematician, [[Micaiah John Muller Hill|M. J. M. Hill]] of [[University College London]], commented that Ramanujan's papers were riddled with holes.<ref>{{Harvnb|Kanigel|1991|p=105}}</ref> 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.<ref>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.</ref> Although Hill did 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.<ref>{{Harvnb|Kanigel|1991|p=106}}</ref>
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| The first two professors, [[H. F. Baker]] and [[E. W. Hobson]], returned Ramanujan's papers without comment.<ref>{{Harvnb|Kanigel|1991|pp=170–171}}</ref> On 16 January 1913, Ramanujan wrote to [[G. H. Hardy]]. Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible "fraud".<ref>{{Cite book|last=Snow|first=C. P. |title= Variety of Men |year= 1966|publisher= [[Charles Scribner's Sons]] |location=New York|isbn= | pages=30–31}}</ref> Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe".<ref>{{Cite journal|last=Hardy |first=G. H.|authorlink=G. H. Hardy|date=June 1920 |title=Obituary, S. Ramanujan |journal=Nature|volume=105 |issue= 7|page=494 |doi=10.1038/105494a0 |bibcode = 1920Natur.105..494H|ref=harv }}</ref> One of the theorems Hardy found so incredible was found on the bottom of page three (valid for 0 < ''a'' < ''b'' + 1/2):
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| : <math>\int_0^\infty \cfrac{1+{x}^2/({b+1})^2}{1+{x}^2/({a})^2} \times\cfrac{1+{x}^2/({b+2})^2}{1+{x}^2/({a+1})^2}\times\cdots\;\;dx = \frac{\sqrt \pi}{2} \times\frac{\Gamma(a+\frac{1}{2})\Gamma(b+1)\Gamma(b-a+\frac{1}{2})}{\Gamma(a)\Gamma(b+\frac{1}{2})\Gamma(b-a+1)}.</math>
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| Hardy was also impressed by some of Ramanujan's other work relating to infinite series:
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| : <math>1 - 5\left(\frac{1}{2}\right)^3 + 9\left(\frac{1\times3}{2\times4}\right)^3 - 13\left(\frac{1\times3\times5}{2\times4\times6}\right)^3 + \cdots = \frac{2}{\pi}</math>
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| : <math>1 + 9\left(\frac{1}{4}\right)^4 + 17\left(\frac{1\times5}{4\times8}\right)^4 + 25\left(\frac{1\times5\times9}{4\times8\times12}\right)^4 + \cdots = \frac{2^\frac{3}{2}}{\pi^\frac{1}{2}\Gamma^2\left(\frac{3}{4}\right)}.</math>
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| The first result had already been determined by a mathematician named Bauer. The second one was new to Hardy, and was derived from a class of functions called a [[hypergeometric series]] which had first been researched by [[Leonhard Euler]] and [[Carl Friedrich Gauss]]. Compared to Ramanujan's work on [[integrals]], Hardy found these results "much more intriguing".<ref>{{Harvnb|Kanigel|1991|p=167}}</ref> After he saw Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy commented that "they [theorems] defeated me completely; I had never seen anything in the least like them before".<ref name="p168">{{Harvnb|Kanigel|1991|p=168}}</ref> He figured that Ramanujan's theorems "must be true, because, if they were not true, no one would have the imagination to invent them".<ref name="p168"/> Hardy asked a colleague, [[John Edensor Littlewood|J. E. Littlewood]], to take a look at the papers. Littlewood was amazed by the mathematical genius of Ramanujan. After discussing the papers with Littlewood, Hardy concluded 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".<ref>Hardy (June 1920), pp494–495.</ref> One colleague, [[Eric Harold Neville|E. H. Neville]], later commented that "not one [theorem] could have been set in the most advanced mathematical examination in the world".<ref name="Neville293">{{Cite journal|last=Neville |first=Eric Harold |date=March 1942 |title=Srinivasa Ramanujan |journal=Nature|volume=149 |issue=3776 |page=293 |doi=10.1038/149292a0 |bibcode = 1942Natur.149..292N|ref=harv }}</ref>
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| On 8 February 1913, Hardy wrote a letter to Ramanujan, expressing his interest for his work. Hardy also added that it was "essential that I should see proofs of some of your assertions".<ref>Letter, Hardy to Ramanujan, 8 February 1913.</ref> Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to plan for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip.<ref>Letter, Ramanujan to Hardy, 22 January 1914.</ref> In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land".<ref>{{Harvnb|Kanigel|1991|p=185}}</ref> Meanwhile, Ramanujan sent a letter packed with theorems to Hardy, writing, "I have found a friend in you who views my labour sympathetically."<ref>Letter, Ramanujan to Hardy, 27 February 1913, [[Cambridge University Library]].</ref>
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| 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.<ref>{{Harvnb|Kanigel|1991|p=175}}</ref> 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".<ref>{{Cite book|last=Ram|first=Suresh |title= Srinivasa Ramanujan |year= 1972|publisher= National Book Trust |location=New Delhi|isbn= | page=29}}</ref> The board agreed to grant Ramanujan a research scholarship of 75 rupees per month for the next two years at the [[University of Madras]].<ref>{{Harvnb|Ranganathan|1967|pp=30–31}}</ref> 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 a few years before, 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.<ref>{{Harvnb|Ranganathan|1967|p=12}}</ref> In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to evaluate formerly unyielding integrals.<ref>{{Harvnb|Kanigel|1991|p=183}}</ref>
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| Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England.<ref>{{Harvnb|Kanigel|1991|p=184}}</ref> Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparently had now accepted the proposal; as Neville put it, "Ramanujan needed no converting and that his parents' opposition had been withdrawn".<ref name="Neville293"/> Apparently, Ramanujan's mother had a vivid dream in which the family Goddess, [[Namagiri Thayar|the deity of Namagiri]], commanded her "to stand no longer between her son and the fulfilment of his life's purpose".<ref name="Neville293"/> Ramanujan then set sail for England, leaving his wife to stay with his parents in India.
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| ==Life in England==
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| [[File:RamanujanCambridge.jpg|thumb|right|Ramanujan (centre) with other scientists at Trinity College]]
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| [[File:Whewell's Court, Trinity College, Cambridge.jpg|thumb|right|upright|Whewell's Court, [[Trinity College, Cambridge|Trinity College]], Cambridge]]
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| Ramanujan boarded the S.S. ''Nevasa'' on 17 March 1914, and at 10 o'clock in the morning, the ship departed from Madras.<ref>{{Harvnb|Kanigel|1991|p=196}}</ref> He arrived in London on 14 April, with E. H. Neville waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and 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.<ref>{{Harvnb|Kanigel|1991|p=202}}</ref> Hardy and Ramanujan began to take a look at Ramanujan's notebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems to be found in the notebooks. Hardy saw that some were wrong, others had already been discovered, while the rest were new breakthroughs.<ref>{{Cite book|last=Hardy |first=G. H. |title= Ramanujan|year= 1940 |publisher= [[Cambridge University Press]] |location=Cambridge|isbn=| page=10}}</ref> Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a [[Carl Gustav Jacob Jacobi|Jacobi]]",<ref>Letter, Littlewood to Hardy, early March 1913.</ref> while Hardy said he "can compare him only with [Leonhard] Euler or Jacobi."<ref>{{Cite book|last=Hardy |first=G. H. |title= Collected Papers of G. H. Hardy|year= 1979 |publisher= [[Oxford University Press|Clarendon Press]] |location=Oxford, England|isbn=| pages=Vol. 7, p720 |nopp=true}}</ref>
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| Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood and published a part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different 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, Hardy tried his best to fill the gaps in Ramanujan's education without interrupting his spell of inspiration.
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| Ramanujan was awarded a B.A. degree by research (this degree was later renamed PhD) in March 1916 for his work on [[highly composite number]]s, the first part of which was published as a paper in the ''Proceedings of the [[London Mathematical Society]]''. The paper was over 50 pages with different properties of such numbers proven. Hardy remarked that this was one of the most unusual papers seen in mathematical research at that time and that Ramanujan showed extraordinary ingenuity in handling it.{{Citation needed|date=October 2012}} On 6 December 1917, he was elected to the London Mathematical Society. He became a [[Fellow of the Royal Society]] in 1918, becoming the second Indian to do so, following [[Ardaseer Cursetjee]] in 1841, and he was one of the youngest Fellows in the history of the Royal Society. He was elected "for his investigation in [[Elliptic function]]s and the Theory of Numbers." On 13 October 1918, he became the first Indian to be elected a [[Trinity College, Cambridge#Notable fellows and alumni|Fellow of Trinity College, Cambridge]].<ref>{{Harvnb|Kanigel|1991|pp=299–300}}</ref>
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| ===Illness and return to India===
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| Plagued by health problems throughout his life, living in a country far away from home, and obsessively involved with his mathematics, Ramanujan's health worsened in England, perhaps exacerbated by [[stress (medicine)|stress]] and by the scarcity of [[vegetarian food]] during the First World War. He was diagnosed with [[tuberculosis]] and a severe [[vitamin]] deficiency and was confined to a sanatorium.
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| Ramanujan returned to Kumbakonam, [[Madras Presidency]] in 1919 and died soon thereafter at the age of 32. His widow, S. Janaki Ammal, moved to Mumbai, but returned to Chennai (formerly Madras) in 1950, where she lived until her death in 1994.<ref name="Janaki"/>
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| 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 spent time. He had two episodes of [[dysentery]] before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis,<ref name="lostnotebook"/> a difficult disease to diagnose, but once diagnosed readily cured.<ref name="lostnotebook"/>
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| ===Personality and spiritual life===
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| Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man with pleasant manners.<ref name="Ramanujan's Personality">{{cite web|url= http://www.imsc.res.in/~rao/ramanujan/newnow/pcm5.htm
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| |title=Ramanujan's Personality}}</ref> He lived a rather Spartan life while at Cambridge. Ramanujan's first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to his [[Kuladevata|family goddess]], [[Namagiri Thayar|Mahalakshmi]] of [[Namakkal]]. He looked to her for inspiration in his work,<ref>{{Harvnb|Kanigel|1991|p=36}}</ref> and claimed to dream of blood drops that symbolised her male consort, [[Narasimha]], after which he would receive visions of scrolls of complex mathematical content unfolding before his eyes.<ref>{{Harvnb|Kanigel|1991|p=281}}</ref> He often said, "An equation for me has no meaning, unless it represents a thought of God."<ref>{{cite web| url = http://lagrange.math.trinity.edu/aholder/misc/quotes.shtml | title = Quote by Srinivasa Ramanujan Iyengar}}</ref><ref>{{Cite journal|title=Less Proof, More Truth|journal=NewScientist |issue=2614 |date=28 July 2007 |page=49 |author=Chaitin, Gregory|ref=harv }}</ref>
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| Hardy cites Ramanujan as remarking that all religions seemed equally true to him.<ref>{{Harvnb|Kanigel|1991|p=283}}</ref> Hardy further argued that Ramanujan's religiousness had been romanticised by Westerners and overstated—in reference to his belief, not practice—by Indian biographers. At the same time, he remarked on Ramanujan's strict observance of vegetarianism.
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| ==Mathematical achievements==
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| In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later. It is said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a by-product, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite [[Series (mathematics)|series]] for [[pi|π]], one of which is given below
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| :<math> \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}.</math>
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| This result is based on the negative [[fundamental discriminant]] ''d'' = −4×58 = −232 with class number ''h''(''d'') = 2 (note that 5×7×13×58 = 26390 and that 9801=99×99; 396=4×99) and is related to the fact that
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| :<math> e^{\pi \sqrt{58}} = 396^4 - 104.000000177\dots. </math>
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| Compare to [[Heegner number]]s, which have class number 1 and yield similar formulae.
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| Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation <math>9801\sqrt{2}/4412</math> for π, which is correct to six decimal places.
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| One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with [[Mahalanobis|P. C. Mahalanobis]] who had a problem, "Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x?" This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a [[continued fraction]]. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. "It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind," Ramanujan replied.<ref>{{Harvnb|Ranganathan|1967|p=82}}</ref><ref>{{Cite book|title=Statistics and truth: putting chance to work|year=1997|publisher=[[World Scientific]]|url=http://books.google.com/?id=jqWd4oe3iwIC&pg=PA185&dq=%22Which+continued+fraction%22|author=Calyampudi Radhakrishna Rao|accessdate=7 June 2010|page=185|isbn=978-981-02-3111-8}}</ref>
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| His intuition also led him to derive some previously unknown [[identity (mathematics)|identities]], such as
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| :<math> \left [ 1+2\sum_{n=1}^\infty \frac{\cos(n\theta)}{\cosh(n\pi)} \right ]^{-2} + \left [1+2\sum_{n=1}^\infty \frac{\cosh(n\theta)}{\cosh(n\pi)} \right ]^{-2} = \frac {2 \Gamma^4 \left ( \frac{3}{4} \right )}{\pi} </math>
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| for all <math>\theta</math>, where <math>\Gamma(z)</math> is the [[gamma function]]. Expanding into series of powers and equating coefficients of <math>\theta^0</math>, <math>\theta^4</math>, and <math>\theta^8</math> gives some deep identities for the [[hyperbolic secant]].
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| In 1918, Hardy and Ramanujan studied the [[Partition function (number theory)#Partition function|partition function]] ''P''(''n'') extensively and gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. [[Hans Rademacher]], in 1937, was able to refine their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae, called the [[Hardy–Littlewood circle method|circle method]].<ref name="Partition Function">{{cite web|url = http://mathworld.wolfram.com/PartitionFunctionP.html |title=Partition Formula}}</ref>
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| He discovered [[mock theta function]]s in the last year of his life.<ref name="Fox">http://www.foxnews.com/science/2012/12/28/mathematician-century-old-secrets-unlocked/</ref> For many years these functions were a mystery, but they are now known to be the holomorphic parts of
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| harmonic weak [[Maass forms]].
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| ===The Ramanujan conjecture===
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| {{Main|Ramanujan–Petersson conjecture}}
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| Although there are numerous statements that could have borne the name ''Ramanujan conjecture'', there is one statement that was very influential on later work. In particular, the connection of this conjecture with conjectures of [[André Weil]] in algebraic geometry opened up new areas of research. That [[Ramanujan conjecture]] is an assertion on the size of the [[Tau-function]], which has as generating function the discriminant modular form Δ(''q''), a typical [[cusp form]] in the theory of [[modular 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 Weil conjectures.<ref>Ono (June–July 2006), p649.</ref>
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| ===Ramanujan's notebooks===
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| {{Further|Ramanujan's lost notebook}}
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| While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of [[loose leaf]] paper. These results were mostly written up without any derivations. This is probably the origin of the misperception that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician [[Bruce C. Berndt]], in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to make the proofs of most of his results, but chose not to.
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| This style of working may have been for several reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on [[slate (writing)|slate]], and then transfer just the results to paper. Using a slate was common for mathematics students in the [[Madras Presidency]] at the time. He was also quite likely to have been influenced by the style of [[G. S. Carr]]'s book studied in his youth, which stated results without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone; and therefore recorded only the results.<ref name = "Bruce Berndt on Ramanujan">{{cite web|url = http://www.amazon.com/Ramanujans-Notebooks-Part-Bruce-Berndt/dp/0387949410 |title = Ramanujans Notebooks}}</ref>
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| The first notebook has 351 pages with 16 somewhat organized chapters and some unorganized material. The second notebook has 256 pages in 21 chapters and 100 unorganised pages, with the third notebook containing 33 unorganised pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself created papers exploring material from Ramanujan's work as did [[G. N. Watson]], B. M. Wilson, and Bruce Berndt.<ref name = "Bruce Berndt on Ramanujan"/> A fourth notebook with 87 unorganised pages, the so-called [[Ramanujan's lost notebook|"lost notebook"]], was rediscovered in 1976 by [[George Andrews (mathematician)|George Andrews]].<ref name="lostnotebook"/>
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| Notebooks 1, 2 and 3 were published as a two-volume set in 1957 by the [http://www.tifr.res.in/ Tata Institute of Fundamental Research] (TIFR), Mumbai, India. This was a photocopy edition of the original manuscripts, in his own handwriting.
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| In December 2011, as part of the celebrations of the 125th anniversary of Ramanujan's birth, TIFR republished the notebooks in a colored two-volume [http://www.math.tifr.res.in/~publ/ramanujan.html collector's edition]. These were produced from scanned and microfilmed images of the original manuscripts by expert archivists of [http://www.lib.uchicago.edu/e/su/southasia/about-rmrl.html Roja Muthiah Research Library], Chennai.
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| ==Ramanujan–Hardy number 1729==
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| {{Main|1729 (number)}}
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| The number 1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician [[G. H. Hardy]] regarding a visit to the hospital to see Ramanujan. In Hardy's words:<ref>{{cite web|url=http://www-gap.dcs.st-and.ac.uk/~history/Quotations/Hardy.html |title=Quotations by Hardy |publisher=Gap.dcs.st-and.ac.uk |date= |accessdate=20 November 2012}}</ref>
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| {{cquote|I remember once going to see him when he was ill at [[Putney]]. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a [[Interesting number paradox|dull one]], and that I hoped it was not an unfavorable [[omen]]. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."}}
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| The two different ways are
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| : 1729 = 1<sup>3</sup> + 12<sup>3</sup> = 9<sup>3</sup> + 10<sup>3</sup>.
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| Generalizations of this idea have created the notion of "[[taxicab number]]s". Coincidentally, 1729 is also a [[Carmichael number]].
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| ==Other mathematicians' views of Ramanujan==
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| Hardy said : "The limitations of his knowledge were as startling as its profundity. Here was a man who could work out [[modular equation]]s 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 [[Riemann zeta function|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 integral theorem|Cauchy's theorem]], and had indeed but the vaguest idea of what a function of a [[complex variable]] was...".<ref>{{cite web|url = http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Ramanujan.html | title = Ramanujan quote}}</ref> When asked about the methods employed by Ramanujan to arrive at his solutions, Hardy said that they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account."<ref name="cola1">[http://www.usna.edu/Users/math/meh/ramanujan.html Srinivasa Ramanujan]. Retrieved 2 December 2010.</ref> He also stated that he had "never met his equal, and can compare him only with Euler or Jacobi."<ref name="cola1" />
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| Quoting K. Srinivasa Rao,<ref name=imscraman>{{cite web| url=http://www.imsc.res.in/~rao/ramanujan.html|author=K Srinivasa Rao |title=Srinivasa Ramanujan (22 December 1887 – 26 April 1920)}}</ref> "As for his place in the world of Mathematics, we quote Bruce C. Berndt: '[[Paul Erdős]] has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, J.E. Littlewood 30, [[David Hilbert]] 80 and Ramanujan 100.'"
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| Professor Bruce C. Berndt of the [[University of Illinois]], during a lecture at [[IIT Madras]] in May 2011, stated that over the last 40 years, as nearly all of Ramanujan's theorems have been proven right, there had been a greater appreciation of Ramanujan's work and brilliance. Further, he stated Ramanujan's work was now pervading many areas of modern mathematics and physics.<ref name="Fox" /><ref>{{cite web|work=youtube.com|title=Bruce Berndt on "Ramanujan's Lost Notebook", IIT Madras, 24th May 2011|url=http://www.youtube.com/watch?v=5uCAuwOQoxA&feature=related}}</ref>
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| In his book ''Scientific Edge'', noted physicist [[Jayant Narlikar]] spoke of "Srinivasa Ramanujan, discovered by the Cambridge mathematician Hardy, whose great mathematical findings were beginning to be appreciated from 1915 to 1919. 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 of investigations in the theory of such numbers."
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| During his lifelong mission in educating and propagating mathematics among the school children in India, Nigeria and elsewhere, [[P.K. Srinivasan]] has continually introduced Ramanujan's mathematical works.
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| ==Recognition==
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| {{further|List of things named after Srinivasa Ramanujan}}
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| Ramanujan'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 was released by the [[Government of India]] in 1962 – the 75th anniversary of Ramanujan's birth – commemorating his achievements in the field of number theory,<ref>{{cite web|title=Stamps released in 1962|publisher=Indian Postage Stamps|url=http://www.indianpostagestamps.com/gallery/1962.html|accessdate=22 May 2012}}</ref> and a new design was issued on December 26, 2011, by the [[India Post]].<ref>{{cite web|title=Stamps 2011|publisher=[[India Post]]|url=http://www.indiapost.gov.in/Stamps2011.aspx|accessdate=22 May 2012}}</ref><ref>{{cite web|title=India Post Issued a Commemorative Stamp on S Ramanujan|publisher=Phila Mirror|url=http://philamirror.info/2011/12/26/india-post-will-issue-a-commemorative-stamp-on-s-ramanujan-today/|date=26 December 2011|accessdate=22 May 2012}}</ref>
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| Since the Centennial year of Ramanujan, every year 22 Dec, is celebrated as Ramanujan Day by the [[Government Arts College, Kumbakonam]] where he had studied and later dropped out. It is celebrated by the Department of Mathematics by organising one-, two-, or three-day seminars by inviting eminent scholars from universities/colleges, and participants are mainly students of mathematics, research scholars, and professors from local colleges. It has been planned to celebrate the 125th birthday in a grand manner by inviting the foreign eminent mathematical scholars of this century viz., G E Andrews. and Bruce C Berndt, who are very familiar with the contributions and works of Ramanujan.
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| Ramanujan's work and life are celebrated on 22 December at the [[Indian Institute of Technology Madras|Indian Institute of Technology (IIT), Madras]] in [[Chennai]]. The Department of Mathematics celebrates this day by organising a National Symposium On Mathematical Methods and Applications (NSMMA) for one day by inviting eminent Indian and foreign scholars.
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| A prize for young mathematicians from developing countries has been created in the name of Ramanujan by the [[International Centre for Theoretical Physics]] (ICTP), in cooperation with the [[International Mathematical Union]], which nominate 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 $10,000 to be given annually to a mathematician not exceeding the age of 32 for outstanding contributions in an area of mathematics influenced by Ramanujan. The age limit refers to the years Ramanujan lived, having nevertheless still achieved many accomplishments. This prize has been awarded annually since 2005, at an international conference conducted by SASTRA in [[Kumbakonam]], Ramanujan's hometown, around Ramanujan's birthday, 22 December.
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| On the 125th anniversary of his birth, India declared the birthday of Ramanujan, December 22, as 'National Mathematics Day.' The declaration was made by [[Dr. Manmohan Singh]] in [[Chennai]] on December 26, 2011.<ref>{{cite web|url=http://ibnlive.in.com/generalnewsfeed/news/singhs-visit-first-to-the-state-after-the-congressdmk/942941.html |title=News / National : |work=CNN IBN |location=India |accessdate=26 December 2011}}</ref> Dr Manmohan Singh also declared that the year 2012 would be celebrated as the [[National Mathematics Year]].
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| ==In popular culture==
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| *''[[Ramanujan (film)|Ramanujan]]'', an Indo-British collaboration film, chronicling the life of the mathematical genius Srinivasa Ramanujan, is being made by the independent film company [[Ramanujan (film)#Camphor Cinema|Camphor Cinema]].<ref>{{cite web|title='Ramanujan' Makers Shoot in His House|url=http://www.indiatimes.com/entertainment/regional/ramanujan-makers-shoot-in-his-house-87409.html|work=Indiatimes|publisher=Times Internet Limited.|accessdate=12 July 2013}}</ref> It consists of a critically acclaimed team of cast and crew, including acclaimed director [[Gnana Rajasekaran]], celebrated cinematographer [[Sunny Joseph]] and renowned editor [[B. Lenin]].<ref>{{cite web|title=Camphor Cinema Presents Their First Film Ramanujan|url=http://www.boxofficeindia.co.in/camphor-cinema-presents-their-first-film-ramanujan/|work=Box Office India|publisher=Select Publishing Company|accessdate=12 July 2013}}</ref><ref>{{cite web|title=Makers of `Ramanujan` shoot in genius` house|url=http://zeenews.india.com/entertainment/regional/makers-of-ramanujan-shoot-in-genius-house_138461.htm|work=http://zeenews.india.com/|publisher=Zee Media Corporation Ltd|accessdate=12 July 2013}}</ref> Popular South Indian and English stars Abhinay Vaddi, [[Suhasini Maniratnam]], [[Bhama]], Kevin McGowan and Michael Lieber have been roped in to star in pivotal roles.<ref>{{cite web|title=Travails of a genius|url=http://www.thehindu.com/features/friday-review/travails-of-a-genius/article4857108.ece|work=The Hindu|publisher=The Hindu|accessdate=12 July 2013}}</ref>
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| * A film, based on the book ''[[The Man Who Knew Infinity]]: A Life of the Genius Ramanujan'' by Robert Kanigel, is being made by Edward Pressman and Matthew Brown with [[R. Madhavan]] playing Ramanujan.<ref>{{cite web|url=http://sify.com/news/othernews/fullstory.php?id=14173864 |title=Two Hollywood movies on Ramanujan |publisher=Sify.com |date=30 March 2006 |accessdate=18 October 2011}}</ref>
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| * A play, ''First Class Man'' by Alter Ego Productions,<ref>{{cite web|url=http://www.alteregoproductions.org/blog/2006/06/alteregos_new_theater_season_b.htm |title=First Class Man |publisher=Alteregoproductions.org |date= |accessdate=20 November 2012}}</ref> was based on David Freeman's ''First Class Man''. The play is centred around Ramanujan and his complex and dysfunctional relationship with Hardy. On 16 October 2011, it was announced that [[Roger Spottiswoode]], best known for his James Bond film ''[[Tomorrow Never Dies]]'', is working on the film version, starring actor [[Siddharth (actor)|Siddharth]]. Like the book and play it is also titled ''The First Class Man''; the film's scripting has been completed and shooting is being planned from 2012.<ref>{{cite news |url=http://www.thehindu.com/news/national/article2541084.ece |title=News / National : James Bond director to make film on Ramanujan |work=The Hindu |location=India |accessdate=18 October 2011 |date=16 October 2011}}</ref>
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| * ''[[A Disappearing Number]]'' is a recent British stage production by the company Complicite that explores the relationship between Hardy and Ramanujan.
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| * The novel ''[[The Indian Clerk]]'' by [[David Leavitt]] explores in fiction the events following Ramanujan's letter to Hardy.<ref>{{cite news|url=http://www.nytimes.com/2007/09/16/books/review/Freudenberger-t.html|title=Lust for Numbers|author=Nell Freudenberger|work=The New York Times|date=16 September 2007|accessdate=4 September 2011}}</ref><ref>{{cite news|title=Adding up to a life|url=http://www.guardian.co.uk/books/2008/jan/26/fiction1|author=DJ Taylor|work=The Guardian |location=UK|date=26 January 2008|accessdate=4 September 2011}}</ref>
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| * On 22 March 1988, the PBS Series ''Nova'' aired a documentary about Ramanujan, "The Man Who Loved Numbers" (Season 15, Episode 19).<ref>{{cite web|url=http://www.pbs.org/wgbh/nova/listseason/15.html |title=The Man Who Loved Numbers |publisher=Pbs.org |accessdate=18 October 2011}}</ref>
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| * Ramanujan is mentioned in the Hollywood Blockbuster [[Good Will Hunting]] starring [[Matt Damon]] a film based on an orphan genius living in the rough part of [[South Boston]].
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| * Popular internet search engine [[Google.com]] honored him on his 125<sup>th</sup> birthday by replacing its logo with a doodle on its home page.<ref>{{cite news|title=Google doodles for Ramanujan’s 125th birthday|url=http://timesofindia.indiatimes.com/tech/tech-news/internet/Google-doodles-for-Ramanujans-125th-birthday/articleshow/17716898.cms|accessdate=22 December 2012|newspaper=Times of India|date=22 December 2012|archiveurl=http://www.webcitation.org/6D6Qvt0Xe|archivedate=22 December 2012}}</ref>
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| ==See also==
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| {{Div col|colwidth=20em}}
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| * [[List of amateur mathematicians]]
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| * [[Ramanujan graph]]
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| * [[Ramanujan summation]]
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| * [[Ramanujan's constant]]
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| * [[Ramanujan's ternary quadratic form]]
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| * [[Rank of a partition]]
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| * [[2719|2719 (number)]]
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| {{Div col end}}
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| ==Notes==
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| {{Reflist|20em}}
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| ==Selected publications by Ramanujan==
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| * {{Cite book|title=Collected Papers of Srinivasa Ramanujan |author=Srinivasa Ramanujan, G. H. Hardy, P. V. Seshu Aiyar, B. M. Wilson, Bruce C. Berndt |publisher=AMS |year=2000 |isbn=0-8218-2076-1}}
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| :This book was originally published in 1927 after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime. The third reprint contains additional commentary by Bruce C. Berndt.
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| * {{Cite book|title=Notebooks (2 Volumes) |author=S. Ramanujan |publisher=Tata Institute of Fundamental Research |place=Bombay |year=1957}}
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| :These books contain photocopies of the original notebooks as written by Ramanujan.
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| * {{Cite book|title=The Lost Notebook and Other Unpublished Papers |author=S. Ramanujan |publisher=Narosa |place=New Delhi |year=1988|isbn=3-540-18726-X}}
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| :This book contains photo copies of the pages of the "Lost Notebook".
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| * [http://www.imsc.res.in/~rao/ramanujan/collectedpapers/question/qJIMS.htm Problems posed by Ramanujan], Journal of the Indian Mathematical Society.
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| * {{Cite book|title=Notebooks (2 Volumes) |author=S. Ramanujan |publisher=Tata Institute of Fundamental Research |place=Bombay |year=2012}}
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| :This was produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai.
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| ==Selected publications about Ramanujan and his work==
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| * {{Cite book |last=Berndt |first=Bruce C. |editor=Butzer, P. L.; Oberschelp, W.; Jongen, H. Th. |year=1998 |title=Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe |publisher=Brepols Verlag |location=Turnhout, Belgium |pages=119–146 |isbn=2-503-50673-9 |url=http://www.math.uiuc.edu/~berndt/articles/aachen.pdf |format=PDF |ref=harv }}
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| * {{Cite book |last1=Berndt |first1=Bruce C. |last2=Andrews |first2=George E. |year=2005 |title=Ramanujan's Lost Notebook |volume=Part I |publisher=Springer |location=New York |isbn=0-387-25529-X |ref=harv }}
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| * {{Cite book |last1=Berndt |first1=Bruce C. |last2=Andrews |first2=George E. |year=2008 |title=Ramanujan's Lost Notebook |volume=Part II |publisher=Springer |location=New York |isbn=978-0-387-77765-8 |ref=harv }}
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| * {{Cite book |last1=Berndt |first1=Bruce C. |last2=Andrews |first2=George E. |year=2012 |title=Ramanujan's Lost Notebook |volume=Part III |publisher=Springer |location=New York |isbn=978-1-4614-3809-0 |ref=harv }}
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| * {{Cite book |last1=Berndt |first1=Bruce C. |last2=Andrews |first2=George E. |year=2013 |title=Ramanujan's Lost Notebook |volume=Part IV |publisher=Springer |location=New York |isbn=978-1-4614-4080-2 |ref=harv }}
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| * {{Cite book |last1=Berndt |first1=Bruce C. |last2=Rankin |first2=Robert A. |year=1995 |title=Ramanujan: Letters and Commentary |volume=9 |publisher=[[American Mathematical Society]] |location=Providence, Rhode Island |isbn=0-8218-0287-9 |ref=harv }}
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| * {{Cite book |last=Berndt |first=Bruce C. |first2=Robert A. |last2=Rankin |author2-link=Robert Alexander Rankin |title=Ramanujan: Essays and Surveys |volume=22 |location=Providence, Rhode Island |publisher=[[American Mathematical Society]] |year=2001 |isbn=0-8218-2624-7 |ref=harv |author-link=Bruce C. Berndt }}
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| * {{Cite book |last=Berndt |first=Bruce C. |year=2006 |title=Number Theory in the Spirit of Ramanujan |volume=9 |publisher=[[American Mathematical Society]] |location=Providence, Rhode Island |isbn=0-8218-4178-5 |ref=harv }}
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| * {{Cite book |last=Berndt |first=Bruce C. |year=1985 |title=Ramanujan's Notebooks |volume=Part I |publisher=Springer |location=New York |isbn=0-387-96110-0 |ref=harv }}
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| * {{Cite book |last=Berndt |first=Bruce C. |year=1999 |title=Ramanujan's Notebooks |volume=Part II |publisher=Springer |location=New York |isbn=0-387-96794-X |ref=harv }}
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| * {{Cite book |last=Berndt |first=Bruce C. |year=2004 |title=Ramanujan's Notebooks |volume=Part III |publisher=Springer |location=New York |isbn=0-387-97503-9 |ref=harv }}
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| * {{Cite book |last=Berndt |first=Bruce C. |year=1993 |title=Ramanujan's Notebooks |volume=Part IV |publisher=Springer |location=New York |isbn=0-387-94109-6 |ref=harv }}
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| * {{Cite book |last=Berndt |first=Bruce C. |year=2005 |title=Ramanujan's Notebooks |volume=Part V |publisher=Springer |location=New York |isbn=0-387-94941-0 |ref=harv }}
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| * {{Cite book |last=Hardy |first=G. H. |year=1978 |title=Ramanujan |publisher=Chelsea Pub. Co. |location=New York |isbn=0-8284-0136-5 |ref=harv }}
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| * {{Cite book |last=Hardy |first=G. H. |year=1999 |title=Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work |publisher=American Mathematical Society |location=Providence, Rhode Island |isbn=0-8218-2023-0 |ref=harv }}
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| * {{Cite book |last=Henderson |first=Harry |year=1995 |title=Modern Mathematicians |publisher=Facts on File Inc. |location=New York |isbn=0-8160-3235-1 |ref=harv }}
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| * {{Cite book |last=Kanigel |first=Robert |title=The Man Who Knew Infinity: a Life of the Genius Ramanujan |location=New York |publisher=[[Charles Scribner's Sons]] |year=1991 |isbn=0-684-19259-4 |ref=harv }}
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| * {{Cite journal |last=Kolata |first=Gina |authorlink=Gina Kolata |date=19 Jun 1987 |title=Remembering a 'Magical Genius' |journal=Science, New Series |volume=236 |issue=4808 |publisher=American Association for the Advancement of Science |pages=1519–1521 |ref=harv }}
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| * {{Cite book |last=Leavitt |first=David |authorlink=David Leavitt |year=2007 |title=The Indian Clerk |edition=paperback |publisher=Bloomsbury |location=London |isbn=978-0-7475-9370-6 |ref=harv }}
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| * {{Cite book |last=Narlikar |first=Jayant V. |authorlink=Jayant V. Narlikar |year=2003 |title=Scientific Edge: the Indian Scientist From Vedic to Modern Times |publisher=[[Penguin Books]] |location=New Delhi, India |isbn=0-14-303028-0 |ref=harv }}
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| * {{Cite journal |last=Sankaran |first=T. M. |authorlink=T. M. Sankaran |year=2005 |title=Srinivasa Ramanujan- Ganitha lokathile Mahaprathibha |publisher=Kerala Sastra Sahithya Parishath |location=Kochi, India |language=Malayalam |ref=harv }}
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| ==External links==
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| {{Sister project links| wikt = no| b = no| q = Srinivasa Ramanujan | s = Author:Srinivasa Ramanujan | commons = Srinivasa Ramanujan | n = no| v = no| species = no| display = Srinivasa Ramanujan | author = no}}
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| ===Media links===
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| * {{Cite news| title=Film to celebrate mathematics genius |work=BBC |
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| url=http://news.bbc.co.uk/2/hi/south_asia/4811920.stm| accessdate=24 August 2006 | date=16 March 2006| first=Soutik| last=Biswas}}
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| * [http://devbenegal.com/2006/03/15/feature-film-on-math-genius-ramanujan/ Feature Film on Mathematics Genius Ramanujan by Dev Benegal and Stephen Fry]
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| * [http://www.bbc.co.uk/radio4/science/further5.shtml BBC radio programme about Ramanujan – episode 5]
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| * [http://www.archive.org/details/Ramanujan A biographical song about Ramanujan's life]
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| * P.B.S. Nova Series: "The Man Who Loved Numbers" (1988)
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| ===Biographical links===
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| * {{MathGenealogy|id=91561}}
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| * {{MacTutor | id=Ramanujan}}
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| * {{ScienceWorldBiography | urlname=Ramanujan | title=Ramanujan, Srinivasa (1887–1920)}}
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| * [http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/History/Rmnjn.htm Srinivasa Aiyangar Ramanujan]
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| * [http://www.usna.edu/Users/math/meh/ramanujan.html A short biography of Ramanujan]
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| * [http://www.srinivasaramanujan.in/ "Our Devoted Site for Great Mathematical Genius"]
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| ===Other links===
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| * A Study Group For Mathematics: [http://groups.yahoo.com/group/srinivasaramanujan/ Srinivasa Ramanujan Iyengar]
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| * [http://www.math.ufl.edu/~frank/ramanujan.html ''The Ramanujan Journal''] – An international journal devoted to Ramanujan
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| * [http://www.mathunion.org/General/Prizes/ International Math Union Prizes], including a Ramanujan Prize.
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| * Hindu.com: [http://www.hindu.com/mag/2004/12/26/stories/2004122600610400.htm Norwegian and Indian mathematical geniuses], [http://www.hindu.com/thehindu/br/2003/08/26/stories/2003082600120300.htm RAMANUJAN – Essays and Surveys], [http://www.hindu.com/2003/12/22/stories/2003122204061100.htm Ramanujan's growing influence], [http://www.hindu.com/thehindu/mag/2002/12/22/stories/2002122200040400.htm Ramanujan's mentor]
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| * Hindu.com: [http://www.thehindu.com/life-and-style/metroplus/article884010.ece The sponsor of Ramanujan]
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| * {{Cite journal
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| | author = Bruce C. Berndt; Robert A. Rankin
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| | title = The Books Studied by Ramanujan in India
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| | journal = [[American Mathematical Monthly]]
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| | volume = 107
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| | year = 2000
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| | issue = 7
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| | pages = 595–601
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| | doi = 10.2307/2589114
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| | mr = 1786233
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| | jstor = 2589114
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| | publisher = Mathematical Association of America
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| | ref = harv}}
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| * [http://www.maa.org/news/030807puzzlesolved.html "Ramanujan's mock theta function puzzle solved"]
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| * [http://www.imsc.res.in/~rao/ramanujan/contentindex.html Ramanujan's papers and notebooks]
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| * [http://www.cecm.sfu.ca/organics/papers/borwein/paper/html/local/ramnotebook.html Sample page from the second notebook]
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| * [http://www.friedeye.com/2011/01/01/ramanujan/ Ramanujan] on ''Fried Eye''
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| * {{cite web|last=Clark|first=Alex|title=163 and Ramanujan Constant|url=http://www.numberphile.com/videos/163.html|work=Numberphile|publisher=[[Brady Haran]]}}
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| {{Indian mathematics}}
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| {{Good article}}
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| {{Authority control|PND=118748955|LCCN=n/50/54441|VIAF=27132864|SELIBR=238676|TSURL=viaf/27132864}}
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| <!-- Metadata: see [[Wikipedia:Persondata]] -->
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| {{Persondata
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| |NAME= Ramanujan, Srinivasa
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| |ALTERNATIVE NAMES=
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| |SHORT DESCRIPTION= [[Mathematician]]
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| |DATE OF BIRTH= 22 December 1887
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| |PLACE OF BIRTH= [[Erode]], Tamil Nadu, India
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| |DATE OF DEATH= 26 April 1920
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| |PLACE OF DEATH= [[Chetput]], ([[Chennai|Madras]]), [[Tamil Nadu]], India
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| }}
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| {{DEFAULTSORT:Ramanujan, Srinivasa}}
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| [[Category:Srinivasa Ramanujan| ]]
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| [[Category:1887 births]]
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| [[Category:1920 deaths]]
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| [[Category:Tamil Nadu scientists]]
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| [[Category:20th-century mathematicians]]
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| [[Category:Indian Hindus]]
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| [[Category:Indian mathematicians]]
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| [[Category:Mental calculators]]
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| [[Category:Combinatorialists]]
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| [[Category:Number theorists]]
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| [[Category:Fellows of Trinity College, Cambridge]]
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| [[Category:Fellows of the Royal Society]]
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| [[Category:Pi]]
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| [[Category:People from Erode district]]
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| {{Link FA|scn}}
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