Chidambaram Padmanabhan Ramanujam

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Srinivasa Ramanujan

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

In thisIndian name,the name Srinivasais apatronymic,not afamily name,and the person should be

referred to by thegiven name,Ramanujan.

Srinivasa Ramanujan

Born

22 December 1887

Erode,Madras Presidency(nowTamil

Nadu)

Died

26 April 1920 (aged 32)

Chetput,Madras,Madras Presidency(nowTamil Nadu)

Residence Kumbakonam,Tamil Nadu

Nationality Indian

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Fields Mathematics

Alma materGovernment Arts College

Pachaiyappa's College

Academic

advisorsG. H. HardyJ. E. Littlewood

Known for

LandauRamanujan constant

Mock theta functions

Ramanujan conjecture

Ramanujan prime

RamanujanSoldner constant

Ramanujan theta function

Ramanujan's sum

RogersRamanujan identitiesRamanujan's master theorem

Influences G. H. Hardy

Signature

Srinivasa RamanujanFRS( pronunciation(helpinfo)) (Tamil:; 22 December 188726 April 1920) was an Indianmathematicianandautodidactwho, with almost no formal training inpure mathematics,made extraordinarycontributions tomathematical analysis,number theory,infinite series,andcontinued fractions.Living in India with no access to the larger mathematical community, which was centred inEurope at the time, Ramanujan developed his own mathematical research in isolation. As aresult, he rediscovered known theorems in addition to producing new work. Ramanujan was saidto be a natural genius by the English mathematicianG. H. Hardy,in the same league asmathematicians such asEulerandGauss.[1]He died at the age of 32.

Ramanujan was born atErode,Madras Presidency(nowTamil Nadu)in aTamilBrahminfamily

ofThenkalaiIyengar sect.[2][3][4]His introduction to formalmathematicsbegan at age 10. Hedemonstrated a natural ability, and was given books on advancedtrigonometrywritten byS. L.Loneythat he mastered by the age of 12; he even discoveredtheoremsof his own, and re-discoveredEuler's identityindependently.[5]He demonstrated unusual mathematical skills atschool, winning accolades and awards. By 17, Ramanujan had conducted his own mathematicalresearch onBernoulli numbersand theEulerMascheroni constant.

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part ofBrahminculture.[20]At the Kangayan Primary School, Ramanujan performed well. Justbefore the age of 10, in November 1897, he passed his primary examinations in English,Tamil,geography and arithmetic. With his scores, he stood first in the district.[21]That year, Ramanujanentered Town Higher Secondary School where he encountered formal mathematics for the firsttime.[21]

By age 11, he had exhausted the mathematical knowledge of two college students who werelodgers at his home. He was later lent a book on advanced trigonometry written byS. L.Loney.[5][22]He completely mastered this book by the age of 13 and discovered sophisticatedtheorems on his own. By 14, he was receiving merit certificates and academic awards whichcontinued throughout his school career and also assisted the school in thelogisticsof assigningits 1200 students (each with their own needs) to its 35-odd teachers.[23]He completedmathematical exams in half the allotted time, and showed a familiarity withgeometryandinfinite series.Ramanujan was shown how to solve cubic equations in 1902 and he went on tofind his own method to solve the quartic. The following year, not knowing that the quintic couldnot be solved by radicals, he tried (and of course failed) to solve the quintic.

In 1903 when he was 16, Ramanujan obtained from a friend a library-loaned copy of a book byG. S. Carr.[24][25]The book was titledA Synopsis of Elementary Results in Pure and AppliedMathematicsand was a collection of 5000 theorems. Ramanujan reportedly studied the contentsof the book in detail.[26]The book is generally acknowledged as a key element in awakening thegenius of Ramanujan.[26]The next year, he had independently developed and investigated theBernoulli numbersand had calculated theEulerMascheroni constantup to 15 decimal places.[27]His peers at the time commented that they "rarely understood him" and "stood in respectful awe"of him.[23]

When he graduated fromTown Higher Secondary Schoolin 1904, Ramanujan was awarded the

K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyerintroduced Ramanujan as an outstanding student who deserved scores higher than the maximumpossible marks.[23]He received a scholarship to study atGovernment Arts College,Kumbakonam,[28][29]However, Ramanujan was so intent on studying mathematics that he couldnot focus on any other subjects and failed most of them, losing his scholarship in the process.[30]In August 1905, he ran away from home, heading towardsVisakhapatnamand stayed inRajahmundry[31]for about a month.[32]He later enrolled atPachaiyappa's Collegein Madras. Heagain excelled in mathematics but performed poorly in other subjects such as physiology.Ramanujan failed hisFellow of Artsexam in December 1906 and again a year later. Without adegree, he left college and continued to pursue independent research in mathematics. At thispoint in his life, he lived in extreme poverty and was often on the brink of starvation.[33]

Adulthood in India

On 14 July 1909, Ramanujan was married to a ten-year old bride, Janakiammal (21 March 189913 April 1994).[34]She came from Rajendram, a village close to Marudur (Karur district)Railway Station. Ramanujan's father did not participate in the marriage ceremony.[35]

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After the marriage, Ramanujan developed ahydrocele testis,an abnormal swelling of thetunicavaginalis,an internal membrane in the testicle.[36]The condition could be treated with a routinesurgical operation that would release the blocked fluid in the scrotal sac. His family did not havethe money for the operation, but in January 1910, a doctor volunteered to do the surgery forfree.[37]

After his successful surgery, Ramanujan searched for a job. He stayed at friends' houses while hewent door to door around the city ofMadras(now Chennai) looking for a clerical position. Tomake some money, he tutored some students at Presidency College who were preparing for theirF.A. exam.[38]

In late 1910, Ramanujan was sick again, possibly as a result of the surgery earlier in the year. Hefeared for his health, and even told his friend, R. Radakrishna Iyer, to "hand these [Ramanujan'smathematical notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor atPachaiyappa's College] or to the British professor Edward B. Ross, of theMadras ChristianCollege."[39]After Ramanujan recovered and got back his notebooks from Iyer, he took a

northbound train from Kumbakonam toVillupuram,a coastal city under French control.

[40][41]

Attention towards mathematics

Ramanujan met deputy collectorV. Ramaswamy Aiyer,who had recently founded the IndianMathematical Society.[42]Ramanujan, wishing for a job at the revenue department whereRamaswamy Aiyer worked, showed him his mathematics notebooks. As Ramaswamy Aiyer laterrecalled:

I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had nomind to smother his genius by an appointment in the lowest rungs of the revenue department.[43]

Ramaswamy Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends inMadras.[42]Some of these friends looked at his work and gave him letters of introduction toR.Ramachandra Rao,the district collector forNelloreand the secretary of the Indian MathematicalSociety.[44][45][46]Ramachandra Rao was impressed by Ramanujan's research but doubted that itwas actually his own work. Ramanujan mentioned a correspondence he had with ProfessorSaldhana, a notableBombaymathematician, in which Saldhana expressed a lack ofunderstanding for his work but concluded that he was not a phony.[47]Ramanujan's friend, C. V.Rajagopalachari, persisted with Ramachandra Rao and tried to quell any doubts overRamanujan's academic integrity. Rao agreed to give him another chance, and he listened asRamanujan discussedelliptic integrals,hypergeometric series,and his theory ofdivergent series,

which Rao said ultimately "converted" him to a belief in Ramanujan's mathematical brilliance.

[47]

When Rao asked him what he wanted, Ramanujan replied that he needed some work andfinancial support. Rao consented and sent him to Madras. He continued his mathematicalresearch with Rao's financial aid taking care of his daily needs. Ramanujan, with the help ofRamaswamy Aiyer, had his work published in theJournal of the Indian MathematicalSociety.[48]

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

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Contents

1 Early life 2 Adulthood in India

o 2.1 Attention towards mathematicso 2.2 Contacting English mathematicians

3 Life in Englando 3.1 Illness and return to Indiao 3.2 Personality and spiritual life

4 Mathematical achievementso 4.1 The Ramanujan conjectureo 4.2 Ramanujan's notebooks

5 RamanujanHardy number 1729 6 Other mathematicians' views of Ramanujan 7 Recognition 8 In popular culture 9 See also 10 Notes 11 Selected publications by Ramanujan 12 Selected publications about Ramanujan and his work 13 External links

o 13.1 Media linkso 13.2 Biographical linkso 13.3 Other links

Early life

Ramanujan's home on Sarangapani Street, Kumbakonam.

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Ramanujan was born on 22 December 1887 inErode,Madras Presidency(nowTamil Nadu), atthe residence of his maternal grandparents.[13]His father, K. Srinivasa Iyengar, worked as a clerkin a sari shop and hailed from the district ofThanjavur.[14]His mother, Komalatammal, was ahousewifeand also sang at a local temple.[15]They lived in Sarangapani Street in a traditionalhome 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 threemonths later. In December 1889, Ramanujan hadsmallpoxand recovered, unlike thousands intheThanjavur Districtwho died from the disease that year.[16]He moved with his mother to herparents' house inKanchipuram,near Madras (nowChennai). In November 1891, and again in1894, his mother gave birth to two children, but both children died in infancy.

On 1 October 1892, Ramanujan was enrolled at the local school.[17]In March 1894, he wasmoved to aTelugu mediumschool. After his maternal grandfather lost his job as a court officialin Kanchipuram,[18]Ramanujan and his mother moved back toKumbakonamand he wasenrolled in the Kangayan Primary School.[19]When his paternal grandfather died, he was sentback 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 heattended school. Within six months, Ramanujan was back in Kumbakonam.[19]

Since Ramanujan's father was at work most of the day, his mother took care of him as a child. Hehad a close relationship with her. From her, he learned about tradition andpuranas.He learned tosing religious songs, to attend pujas at the temple and particular eating habitsall of which arepart ofBrahminculture.[20]At the Kangayan Primary School, Ramanujan performed well. Justbefore the age of 10, in November 1897, he passed his primary examinations in English,Tamil,geography and arithmetic. With his scores, he stood first in the district.[21]That year, Ramanujanentered Town Higher Secondary School where he encountered formal mathematics for the firsttime.[21]

By age 11, he had exhausted the mathematical knowledge of two college students who werelodgers at his home. He was later lent a book on advanced trigonometry written byS. L.Loney.[5][22]He completely mastered this book by the age of 13 and discovered sophisticatedtheorems on his own. By 14, he was receiving merit certificates and academic awards whichcontinued throughout his school career and also assisted the school in thelogisticsof assigningits 1200 students (each with their own needs) to its 35-odd teachers.[23]He completedmathematical exams in half the allotted time, and showed a familiarity withgeometryandinfinite series.Ramanujan was shown how to solve cubic equations in 1902 and he went on tofind his own method to solve the quartic. The following year, not knowing that the quintic couldnot be solved by radicals, he tried (and of course failed) to solve the quintic.

In 1903 when he was 16, Ramanujan obtained from a friend a library-loaned copy of a book byG. S. Carr.[24][25]The book was titledA Synopsis of Elementary Results in Pure and AppliedMathematicsand was a collection of 5000 theorems. Ramanujan reportedly studied the contentsof the book in detail.[26]The book is generally acknowledged as a key element in awakening thegenius of Ramanujan.[26]The next year, he had independently developed and investigated theBernoulli numbersand had calculated theEulerMascheroni constantup to 15 decimal places.[27]

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His peers at the time commented that they "rarely understood him" and "stood in respectful awe"of him.[23]

When he graduated fromTown Higher Secondary Schoolin 1904, Ramanujan was awarded theK. 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 maximumpossible marks.[23]He received a scholarship to study atGovernment Arts College,Kumbakonam,[28][29]However, Ramanujan was so intent on studying mathematics that he couldnot focus on any other subjects and failed most of them, losing his scholarship in the process.[30]In August 1905, he ran away from home, heading towardsVisakhapatnamand stayed inRajahmundry[31]for about a month.[32]He later enrolled atPachaiyappa's Collegein Madras. Heagain excelled in mathematics but performed poorly in other subjects such as physiology.Ramanujan failed hisFellow of Artsexam in December 1906 and again a year later. Without adegree, he left college and continued to pursue independent research in mathematics. At thispoint in his life, he lived in extreme poverty and was often on the brink of starvation.[33]

Adulthood in India

On 14 July 1909, Ramanujan was married to a ten-year old bride, Janakiammal (21 March 189913 April 1994).[34]She came from Rajendram, a village close to Marudur (Karur district)Railway Station. Ramanujan's father did not participate in the marriage ceremony.[35]

After the marriage, Ramanujan developed ahydrocele testis,an abnormal swelling of thetunicavaginalis,an internal membrane in the testicle.[36]The condition could be treated with a routinesurgical operation that would release the blocked fluid in the scrotal sac. His family did not havethe money for the operation, but in January 1910, a doctor volunteered to do the surgery forfree.[37]

After his successful surgery, Ramanujan searched for a job. He stayed at friends' houses while hewent door to door around the city ofMadras(now Chennai) looking for a clerical position. Tomake some money, he tutored some students at Presidency College who were preparing for theirF.A. exam.[38]

In late 1910, Ramanujan was sick again, possibly as a result of the surgery earlier in the year. Hefeared for his health, and even told his friend, R. Radakrishna Iyer, to "hand these [Ramanujan'smathematical notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor atPachaiyappa's College] or to the British professor Edward B. Ross, of theMadras ChristianCollege."[39]After Ramanujan recovered and got back his notebooks from Iyer, he took a

northbound train from Kumbakonam toVillupuram,a coastal city under French control.[40][41]

Attention towards mathematics

Ramanujan met deputy collectorV. Ramaswamy Aiyer,who had recently founded the IndianMathematical Society.[42]Ramanujan, wishing for a job at the revenue department whereRamaswamy Aiyer worked, showed him his mathematics notebooks. As Ramaswamy Aiyer laterrecalled:

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I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had nomind to smother his genius by an appointment in the lowest rungs of the revenue department.[43]

Ramaswamy Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends inMadras.[42]Some of these friends looked at his work and gave him letters of introduction toR.

Ramachandra Rao,the district collector forNelloreand the secretary of the Indian MathematicalSociety.[44][45][46]Ramachandra Rao was impressed by Ramanujan's research but doubted that itwas actually his own work. Ramanujan mentioned a correspondence he had with ProfessorSaldhana, a notableBombaymathematician, in which Saldhana expressed a lack ofunderstanding for his work but concluded that he was not a phony.[47]Ramanujan's friend, C. V.Rajagopalachari, persisted with Ramachandra Rao and tried to quell any doubts overRamanujan's academic integrity. Rao agreed to give him another chance, and he listened asRamanujan discussedelliptic integrals,hypergeometric series,and his theory ofdivergent series,which Rao said ultimately "converted" him to a belief in Ramanujan's mathematical brilliance.[47]When Rao asked him what he wanted, Ramanujan replied that he needed some work andfinancial 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 ofRamaswamy Aiyer, had his work published in theJournal of the Indian MathematicalSociety.[48]

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 an equation that could be used to solve the infinitelynested radicalsproblem.

Using this equation, the answer to the question posed in theJournalwas simply 3.[49]Ramanujanwrote his first formal paper for theJournalon the properties ofBernoulli numbers.One propertyhe discovered was that the denominators (sequenceA027642inOEIS)of the fractions ofBernoulli numbers were always divisible by six. He also devised a method of calculatingBnbased on previous Bernoulli numbers. One of these methods went as follows:

It will be observed that if nis even but not equal to zero,

(i)Bnis a fraction and the numerator of in its lowest terms is a prime number,(ii) the denominator ofBncontains each of the factors 2 and 3 once and only once,

(iii) is an integer and consequently is an oddinteger.

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In his 17-page paper, "Some Properties of Bernoulli's Numbers", Ramanujan gave three proofs,two corollaries and three conjectures.[50]Ramanujan's writing initially had many flaws. AsJournaleditor M. T. Narayana Iyengar noted:

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 intellectualgymnastics, could hardly follow him.[51]

Ramanujan later wrote another paper and also continued to provide problems in theJournal.[52]In early 1912, he got a temporary job in the MadrasAccountant General's office, with a salary of20 rupees per month. He lasted for only a few weeks.[53]Toward the end of that assignment heapplied for a position under the Chief Accountant of the Madras Port Trust. In a letter dated 9February 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 the Matriculation Examination and studied up to the F.A. but was prevented frompursuing my studies further owing to several untoward circumstances. I have, however, beendevoting all my time to Mathematics and developing the subject. I can say I am quite confident Ican do justice to my work if I am appointed to the post. I therefore beg to request that you will begood enough to confer the appointment on me.[54]

Attached to his application was a recommendation fromE. W. Middlemast,a mathematicsprofessor at thePresidency College,who wrote that Ramanujan was "a young man of quiteexceptional capacity in Mathematics".[55]Three weeks after he had applied, on 1 March,Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making30 rupees per month.[56]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,Sir FrancisSpring,and S. Narayana Iyer, a colleague who was also treasurer of the Indian MathematicalSociety, encouraged Ramanujan in his mathematical pursuits.

Contacting English mathematicians

In the spring of 1913, Narayana Iyer, Ramachandra Rao andE. W. Middlemasttried to presentRamanujan's work to British mathematicians. One mathematician,M. J. M. HillofUniversityCollege London,commented that Ramanujan's papers were riddled with holes.[57]He said thatalthough Ramanujan had "a taste for mathematics, and some ability", he lacked the educationalbackground and foundation needed to be accepted by mathematicians.[58]Although Hill did not

offer to take Ramanujan on as a student, he did give thorough and serious professional advice onhis work. With the help of friends, Ramanujan drafted letters to leading mathematicians atCambridge University.[59]

The first two professors,H. F. BakerandE. W. Hobson,returned Ramanujan's papers withoutcomment.[60]On 16 January 1913, Ramanujan wrote toG. H. Hardy.Coming from an unknownmathematician, the nine pages of mathematics made Hardy initially view Ramanujan'smanuscripts as a possible "fraud".[61]Hardy recognised some of Ramanujan's formulae but others

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Whewell's Court,Trinity College,Cambridge

Ramanujan boarded the S.S.Nevasaon 17 March 1914, and at 10 o'clock in the morning, theship departed from Madras.[77]He arrived in London on 14 April, with E. H. Neville waiting forhim 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.[78]Hardy and Ramanujan began to take a look at Ramanujan'snotebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, butthere were many more results and theorems to be found in the notebooks. Hardy saw that somewere wrong, others had already been discovered, while the rest were new breakthroughs.[79]Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can

believe that he's at least aJacobi",

[80]

while Hardy said he "can compare him only with[Leonhard] Euler or Jacobi."[81]

Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood andpublished 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 anatheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeplyreligious man and relied very strongly on his intuition. While in England, Hardy tried his best tofill 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 work onhighly composite numbers,the first part of which was published as apaper in theProceedings of theLondon Mathematical Society.The paper was over 50 pages withdifferent properties of such numbers proven. Hardy remarked that this was one of the mostunusual papers seen in mathematical research at that time and that Ramanujan showedextraordinary ingenuity in handling it.[citation needed]On 6 December 1917, he was elected to theLondon Mathematical Society. He became aFellow of the Royal Societyin 1918, becoming thesecond Indian to do so, followingArdaseer Cursetjeein 1841, and he was one of the youngestFellows in the history of the Royal Society. He was elected "for his investigation inElliptic

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functionsand the Theory of Numbers." On 13 October 1918, he became the first Indian to beelected aFellow of Trinity College, Cambridge.[82]

Illness and return to India

Plagued by health problems throughout his life, living in a country far away from home, andobsessively involved with his mathematics, Ramanujan's health worsened in England, perhapsexacerbated bystressand by the scarcity ofvegetarian foodduring the First World War. He wasdiagnosed withtuberculosisand a severevitamindeficiency and was confined to a sanatorium.

Ramanujan returned to Kumbakonam,Madras Presidencyin 1919 and died soon thereafter at theage of 32. His widow, S. Janaki Ammal, moved to Mumbai, but returned to Chennai (formerlyMadras) in 1950, where she lived until her death in 1994.[35]

A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D.A.B. Young concludedthat it was much more likely he had hepaticamoebiasis,a parasitic infection of the liver

widespread in Madras, where Ramanujan had spent time. He had two episodes ofdysenterybefore he left India. When not properly treated, dysentery can lie dormant for years and lead tohepatic amoebiasis,[6]a difficult disease to diagnose, but once diagnosed readily cured.[6]

Personality and spiritual life

Ramanujan has been described as a person with a somewhat shy and quiet disposition, adignified man with pleasant manners.[83]He lived a rather Spartan life while at Cambridge.Ramanujan's first Indian biographers describe him as rigorously orthodox. Ramanujan creditedhis acumen to hisfamily Goddess,NamagiriofNamakkal.He looked to her for inspiration in hiswork,[84]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 unfoldingbefore his eyes.[85]He often said, "An equation for me has no meaning, unless it represents athought of God."[86][87]

Hardy cites Ramanujan as remarking that all religions seemed equally true to him.[88]Hardyfurther argued that Ramanujan's religiousness had been romanticised by Westerners andoverstatedin reference to his belief, not practiceby Indian biographers. At the same time, heremarked on Ramanujan's strict observance of vegetarianism.

Mathematical achievements

In mathematics, there is a distinction between having an insight and having a proof. Ramanujan'stalent suggested a plethora of formulae that could then be investigated in depth later. It is saidthat Ramanujan's discoveries are unusually rich and that there is often more to them than initiallymeets the eye. As a by-product, new directions of research were opened up. Examples of themost interesting of these formulae include the intriguing infiniteseriesfor,one of which isgiven below