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Srinivasa Ramanujan

From Wikipedia, the free encyclopedia"Ramanujan" redirects here. For other uses, see Ramanujan (disambiguation).In this Indian name, the name Srinivasa is a patronymic, not a family name, and the person should

 be referred to by the gien name, Ramanujan.

Srinivasa Ramanujan

Native name ஸனவச ரமஜ

Born !! ecember #$$%&rode, 'adras residency (no *amil +adu)

Died! -pril #!/ (aged 0!)1hetput, 'adras, 'adras residency (no *amil +adu)

Residence 2umbakonam, *amil +adu

Nationality Indian

Fields 'athematics

Alma mater3oernment -rts 1ollegeachaiyappa4s 1ollege

Academic advisors 3. 5. 5ardy6. &. 7ittleood

Known for

7andau8Ramanujan constant'ock theta functionsRamanujan conjectureRamanujan primeRamanujan89oldner constantRamanujan theta functionRamanujan4s sumRogers8Ramanujan identitiesRamanujan4s master theorem

Influences 3. 5. 5ardy

Signature 

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Srinivasa Ramanujan FR9 ( pronunciation:  i ;s r i ː.n i. ɑ ː.s uickly recogni?ed by Indian mathematicians. When his skills became apparent to the ider mathematical community, centered in &urope at the time, he began afamous partnership ith the &nglish mathematician 3. 5. 5ardy. 5e rediscoered preiouslyknon theorems in addition to producing ne ork. Ramanujan as said to be a natural genius, inthe same league as mathematicians such as &uler  and 3auss.@#A

uring his short life, Ramanujan independently compiled nearly 0// results (mostly identities ande>uations).@!A +early all his claims hae no been proen correct, although a small number ofthese results ere actually false and some ere already knon.@0A 5e stated results that ere bothoriginal and highly unconentional, such as the Ramanujan prime and the Ramanujan thetafunction, and these hae inspired a ast amount of further research.@BA *he Ramanujan Journal , an

international publication, as launched to publish ork in all areas of mathematics influenced byhis ork.@CA

Contents

• # &arly life 

• ! -dulthood in India 

• !.# -ttention toards mathematics 

• !.! 1ontacting &nglish mathematicians 

• 0 7ife in &ngland • 0.# Illness and return to India 

• 0.! ersonality and spiritual life 

• B 'athematical achieements 

• B.# *he Ramanujan conjecture 

• B.! Ramanujan4s notebooks 

• C 5ardyDRamanujan number #%! 

• Ether mathematicians4 ies of Ramanujan 

• % Recognition 

• $ In popular culture 

• 9ee also • #/ +otes 

• ## 9elected publications by Ramanujan 

• #! 9elected publications about Ramanujan and his ork  

• #0 &=ternal links 

• #0.# 'edia links 

• #0.! iographical links 

• #0.0 Ether links 

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Early life

Ramanujan4s home on 9arangapani 9treet, 2umbakonam

Ramanujan as born on !! ecember #$$% in &rode, 'adras residency (no allipalayam, &rode, *amil +adu), at the residence of his maternal grandparents.@A 5is father, 2. 9riniasaIyengar, orked as a clerk in a sari shop and hailed from the district of *hanjaur .@%A 5is mother,2omalatammal, as a houseife and also sang at a local temple.@$A *hey lied in 9arangapani9treet in a traditional home in the ton of 2umbakonam. *he family home is no a museum. WhenRamanujan as a year and a half old, his mother gae birth to a son named 9adagopan, ho diedless than three months later. In ecember #$$, Ramanujan had smallpo= and recoered, unlikethousands in the *hanjaur istrict ho died from the disease that year.@A 5e moed ith his

mother to her parents4 house in 2anchipuram, near 'adras (no 1hennai). In +oember #$#, andagain in #$B, his mother gae birth to to children, but both children died in infancy.

En # Ectober #$!, Ramanujan as enrolled at the local school.@#/A In 'arch #$B, he as moedto a *elugu medium school. -fter his maternal grandfather lost his job as a court official in2anchipuram,@##A Ramanujan and his mother moed back to 2umbakonam and he as enrolled inthe 2angayan rimary 9chool.@#!A When his paternal grandfather died, he as sent back to hismaternal grandparents, ho ere no liing in 'adras. 5e did not like school in 'adras, and hetried to aoid attending. 5is family enlisted a local constable to make sure he attended school.Within si= months, Ramanujan as back in 2umbakonam.@#!A

9ince Ramanujan4s father as at ork most of the day, his mother took care of him as a child. 5e

had a close relationship ith her. From her, he learned about tradition and puranas. 5e learned tosing religious songs, to attend pujas at the temple and particular eating habits 8 all of hich are partof rahmin culture.@#0A -t the 2angayan rimary 9chool, Ramanujan performed ell. 6ust beforethe age of #/, in +oember #$%, he passed his primary e=aminations in &nglish, *amil, geographyand arithmetic. With his scores, he stood first in the district.@#BA *hat year, Ramanujan entered*on 5igher 9econdary 9chool here he encountered formal mathematics for the first time. @#BA

y age ##, he had e=hausted the mathematical knoledge of to college students ho ere lodgersat his home. 5e as later lent a book on adanced trigonometry ritten by 9. 7. 7oney.@#CA@#A 5ecompletely mastered this book by the age of #0 and discoered sophisticated theorems on his on.y #B, he as receiing merit certificates and academic aards hich continued throughout his

school career and also assisted the school in the logistics of assigning its #!// students (each iththeir on needs) to its 0CDodd teachers.@#%A 5e completed mathematical e=ams in half the allottedtime, and shoed a familiarity ith geometry and infinite series. Ramanujan as shon ho tosole cubic e>uations in #/! and he ent on to find his on method to sole the >uartic. *he

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folloing year, not knoing that the >uintic could not be soled by radicals, he tried (and of coursefailed) to sole the >uintic.

In #/0 hen he as #, Ramanujan obtained from a friend a libraryDloaned copy of a book by 3.9. 1arr .@#$A@#A *he book as titled  A Synopsis of Elementary Results in Pure and Applied

 Mathematics and as a collection of C/// theorems. Ramanujan reportedly studied the contents ofthe book in detail.@!/A *he book is generally acknoledged as a key element in aakening the

genius of Ramanujan.@!/A *he ne=t year, he had independently deeloped and inestigated theernoulli numbers and had calculated the &uler8'ascheroni constant up to #C decimal places.@!#A 5is peers at the time commented that they "rarely understood him" and "stood in respectful ae" ofhim.@#%A

When he graduated from *on 5igher 9econdary 9chool in #/B, Ramanujan as aarded the 2.Ranganatha Rao pri?e for mathematics by the school4s headmaster, 2rishnasami Iyer. Iyerintroduced Ramanujan as an outstanding student ho desered scores higher than the ma=imum

 possible marks.@#%A 5e receied a scholarship to study at 3oernment -rts 1ollege, 2umbakonam,@!!A@!0A 5oeer, Ramanujan as so intent on studying mathematics that he could not focus onany other subjects and failed most of them, losing his scholarship in the process.@!BA In -ugust

#/C, he ran aay from home, heading toards Gisakhapatnam and stayed in Rajahmundry@!CA forabout a month.@!A 5e later enrolled at achaiyappa4s 1ollege in 'adras. 5e again e=celled inmathematics but performed poorly in other subjects such as physiology. Ramanujan failed hisFello of -rts e=am in ecember #/ and again a year later. Without a degree, he left college andcontinued to pursue independent research in mathematics. -t this point in his life, he lied ine=treme poerty and as often on the brink of staration.@!%A

Adultood in India

En #B 6uly #/, Ramanujan as married to a tenDyear old bride, 6anakiammal (!# 'arch #$ 8#0 -pril #B).@!$A 9he came from Rajendram, a illage close to 'arudur (2arur district) Railay9tation. Ramanujan4s father did not participate in the marriage ceremony.@!A

-fter the marriage, Ramanujan deeloped a hydrocele testis, an abnormal selling of the tunicaaginalis, an internal membrane in the testicle.@0/A *he condition could be treated ith a routinesurgical operation that ould release the blocked fluid in the scrotal sac. 5is family did not hae themoney for the operation, but in 6anuary ##/, a doctor olunteered to do the surgery for free.@0#A

-fter his successful surgery, Ramanujan searched for a job. 5e stayed at friends4 houses hile heent door to door around the city of 'adras (no 1hennai) looking for a clerical position. *o makesome money, he tutored some students at residency 1ollege ho ere preparing for their F.-.e=am.@0!A

In late ##/, Ramanujan as sick again, possibly as a result of the surgery earlier in the year. 5efeared for his health, and een told his friend, R. Radakrishna Iyer, to "hand these @Ramanujan4smathematical notebooksA oer to rofessor 9ingaraelu 'udaliar @the mathematics professor atachaiyappa4s 1ollegeA or to the ritish professor &dard . Ross, of the 'adras 1hristian1ollege."@00A -fter Ramanujan recoered and got back his notebooks from Iyer, he took anorthbound train from 2umbakonam to Gillupuram, a coastal city under French control.@0BA@0CA

Attention towards matematics

Ramanujan met deputy collector G. Ramasamy -iyer , ho had recently founded the Indian'athematical 9ociety.@0A Ramanujan, ishing for a job at the reenue department here

Ramasamy -iyer orked, shoed him his mathematics notebooks. -s Ramasamy -iyer laterrecalled:

I as struck by the e=traordinary mathematical results contained in it @the notebooksA. I

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had no mind to smother his genius by an appointment in the loest rungs of the reenuedepartment.@0%A

Ramasamy -iyer sent Ramanujan, ith letters of introduction, to his mathematician friends in'adras.@0A 9ome of these friends looked at his ork and gae him letters of introduction to R.Ramachandra Rao, the district collector for +ellore and the secretary of the Indian 'athematical

9ociety.@0$A@0A@B/A Ramachandra Rao as impressed by Ramanujan4s research but doubted that itas actually his on ork. Ramanujan mentioned a correspondence he had ith rofessor9aldhana, a notable ombay mathematician, in hich 9aldhana e=pressed a lack of understandingof his ork but concluded that he as not a phoney.@B#A Ramanujan4s friend, 1. G. Rajagopalachari,

 persisted ith Ramachandra Rao and tried to >uell any doubts oer Ramanujan4s academic integrity.Rao agreed to gie him another chance, and he listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of diergent series, hich Rao said ultimately "conerted"him to a belief in Ramanujan4s mathematical brilliance.@B#A When Rao asked him hat he anted,Ramanujan replied that he needed some ork and financial support. Rao consented and sent him to'adras. 5e continued his mathematical research ith Rao4s financial aid taking care of his dailyneeds. Ramanujan, ith the help of Ramasamy -iyer, had his ork published in the Journal of

the Indian Mathematical Society.@B!A

Ene of the first problems he posed in the journal as:

 

5e aited for a solution to be offered in three issues, oer si= months, but failed to receie any. -tthe end, Ramanujan supplied the solution to the problem himself. En page #/C of his first notebook,he formulated an e>uation that could be used to sole the infinitely nested radicals problem.

 

Hsing this e>uation, the anser to the >uestion posed in the Journal  as simply 0.@B0A Ramanujanrote his first formal paper for the Journal  on the properties of ernoulli numbers. Ene property hediscoered as that the denominators (se>uence -/!%B! in E&I9) of the fractions of ernoullinumbers ere alays diisible by si=. 5e also deised a method of calculating B

n based on

 preious ernoulli numbers. Ene of these methods ent as follos:

It ill be obsered that if n is een but not e>ual to ?ero,

(i) Bn

 is a fraction and the numerator of in its loest terms is a prime number,

(ii) the denominator of Bn contains each of the factors ! and 0 once and only once,

(iii) is an integer and conse>uently is an odd  integer.

In his #%Dpage paper, "9ome roperties of ernoulli4s +umbers", Ramanujan gae three proofs, tocorollaries and three conjectures.@BBA Ramanujan4s riting initially had many flas. -s Journal  editor '. *. +arayana Iyengar noted:

'r. Ramanujan4s methods ere so terse and noel and his presentation so lacking inclearness and precision, that the ordinary @mathematical readerA, unaccustomed to suchintellectual gymnastics, could hardly follo him.@BCA

Ramanujan later rote another paper and also continued to proide problems in the  Journal .@BA Inearly ##!, he got a temporary job in the 'adras -ccountant 3eneral4s office, ith a salary of !/

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rupees per month. 5e lasted for only a fe eeks.@B%A *oard the end of that assignment he appliedfor a position under the 1hief -ccountant of the 'adras ort *rust. In a letter dated February##!, Ramanujan rote:

9ir,I understand there is a clerkship acant in your office, and I beg to apply for the same. Ihae passed the 'atriculation &=amination and studied up to the F.-. but as preented

from pursuing my studies further oing to seeral untoard circumstances. I hae,hoeer, been deoting all my time to 'athematics and deeloping the subject. I cansay I am >uite confident I can do justice to my ork if I am appointed to the post. Itherefore beg to re>uest that you ill be good enough to confer the appointment on me.@B$A

-ttached to his application as a recommendation from &. W. 'iddlemast, a mathematics professor at the residency 1ollege, ho rote that Ramanujan as "a young man of >uite e=ceptionalcapacity in 'athematics".@BA *hree eeks after he had applied, on # 'arch, Ramanujan learnedthat he had been accepted as a 1lass III, 3rade IG accounting clerk, making 0/ rupees per month.

@C/A -t his office, Ramanujan easily and >uickly completed the ork he as gien, so he spent hisspare time doing mathematical research. Ramanujan4s boss, 9ir Francis 9pring, and 9. +arayanaIyer, a colleague ho as also treasurer of the Indian 'athematical 9ociety, encouragedRamanujan in his mathematical pursuits.

Contacting Englis matematicians

In the spring of ##0, +arayana Iyer, Ramachandra Rao and &. W. 'iddlemast tried to presentRamanujan4s ork to ritish mathematicians. Ene mathematician, '. 6. '. 5ill of Hniersity1ollege 7ondon, commented that Ramanujan4s papers ere riddled ith holes.@C#A 5e said thatalthough Ramanujan had "a taste for mathematics, and some ability", he lacked the educational

 background and foundation needed to be accepted by mathematicians.@C!A -lthough 5ill did notoffer to take Ramanujan on as a student, he did gie thorough and serious professional adice on hisork. With the help of friends, Ramanujan drafted letters to leading mathematicians at 1ambridgeHniersity.@C0A

*he first to professors, 5. F. aker  and &. W. 5obson, returned Ramanujan4s papers ithoutcomment.@CBA En # 6anuary ##0, Ramanujan rote to 3. 5. 5ardy. 1oming from an unknonmathematician, the nine pages of mathematics made 5ardy initially ie Ramanujan4s manuscriptsas a possible "fraud".@CCA 5ardy recognised some of Ramanujan4s formulae but others "seemedscarcely possible to beliee".@CA Ene of the theorems 5ardy found scarcely possible to beliee asfound on the bottom of page three (alid for / a  b J #;!):

 

5ardy as also impressed by some of Ramanujan4s other ork relating to infinite series:

 

*he first result had already been determined by a mathematician named auer. *he second one asne to 5ardy, and as deried from a class of functions called a hypergeometric series hich hadfirst been researched by 7eonhard &uler  and 1arl Friedrich 3auss. 1ompared to Ramanujan4s ork

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on integrals, 5ardy found these results "much more intriguing".@C%A -fter he sa Ramanujan4stheorems on continued fractions on the last page of the manuscripts, 5ardy commented that "they@theoremsA defeated me completelyK I had neer seen anything in the least like them before".@C$A 5efigured that Ramanujan4s theorems "must be true, because, if they ere not true, no one ould haethe imagination to inent them".@C$A 5ardy asked a colleague, 6. &. 7ittleood, to take a look at the

 papers. 7ittleood as ama?ed by the mathematical genius of Ramanujan. -fter discussing the

 papers ith 7ittleood, 5ardy concluded that the letters ere "certainly the most remarkable I haereceied" and commented that Ramanujan as "a mathematician of the highest >uality, a man ofaltogether e=ceptional originality and poer".@CA Ene colleague, &. 5. +eille, later commentedthat "not one @theoremA could hae been set in the most adanced mathematical e=amination in theorld".@/A

En $ February ##0, 5ardy rote a letter to Ramanujan, e=pressing his interest for his ork. 5ardyalso added that it as "essential that I should see proofs of some of your assertions".@#A efore hisletter arried in 'adras during the third eek of February, 5ardy contacted the Indian Effice to

 plan for Ramanujan4s trip to 1ambridge. 9ecretary -rthur aies of the -disory 1ommittee forIndian 9tudents met ith Ramanujan to discuss the oerseas trip.@!A In accordance ith his

rahmin upbringing, Ramanujan refused to leae his country to "go to a foreign land".@0A 'eanhile, Ramanujan sent a letter packed ith theorems to 5ardy, riting, "I hae found a friendin you ho ies my labour sympathetically."@BA

*o supplement 5ardy4s endorsement, a former mathematical lecturer at *rinity 1ollege, 1ambridge,3ilbert Walker , looked at Ramanujan4s ork and e=pressed ama?ement, urging him to spend time at1ambridge.@CA -s a result of Walker4s endorsement, . 5anumantha Rao, a mathematics professorat an engineering college, inited Ramanujan4s colleague +arayana Iyer to a meeting of the oardof 9tudies in 'athematics to discuss "hat e can do for 9. Ramanujan". @A *he board agreed togrant Ramanujan a research scholarship of %C rupees per month for the ne=t to years at theHniersity of 'adras.@%A While he as engaged as a research student, Ramanujan continued tosubmit papers to the Journal of the Indian Mathematical Society. In one instance, +arayana Iyer

submitted some theorems of Ramanujan on summation of series to the aboe mathematical journaladding "*he folloing theorem is due to 9. Ramanujan, the mathematics student of 'adrasHniersity". 7ater in +oember, ritish rofessor &dard . Ross of 'adras 1hristian 1ollege,hom Ramanujan had met a fe years before, stormed into his class one day ith his eyes gloing,asking his students, "oes Ramanujan kno olishL" *he reason as that in one paper, Ramanujanhad anticipated the ork of a olish mathematician hose paper had just arried by the day4s mail.@$A In his >uarterly papers, Ramanujan dre up theorems to make definite integrals more easilysolable. Working off 3iuliano Frullani4s #$!# integral theorem, Ramanujan formulatedgeneralisations that could be made to ealuate formerly unyielding integrals.@A

5ardy4s correspondence ith Ramanujan soured after Ramanujan refused to come to &ngland.

5ardy enlisted a colleague lecturing in 'adras, &. 5. +eille, to mentor and bring Ramanujan to&ngland.@%/A +eille asked Ramanujan hy he ould not go to 1ambridge. Ramanujan apparentlyhad no accepted the proposalK as +eille put it, "Ramanujan needed no conerting and that his

 parents4 opposition had been ithdran".@/A -pparently, Ramanujan4s mother had a iid dream inhich the family 3oddess, the deity of +amagiri, commanded her "to stand no longer beteen herson and the fulfilment of his life4s purpose".@/A Ramanujan then set sail for &ngland, leaing hisife to stay ith his parents in India.

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!ife in England

Ramanujan (centre) ith other scientists at *rinity 1ollege

Wheell4s 1ourt, *rinity 1ollege, 1ambridge

Ramanujan boarded the 9.9. Nevasa on #% 'arch ##B, and at #/ o4clock in the morning, the shipdeparted from 'adras.@%#A 5e arried in 7ondon on #B -pril, ith &. 5. +eille aiting for himith a car. Four days later, +eille took him to his house on 1hesterton Road in 1ambridge.

Ramanujan immediately began his ork ith 7ittleood and 5ardy. -fter si= eeks, Ramanujanmoed out of +eille4s house and took up residence on Wheell4s 1ourt, just a fieDminute alkfrom 5ardy4s room.@%!A 5ardy and Ramanujan began to take a look at Ramanujan4s notebooks.5ardy had already receied #!/ theorems from Ramanujan in the first to letters, but there eremany more results and theorems to be found in the notebooks. 5ardy sa that some ere rong,others had already been discoered, hile the rest ere ne breakthroughs.@%0A Ramanujan left adeep impression on 5ardy and 7ittleood. 7ittleood commented, "I can beliee that he4s at least a6acobi",@%BA hile 5ardy said he "can compare him only ith @7eonhardA &uler or 6acobi."@%CA

Ramanujan spent nearly fie years in 1ambridge collaborating ith 5ardy and 7ittleood and published a part of his findings there. 5ardy and Ramanujan had highly contrasting personalities.

*heir collaboration as a clash of different cultures, beliefs and orking styles. 5ardy as anatheist and an apostle of proof and mathematical rigour, hereas Ramanujan as a deeply religiousman and relied ery strongly on his intuition. While in &ngland, 5ardy tried his best to fill the gapsin Ramanujan4s education ithout interrupting his spell of inspiration.

Ramanujan as aarded a achelor of 9cience degree by research (this degree as later renamedh) in 'arch ## for his ork on highly composite numbers, the first part of hich as

 published as a paper in the Proceedins of the !ondon Mathematical Society. *he paper as oerC/ pages ith different properties of such numbers proen. 5ardy remarked that this as one of themost unusual papers seen in mathematical research at that time and that Ramanujan shoede=traordinary ingenuity in handling it.@citation needed A En ecember ##%, he as elected to the7ondon 'athematical 9ociety. 5e became a Fello of the Royal 9ociety in ##$, becoming thesecond Indian to do so, folloing -rdaseer 1ursetjee in #$B#, and he as one of the youngestFellos in the history of the Royal 9ociety. 5e as elected "for his inestigation in &llipticfunctions and the *heory of +umbers." En #0 Ectober ##$, he became the first Indian to be

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elected a Fello of *rinity 1ollege, 1ambridge.@%A

Illness and return to India

lagued by health problems throughout his life, liing in a country far aay from home, andobsessiely inoled ith his mathematics, Ramanujan4s health orsened in &ngland, perhaps

e=acerbated by stress and by the scarcity of egetarian food during the First World War. 5e asdiagnosed ith tuberculosis and a seere itamin deficiency and as confined to a sanatorium.

Ramanujan returned to 2umbakonam, 'adras residency in ## and died soon thereafter at theage of 0! in #!/. 5is ido, 9. 6anaki -mmal, moed to 'umbai, but returned to 1hennai(formerly 'adras) in #C/, here she lied until her death at age B in #B.@!A

- #B analysis of Ramanujan4s medical records and symptoms by r. .-.. Moung concludedthat it as much more likely he had hepatic amoebiasis, a parasitic infection of the lier idespreadin 'adras, here Ramanujan had spent time. 5e had to episodes of  dysentery  before he left India.When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis, @%%A a difficult disease to diagnose, but once diagnosed readily cured.@%%A

"ersonality and s#iritual life

Ramanujan has been described as a person ith a somehat shy and >uiet disposition, a dignifiedman ith pleasant manners.@%$A 5e lied a rather spartan life hile at 1ambridge. Ramanujan4s firstIndian biographers describe him as rigorously orthodo=. Ramanujan credited his acumen to hisfamily goddess, 'ahalakshmi of +amakkal. 5e looked to her for inspiration in his ork,@%A andclaimed to dream of blood drops that symbolised her male consort,  +arasimha, after hich heould receie isions of scrolls of comple= mathematical content unfolding before his eyes. @$/A 5eoften said, "-n e>uation for me has no meaning, unless it represents a thought of 3od."@$#A@$!A

5ardy cites Ramanujan as remarking that all religions seemed e>ually true to him.@$0A 5ardyfurther argued that Ramanujan4s religiousness had been romanticised by Westerners and oerstated

 Nin reference to his belief, not practiceNby Indian biographers. -t the same time, he remarked onRamanujan4s strict obserance of egetarianism.

$atematical acievements

In mathematics, there is a distinction beteen haing an insight and haing a proof. Ramanujan4stalent suggested a plethora of formulae that could then be inestigated in depth later. It is said by 3.5. 5ardy that Ramanujan4s discoeries are unusually rich and that there is often more to them thaninitially meets the eye. -s a byDproduct, ne directions of research ere opened up. &=amples of

the most interesting of these formulae include the intriguing infinite series for  O, one of hich isgien belo

 

*his result is based on the negatie fundamental discriminant d  P QBC$ P Q!0! ith class numberh(d ) P ! (note that C%#0C$ P !0/ and that $/#PK 0PB) and is related to the factthat

 

1ompare to 5eegner numbers, hich hae class number # and yield similar formulae. Ramanujan4sseries for O conerges e=traordinarily rapidly (e=ponentially) and forms the basis of some of the

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fastest algorithms currently used to calculate O. *runcating the sum to the first term also gies the

appro=imation for O, hich is correct to si= decimal places. 9ee also the more generalRamanujan89ato series.

Ene of his remarkable capabilities as the rapid solution for problems. 5e as sharing a room ith

. 1. 'ahalanobis ho had a problem, "Imagine that you are on a street ith houses marked #through n. *here is a house in beteen (=) such that the sum of the house numbers to the left of ite>uals the sum of the house numbers to its right. If n is beteen C/ and C//, hat are n and =L"*his is a biariate problem ith multiple solutions. Ramanujan thought about it and gae theanser ith a tist: 5e gae a continued fraction. *he unusual part as that it as the solution tothe hole class of problems. 'ahalanobis as astounded and asked ho he did it. "It is simple. *heminute I heard the problem, I kne that the anser as a continued fraction. Which continuedfraction, I asked myself. *hen the anser came to my mind", Ramanujan replied. @$BA@$CA

5is intuition also led him to derie some preiously unknon identities, such as

 

for all , here is the gamma function. &=panding into series of poers and

e>uating coefficients of , , and gies some deep identities forthe hyperbolic secant.

In ##$, 5ardy and Ramanujan studied the partition function  P (n) e=tensiely and gae a nonDconergent asymptotic series that permits e=act computation of the number of partitions of aninteger. 5ans Rademacher , in #0%, as able to refine their formula to find an e=act conergent

series solution to this problem. Ramanujan and 5ardy4s ork in this area gae rise to a poerfulne method for finding asymptotic formulae, called the circle method.@$A

5e discoered mock theta functions in the last year of his life.@$%A For many years these functionsere a mystery, but they are no knon to be the holomorphic parts of harmonic eak 'aassforms.

%e Ramanujan conjecture

'ain article: Ramanujan8etersson conjecture

-lthough there are numerous statements that could hae borne the name Ramanujan conjecture,

there is one statement that as ery influential on later ork. In particular, the connection of thisconjecture ith conjectures of -ndrS Weil in algebraic geometry opened up ne areas of research.*hat Ramanujan conjecture is an assertion on the si?e of the tauDfunction, hich has as generatingfunction the discriminant modular form T("), a typical cusp form in the theory of modular forms. Itas finally proen in #%0, as a conse>uence of ierre eligne4s proof of the Weil conjectures. *hereduction step inoled is complicated. eligne on a Fields 'edal in #%$ for his ork on Weilconjectures.@$$A

Ramanujan&s note'oo(s

Further information: Ramanujan4s lost notebook 

While still in 'adras, Ramanujan recorded the bulk of his results in four notebooks of loose leaf   paper. *hese results ere mostly ritten up ithout any deriations. *his is probably the origin of

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the misperception that Ramanujan as unable to proe his results and simply thought up the finalresult directly. 'athematician ruce 1. erndt, in his reie of these notebooks and Ramanujan4sork, says that Ramanujan most certainly as able to make the proofs of most of his results, butchose not to.

*his style of orking may hae been for seeral reasons. 9ince paper as ery e=pensie,Ramanujan ould do most of his ork and perhaps his proofs on slate, and then transfer just the

results to paper. Hsing a slate as common for mathematics students in the 'adras residency atthe time. 5e as also >uite likely to hae been influenced by the style of  3. 9. 1arr 4s book studiedin his youth, hich stated results ithout proofs. Finally, it is possible that Ramanujan consideredhis orkings to be for his personal interest aloneK and therefore recorded only the results. @$A

*he first notebook has 0C# pages ith # somehat organised chapters and some unorganisedmaterial. *he second notebook has !C pages in !# chapters and #// unorganised pages, ith thethird notebook containing 00 unorganised pages. *he results in his notebooks inspired numerous

 papers by later mathematicians trying to proe hat he had found. 5ardy himself created paperse=ploring material from Ramanujan4s ork as did 3. +. Watson, . '. Wilson, and ruce erndt.@$A - fourth notebook ith $% unorganised pages, the soDcalled "lost notebook", as rediscoered

in #% by 3eorge -ndres.@%%A

 +otebooks #, ! and 0 ere published as a toDolume set in #C% by the *ata Institute ofFundamental Research (*IFR), 'umbai, India. *his as a photocopy edition of the originalmanuscripts, in his on handriting.

In ecember !/##, as part of the celebrations of the #!Cth anniersary of Ramanujan4s birth, *IFRrepublished the notebooks in a coloured toDolume collector4s edition. *hese ere produced fromscanned and microfilmed images of the original manuscripts by e=pert archiists of Roja 'uthiahResearch 7ibrary, 1hennai.

)ardy*Ramanujan num'er +,-.'ain article: #%! (number)

*he number #%! is knon as the 5ardy8Ramanujan number after a famous anecdote of the ritishmathematician 3. 5. 5ardy regarding a isit to the hospital to see Ramanujan. In 5ardy4s ords:@/A

U

I remember once going to see him hen he as ill at utney. I had ridden in ta=i cab number#%! and remarked that the number seemed to me rather a dull one, and that I hoped it as notan unfaorable omen. "+o", he replied, "it is a ery interesting numberK it is the smallestnumber e=pressible as the sum of to cubes in to different ays."

V

*he to different ays are#%! P #0 J #!0 P 0 J #/0.

3enerali?ations of this idea hae created the notion of "ta=icab numbers". 1oincidentally, #%! isalso a 1armichael number .

/ter matematicians& views of Ramanujan

5ardy said : "5e combined a poer of generali?ation, a feeling for form, and a capacity for rapidmodification of his hypotheses, that ere often really startling, and made him, in his on peculiar

field, ithout a rial in his day. *he limitations of his knoledge ere as startling as its profundity.5ere as a man ho could ork out modular e>uations and theorems... to orders unheard of, hosemastery of continued fractions as... beyond that of any mathematician in the orld, ho had

http://en.wikipedia.org/wiki/Bruce_C._Berndthttp://en.wikipedia.org/wiki/Slate_(writing)http://en.wikipedia.org/wiki/Madras_Presidencyhttp://en.wikipedia.org/wiki/G._S._Carrhttp://en.wikipedia.org/wiki/G._S._Carrhttp://en.wikipedia.org/wiki/G._S._Carrhttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-Bruce_Berndt_on_Ramanujan-89http://en.wikipedia.org/wiki/G._N._Watsonhttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-Bruce_Berndt_on_Ramanujan-89http://en.wikipedia.org/wiki/Ramanujan's_lost_notebookhttp://en.wikipedia.org/wiki/George_Andrews_(mathematician)http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-lostnotebook-77http://www.tifr.res.in/http://www.tifr.res.in/http://www.tifr.res.in/http://www.tifr.res.in/http://www.math.tifr.res.in/~publ/ramanujan.htmlhttp://www.lib.uchicago.edu/e/su/southasia/about-rmrl.htmlhttp://www.lib.uchicago.edu/e/su/southasia/about-rmrl.htmlhttp://en.wikipedia.org/wiki/1729_(number)http://en.wikipedia.org/wiki/G._H._Hardyhttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-90http://en.wikipedia.org/wiki/Putneyhttp://en.wikipedia.org/wiki/Interesting_number_paradoxhttp://en.wikipedia.org/wiki/Taxicab_numberhttp://en.wikipedia.org/wiki/Carmichael_numberhttp://en.wikipedia.org/wiki/Carmichael_numberhttp://en.wikipedia.org/wiki/Modular_equationhttp://en.wikipedia.org/wiki/Bruce_C._Berndthttp://en.wikipedia.org/wiki/Slate_(writing)http://en.wikipedia.org/wiki/Madras_Presidencyhttp://en.wikipedia.org/wiki/G._S._Carrhttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-Bruce_Berndt_on_Ramanujan-89http://en.wikipedia.org/wiki/G._N._Watsonhttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-Bruce_Berndt_on_Ramanujan-89http://en.wikipedia.org/wiki/Ramanujan's_lost_notebookhttp://en.wikipedia.org/wiki/George_Andrews_(mathematician)http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-lostnotebook-77http://www.tifr.res.in/http://www.tifr.res.in/http://www.math.tifr.res.in/~publ/ramanujan.htmlhttp://www.lib.uchicago.edu/e/su/southasia/about-rmrl.htmlhttp://www.lib.uchicago.edu/e/su/southasia/about-rmrl.htmlhttp://en.wikipedia.org/wiki/1729_(number)http://en.wikipedia.org/wiki/G._H._Hardyhttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-90http://en.wikipedia.org/wiki/Putneyhttp://en.wikipedia.org/wiki/Interesting_number_paradoxhttp://en.wikipedia.org/wiki/Taxicab_numberhttp://en.wikipedia.org/wiki/Carmichael_numberhttp://en.wikipedia.org/wiki/Modular_equation

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found for himself the functional e>uation of the ?eta function and the dominant terms of many ofthe most famous problems in the analytic theory of numbersK and yet he had neer heard of a doubly

 periodic function or of 1auchy4s theorem, and had indeed but the aguest idea of hat a function ofa comple= ariable as...".@#A When asked about the methods employed by Ramanujan to arrie athis solutions, 5ardy said that they ere "arried at by a process of mingled argument, intuition, andinduction, of hich he as entirely unable to gie any coherent account."@!A 5e also stated that he

had "neer met his e>ual, and can compare him only ith &uler or 6acobi."@!Auoting 2. 9riniasa Rao,@0A "-s for his place in the orld of 'athematics, e >uote ruce 1.erndt: 4aul &rdXs has passed on to us 5ardy4s personal ratings of mathematicians. 9uppose thate rate mathematicians on the basis of pure talent on a scale from / to #//, 5ardy gae himself ascore of !C, 6.&. 7ittleood 0/, aid 5ilbert $/ and Ramanujan #//.4"

rofessor ruce 1. erndt of the Hniersity of Illinois, during a lecture at II* 'adras in 'ay !/##,stated that oer the last B/ years, as nearly all of Ramanujan4s theorems hae been proen right,there had been a greater appreciation of Ramanujan4s ork and brilliance. Further, he statedRamanujan4s ork as no perading many areas of modern mathematics and physics.@$%A@BA

In his book Scientific Ede, the physicist 6ayant +arlikar  spoke of "9riniasa Ramanujan,discoered by the 1ambridge mathematician 5ardy, hose great mathematical findings ere

 beginning to be appreciated from ##C to ##. 5is achieements ere to be fully understood muchlater, ell after his untimely death in #!/. For e=ample, his ork on the highly composite numbers(numbers ith a large number of factors) started a hole ne line of inestigations in the theory ofsuch numbers."

uring his lifelong mission in educating and propagating mathematics among the school children inIndia, +igeria and elsehere, .2. 9riniasan has continually introduced Ramanujan4s mathematicalorks.

RecognitionFurther information: 7ist of things named after 9riniasa Ramanujan

 ust of Ramanujan in the garden of irla Industrial Y *echnological 'useum.

Ramanujan4s home state of *amil +adu celebrates !! ecember (Ramanujan4s birthday) as 49tate I*ay4, memorialising both the man and his achieements, as a natie of *amil +adu. - stamp

 picturing Ramanujan as released by the 3oernment of India in #! 8 the %Cth anniersary ofRamanujan4s birth 8 commemorating his achieements in the field of number theory,@CA and a nedesign as issued on ! ecember !/##, by the India ost.@A@%A

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9ince the 1entennial year of Ramanujan, eery year !! ec, is celebrated as Ramanujan ay by the3oernment -rts 1ollege, 2umbakonam here he had studied and later dropped out. It iscelebrated by the epartment of 'athematics by organising oneD, toD, or threeDday seminars byiniting eminent scholars from uniersities;colleges, and participants are mainly students ofmathematics, research scholars, and professors from local colleges. It as planned to celebrate the#!Cth birthday in a grand manner by initing the foreign eminent mathematical scholars of this

century i?., 3 & -ndres. and ruce 1 erndt, ho are ery familiar ith the contributions andorks of Ramanujan.

Ramanujan4s ork and life are celebrated on !! ecember at the Indian Institute of *echnology(II*), 'adras in 1hennai. *he epartment of 'athematics celebrates this day by organising a

 +ational 9ymposium on 'athematical 'ethods and -pplications (+9''-) for one day byiniting eminent Indian and foreign scholars.

- pri?e for young mathematicians from deeloping countries has been created in the name ofRamanujan by the International 1entre for *heoretical hysics (I1*), in coDoperation ith theInternational 'athematical Hnion, hich nominate members of the pri?e committee. *he9hanmugha -rts, 9cience, *echnology Y Research -cademy (9-9*R-), based in the state of *amil

 +adu in 9outh India, has instituted the 9-9*R- Ramanujan ri?e of Z#/,/// to be gien annuallyto a mathematician not e=ceeding the age of 0! for outstanding contributions in an area ofmathematics influenced by Ramanujan. *he age limit refers to the years Ramanujan lied, haingneertheless still achieed many accomplishments. *his pri?e has been aarded annually since!//C, at an international conference conducted by 9-9*R- in 2umbakonam, Ramanujan4shometon, around Ramanujan4s birthday, !! ecember.

En the #!Cth anniersary of his birth, India declared the birthday of Ramanujan, !! ecember, as4+ational 'athematics ay.4 *he declaration as made by r. 'anmohan 9ingh in 1hennai on !ecember !/##.@$A r 'anmohan 9ingh also declared that the year !/#! ould be celebrated asthe +ational 'athematics Mear . 5is residence is no presered by 9-9*R- uniersity in

2umbakonam.

In #o#ular culture

•  Ramanujan, an IndoDritish collaboration film, chronicling the life of Ramanujan, is being

made by the independent film company 1amphor 1inema.@A *he cast and cre includedirector 3nana Rajasekaran, cinematographer 9unny 6oseph and editor  . 7enin.@#//A@#/#A opular Indian and &nglish stars -bhinay Gaddi, 9uhasini 'aniratnam, hama, 2ein'c3oan and 'ichael 7ieber star in piotal roles.@#/!A 

• - film, based on the book #he Man $ho %ne& Infinity' A !ife of the (enius Ramanujan by

Robert 2anigel, is being made by &dard ressman and 'atthe ron ith R. 'adhaan

 playing Ramanujan.@#/0A • - play, )irst *lass Man by -lter &go roductions,@#/BA as based on aid Freeman4s

 )irst *lass Man. *he play is centred around Ramanujan and his comple= and dysfunctionalrelationship ith 5ardy. En # Ectober !/##, it as announced that Roger 9pottisoode,

 best knon for his 6ames ond film #omorro& Never +ies, is orking on the film ersion,starring actor 9iddharth. 7ike the book and play it is also titled #he )irst *lass Man.@#/CA 

•  A +isappearin Number  is a recent ritish stage production by the company 1omplicite that

e=plores the relationship beteen 5ardy and Ramanujan.• *he noel #he Indian *ler,  by aid 7eaitt e=plores in fiction the eents folloing

Ramanujan4s letter to 5ardy.@#/A@#/%A •

En !! 'arch #$$, the 9 9eries Nova aired a documentary about Ramanujan, "*he 'anWho 7oed +umbers" (9eason #C, &pisode #).@#/$A 

• Ramanujan as mentioned in the #% film 3ood Will 5unting, in a scene here professor

http://en.wikipedia.org/wiki/Government_Arts_College,_Kumbakonamhttp://en.wikipedia.org/wiki/Indian_Institute_of_Technology_Madrashttp://en.wikipedia.org/wiki/Indian_Institute_of_Technology_Madrashttp://en.wikipedia.org/wiki/Chennaihttp://en.wikipedia.org/wiki/International_Centre_for_Theoretical_Physicshttp://en.wikipedia.org/wiki/International_Mathematical_Unionhttp://en.wikipedia.org/wiki/Shanmugha_Arts,_Science,_Technology_%26_Research_Academyhttp://en.wikipedia.org/wiki/SASTRA_Ramanujan_Prizehttp://en.wikipedia.org/wiki/Kumbakonamhttp://en.wikipedia.org/wiki/Dr._Manmohan_Singhhttp://en.wikipedia.org/wiki/Chennaihttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-98http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-98http://en.wikipedia.org/wiki/National_Mathematics_Yearhttp://en.wikipedia.org/wiki/National_Mathematics_Yearhttp://en.wikipedia.org/wiki/Ramanujan_(film)http://en.wikipedia.org/wiki/Ramanujan_(film)#Camphor_Cinemahttp://en.wikipedia.org/wiki/Ramanujan_(film)#Camphor_Cinemahttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-99http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-99http://en.wikipedia.org/wiki/Gnana_Rajasekaranhttp://en.wikipedia.org/wiki/Gnana_Rajasekaranhttp://en.wikipedia.org/wiki/Sunny_Josephhttp://en.wikipedia.org/wiki/Sunny_Josephhttp://en.wikipedia.org/wiki/B._Leninhttp://en.wikipedia.org/wiki/B._Leninhttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-100http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-100http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-101http://en.wikipedia.org/wiki/Suhasini_Maniratnamhttp://en.wikipedia.org/wiki/Bhamahttp://en.wikipedia.org/wiki/Bhamahttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-102http://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinityhttp://en.wikipedia.org/wiki/R._Madhavanhttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-103http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-104http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-104http://en.wikipedia.org/wiki/Roger_Spottiswoodehttp://en.wikipedia.org/wiki/Tomorrow_Never_Dieshttp://en.wikipedia.org/wiki/Siddharth_(actor)http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-105http://en.wikipedia.org/wiki/A_Disappearing_Numberhttp://en.wikipedia.org/wiki/The_Indian_Clerkhttp://en.wikipedia.org/wiki/David_Leavitthttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-106http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-107http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-108http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-108http://en.wikipedia.org/wiki/Good_Will_Huntinghttp://en.wikipedia.org/wiki/Government_Arts_College,_Kumbakonamhttp://en.wikipedia.org/wiki/Indian_Institute_of_Technology_Madrashttp://en.wikipedia.org/wiki/Indian_Institute_of_Technology_Madrashttp://en.wikipedia.org/wiki/Chennaihttp://en.wikipedia.org/wiki/International_Centre_for_Theoretical_Physicshttp://en.wikipedia.org/wiki/International_Mathematical_Unionhttp://en.wikipedia.org/wiki/Shanmugha_Arts,_Science,_Technology_%26_Research_Academyhttp://en.wikipedia.org/wiki/SASTRA_Ramanujan_Prizehttp://en.wikipedia.org/wiki/Kumbakonamhttp://en.wikipedia.org/wiki/Dr._Manmohan_Singhhttp://en.wikipedia.org/wiki/Chennaihttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-98http://en.wikipedia.org/wiki/National_Mathematics_Yearhttp://en.wikipedia.org/wiki/Ramanujan_(film)http://en.wikipedia.org/wiki/Ramanujan_(film)#Camphor_Cinemahttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-99http://en.wikipedia.org/wiki/Gnana_Rajasekaranhttp://en.wikipedia.org/wiki/Sunny_Josephhttp://en.wikipedia.org/wiki/B._Leninhttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-100http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-101http://en.wikipedia.org/wiki/Suhasini_Maniratnamhttp://en.wikipedia.org/wiki/Bhamahttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-102http://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinityhttp://en.wikipedia.org/wiki/R._Madhavanhttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-103http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-104http://en.wikipedia.org/wiki/Roger_Spottiswoodehttp://en.wikipedia.org/wiki/Tomorrow_Never_Dieshttp://en.wikipedia.org/wiki/Siddharth_(actor)http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-105http://en.wikipedia.org/wiki/A_Disappearing_Numberhttp://en.wikipedia.org/wiki/The_Indian_Clerkhttp://en.wikipedia.org/wiki/David_Leavitthttp://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-106http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-107http://en.wikipedia.org/wiki/Srinivasa_Ramanujan#cite_note-108http://en.wikipedia.org/wiki/Good_Will_Hunting

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3erald 7ambeau (9tellan 9karsgard) e=plains to 9ean 'aguire (Robin Williams) the geniusof Will 5unting ('att amon) by comparing him to Ramanujan.

• 3oogle honoured him on his #!Cth birth anniersary by replacing its logo ith a doodle on

its home page.@#/A • *he teleision series +umb0rs has the character r. -mita Ramanujan, a professor of

applied mathematics, named after Ramanujan@##/A 

• Ramanujan4s story is both referenced and echoed in 1yril '. 2ornbluth4s "3ome?".

See also

• 7ist of amateur mathematicians 

• Ramanujan graph 

• Ramanujan summation 

• Ramanujan4s constant 

• Ramanujan4s ternary >uadratic form 

• Rank of a partition 

• !%# (number) • 7ist of Indian mathematicians 

Notes

#.

• 1.. 9no Foreord to "- 'athematician4s -pology" by 3.5. 5ardy 

• erndt, ruce 1. (!//C). Ramanujan-s Noteboo,s Part . . 9pringer7ink . p. B. I9+ /D0$%D

BB#D/.• "Rediscoering Ramanujan".  )rontline +0 (#%): C/. -ugust #. Retrieed !/ ecember

!/#!.• Eno, 2en (6une86uly !//). "5onoring a 3ift from 2umbakonam" (F). Notices of the

 American Mathematical Society ('athematical -ssociation of -merica) 12 (): C/.Retrieed !0 6une !//%.

• -lladi, 2rishnasami (#$). Analytic and Elementary Number #heory' A #ribute to

 Mathematical !eend Paul Erd/s. +orell, 'assachusetts: 2luer -cademic ublishers. p. . I9+ /D%!0D$!%0D/.

• 2anigel ##, p. ##

• 2anigel ##, pp. #%8#$

• erndt Y Rankin !//#, p. $

• 2anigel ##, p. #!• 2anigel ##, p. #0

• 2anigel ##, p. #

• 2anigel ##, p. #B