Srinivasa Ramanujan FRS

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Srinivasa Ramanujan FRS ( pronunciation (helpinfo)) (22 December 188726 April 1920)was an Indian mathematician and autodidact who, with almost no formal training in puremathematics, made extraordinary contributions to mathematical analysis, number theory, infiniteseries, and continued fractions. Living in India with no access to the larger mathematicalcommunity, which was centered in Europe at the time, Ramanujan developed his own

mathematical research in isolation. As a result, he sometimes rediscovered known theorems inaddition to producing new work. Ramanujan was said to be a natural genius by the Englishmathematician G. H. Hardy, in the same league as mathematicians such as Euler and Gauss.[1].He died at the age of 32.

Born at Erode, Madras Presidency (now Tamil Nadu) in a poor Hindu Brahmin family,Ramanujan's introduction to formal mathematics began at age 10. He demonstrated a naturalability, and was given books on advanced trigonometry written by S. L. Loney that he masteredby the age of 12; he even discovered theorems of his own, and re-discovered Euler's identityindependently.[2]He demonstrated unusual mathematical skills at school, winning accolades andawards. By 17, Ramanujan had conducted his own mathematical research on Bernoulli numbers

and the EulerMascheroni constant.

Ramanujan received a scholarship to study at Government College in Kumbakonam, but lost itwhen he failed his non-mathematical coursework. He joined another college to pursueindependent mathematical research, working as a clerk in the Accountant-General's office at theMadras Port Trust Office to support himself.[3]In 19121913, he sent samples of his theorems tothree academics at the University of Cambridge. G. H. Hardy, recognizing the brilliance of hiswork, invited Ramanujan to visit and work with him at Cambridge. He became a Fellow of theRoyal Society and a Fellow ofTrinity College, Cambridge. Ramanujan died of illness,malnutrition, and possibly liver infection in 1920 at the age of 32.

During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostlyidentities and equations).[4]

Nearly all his claims have now been proven correct, although a smallnumber of these results were actually false and some were already known.

[5]He stated results

that were both original and highly unconventional, such as the Ramanujan prime and theRamanujan theta function, and these have inspired a vast amount of further research.

[6]However,

the mathematical mainstream has been rather slow in absorbing some of his major discoveries.TheRamanujan Journal, an international publication, was launched to publish work in all areasof mathematics influenced by his work.[7]

In December 2011, in recognition of his contribution to mathematics, the Government of Indiadeclared that Ramanujan's birthday (22 December) should be celebrated every year as NationalMathematics Day, and also declared 2012 the National Mathematics Year.

[8][9]

Early life

Ramanujan was born on 22 December 1887 in Erode, Madras Presidency (now Tamil Nadu), atthe residence of his maternal grandparents.

[10]His father, K. Srinivasa Iyengar, worked as a clerk

in a sari shop and hailed from the district ofThanjavur.[11]

His mother, Komalatammal, was ahousewife and also sang at a local temple.[12]They lived in Sarangapani Street in a traditional
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home in the town of Kumbakonam. The family home is now a museum. When Ramanujan was ayear and a half old, his mother gave birth to a son named Sadagopan, who died less than threemonths later. In December 1889, Ramanujan had smallpox and recovered, unlike thousands inthe Thanjavur District who died from the disease that year.

[13]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.

On 1 October 1892, Ramanujan was enrolled at the local school.[14]In March 1894, he wasmoved to a Telugu medium school. After his maternal grandfather lost his job as a court officialin Kanchipuram,

[15]Ramanujan and his mother moved back to Kumbakonam and he was

enrolled in the Kangayan Primary School.[16]When his paternal grandfather died, he was sentback to his maternal grandparents, who were now living in Madras. He did not like school inMadras, 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.

[16]

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 tosing religious songs, to attend pujas at the temple and particular eating habitsall of which arepart ofBrahmin culture.

[17]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,geography and arithmetic. With his scores, he stood first in the district.[18]That year, Ramanujanentered Town Higher Secondary School where he encountered formal mathematics for the firsttime.

[18]

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 by S. L.Loney.[19][2]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 whichcontinued throughout his school career and also assisted the school in the logistics of assigningits 1200 students (each with their own needs) to its 35-odd teachers.

[20]He completed

mathematical exams in half the allotted time, and showed a familiarity with geometry andinfinite 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 hewas 16, Ramanujan obtained from a friend a library-loaned copy of a book by G. S. Carr.[21][22]The book was titledA Synopsis of Elementary Results in Pure and Applied Mathematicsand wasa collection of 5000 theorems. Ramanujan reportedly studied the contents of the book indetail.[23]The book is generally acknowledged as a key element in awakening the genius ofRamanujan.[23]The next year, he had independently developed and investigated the Bernoullinumbers and had calculated Euler's constant up to 15 decimal places.[24]His peers at the timecommented that they "rarely understood him" and "stood in respectful awe" of him.[20]

When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded theK. 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.

[20]He received a scholarship to study at Government Arts College,
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Kumbakonam,[25][26]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.[27]In August 1905, he ran away from home, heading towards Visakhapatnam and stayed inRajahmundry for about a month.

[28]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 First Arts exam 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.[29]

Adulthood in India

On 14 July 1909, Ramanujan was married to a ten-year old bride, Janakiammal (21 March 189913 April 1994).

[30]She came from Rajendram, a village close to Marudur (Karur district)

Railway Station. Ramanujan's father did not participate in the marriage ceremony.[31]

After the marriage, Ramanujan developed a hydrocele testis, an abnormal swelling of the tunicavaginalis, an internal membrane in the testicle.

[32]The condition could be treated with a routine

surgical 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.

[33]

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.[34]

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 the Madras ChristianCollege."[35]After Ramanujan recovered and got back his notebooks from Iyer, he took anorthbound train from Kumbakonam to Villupuram, a coastal city under French control.

[36][37]

Attention from mathematicians

Ramanujan met deputy collector V. Ramaswamy Aiyer, who had recently founded the Indian

Mathematical Society.[38]

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.[39]
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Ramaswamy Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends inMadras.[38]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 MathematicalSociety.

[40][41][42]Ramachandra Rao was impressed by Ramanujan's research but doubted that it

was actually his own work. Ramanujan mentioned a correspondence he had with Professor

Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack ofunderstanding for his work but concluded that he was not a phony.[43]

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 discussed elliptic integrals, hypergeometric series, and his theory ofdivergent series,which Rao said ultimately "converted" him to a belief in Ramanujan's mathematical brilliance.[43]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 Indian Mathematical Society.[44]

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 firstnotebook, he formulated an equation that could be used to solve the infinitely nested radicalsproblem.

Using this equation, the answer to the question posed in theJournal was simply 3.[45]Ramanujanwrote his first formal paper for theJournal on the properties ofBernoulli numbers. One propertyhe discovered was that the denominators (sequence A027642 in OEIS) 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 ifn is even but not equal to zero,

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

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

In his 17-page paper, "Some Properties of Bernoulli's Numbers", Ramanujan gave three proofs,two corollaries and three conjectures.

[46]Ramanujan's writing initially had many flaws. As

Journal editor M. T. Narayana Iyengar noted:
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5/13

Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearnessand precision, that the ordinary [mathematical reader], unaccustomed to such intellectualgymnastics, could hardly follow him.

[47]

Ramanujan later wrote another paper and also continued to provide problems in theJournal.[48]

In early 1912, he got a temporary job in the Madras Accountant General's office, with a salary of20 rupees per month. He lasted for only a few weeks.[49]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 havepassed 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 be

good enough to confer the appointment on me.

[50]

Attached to his application was a recommendation from E. W. Middlemast, a mathematicsprofessor at the Presidency College, who wrote that Ramanujan was "a young man of quiteexceptional capacity in Mathematics".[51]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.[52]At his office, Ramanujan easily and quickly completed the work he wasgiven, 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 and E. W. Middlemast tried to presentRamanujan's work to British mathematicians. One mathematician, M. J. M. Hill ofUniversityCollege London, commented that Ramanujan's papers were riddled with holes.

[53]He said that

although Ramanujan had "a taste for mathematics, and some ability", he lacked the educationalbackground and foundation needed to be accepted by mathematicians.[54]Although Hill did notoffer 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.[55]

The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers withoutcomment.[56]On 16 January 1913, Ramanujan wrote to G. H. Hardy. Coming from an unknownmathematician, the nine pages of mathematics made Hardy initially view Ramanujan'smanuscripts as a possible "fraud".

[57]Hardy recognised some of Ramanujan's formulae but others

"seemed scarcely possible to believe".[58]One of the theorems Hardy found so incredible wasfound on the bottom of page three (valid for 0 < a < b + 1/2):
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6/13

Hardy was also impressed by some of Ramanujan's other work relating to infinite series:

The first result had already been determined by a mathematician named Bauer. The second onewas new to Hardy, and was derived from a class of functions called a hypergeometric serieswhich had first been researched by Leonhard Euler and Carl Friedrich Gauss. Compared toRamanujan's work on integrals, Hardy found these results "much more intriguing".[59]After hesaw Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardycommented that the "[theorems] defeated me completely; I had never seen anything in the leastlike them before".

[60]He figured that Ramanujan's theorems "must be true, because, if they were

not true, no one would have the imagination to invent them".[60]

Hardy asked a colleague, J. E.Littlewood, to take a look at the papers. Littlewood was amazed by the mathematical genius ofRamanujan. After discussing the papers with Littlewood, Hardy concluded that the letters were"certainly the most remarkable I have received" and commented that Ramanujan was "amathematician of the highest quality, a man of altogether exceptional originality and power".[61]One colleague, E. H. Neville, later commented that "not one [theorem] could have been set in themost advanced mathematical examination in the world".

[62]

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".

[63]

Before his letter arrived in Madras during the third week of February, Hardy contacted the IndianOffice to plan for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the AdvisoryCommittee for Indian Students met with Ramanujan to discuss the overseas trip.

[64]In

accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to aforeign land".[65]Meanwhile, Ramanujan sent a letter packed with theorems to Hardy, writing, "Ihave found a friend in you who views my labour sympathetically."

[66]

To supplement Hardy's endorsement, a former mathematical lecturer at Trinity College,Cambridge, Gilbert Walker, looked at Ramanujan's work and expressed amazement, urging himto spend time at Cambridge.

[67]As a result of Walker's endorsement, B. Hanumantha Rao, a

mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer toa meeting of the Board of Studies in Mathematics to discuss "what we can do for S.Ramanujan".

[68]The board agreed to grant Ramanujan a research scholarship of 75 rupees per

month for the next two years at the University of Madras.[69]

While he was engaged as a researchstudent, Ramanujan continued to submit papers to theJournal of the Indian MathematicalSociety. In one instance, Narayana Iyer submitted some theorems of Ramanujan on summation
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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 ofMadras Christian College, whom Ramanujan had met a few yearsbefore, stormed into his class one day with his eyes glowing, asking his students, DoesRamanujan 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 days mail.

[70]

In hisquarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable.Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisationsthat could be made to evaluate formerly unyielding integrals.[71]

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 toEngland.

[72]Neville asked Ramanujan why he would not go to Cambridge. Ramanujan

apparently had now accepted the proposal; as Neville put it, "Ramanujan needed no convertingand that his parents' opposition had been withdrawn".[62]Apparently, Ramanujan's mother had avivid dream in which the family Goddess Namagiri commanded her "to stand no longer between

her son and the fulfilment of his life's purpose".

[62]

Ramanujan then set sail for England, leavinghis wife to stay with his parents in India.

Life in England

Ramanujan boarded the S.S.Nevasa on 17 March 1914, and at 10 o'clock in the morning, theship departed from Madras.[73]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.[74]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, 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.

[75]

Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I canbelieve that he's at least a Jacobi",

[76]while Hardy said he "can compare him only with

[Leonhard] Euler or Jacobi."[77]

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) inMarch 1916 for his work on highly composite numbers, the first part of which was published as apaper in the Proceedings of the London 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 showed
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extraordinary ingenuity in handling it.[citation needed] On 6 December 1917, he was elected to theLondon Mathematical Society. He became a Fellow of the Royal Society in 1918, becoming thesecond Indian to do so, following Ardaseer Cursetjee in 1841, and he was one of the youngestFellows in the history of the Royal Society. He was elected "for his investigation in Ellipticfunctions and the Theory of Numbers." On 13 October 1918, he became the first Indian to be

elected a Fellow of Trinity College, Cambridge.

[78]

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 by stress and by the scarcity ofvegetarian food during the First World War. He wasdiagnosed with tuberculosis and a severe vitamin deficiency and was confined to a sanatorium.

Ramanujan returned to Kumbakonam, Madras Presidency in 1919 and died soon thereafter at theage 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.[31]

A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D.A.B. Young concludedthat it was much more likely he had hepatic amoebiasis, a parasitic infection of the liverwidespread 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,[3]a difficult disease to diagnose, but once diagnosed readily cured.[3]

Personality and spiritual life

Ramanujan has been described as a person with a somewhat shy and quiet disposition, a

dignified man with pleasant manners.[79]

He lived a rather Spartan life while at Cambridge.Ramanujan's first Indian biographers describe him as rigorously orthodox. Ramanujan creditedhis acumen to his family Goddess, Namagiri ofNamakkal. He looked to her for inspiration in hiswork,

[80]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.[81]He often said, "An equation for me has no meaning, unless it represents athought of God."

[82][83]

Hardy cites Ramanujan as remarking that all religions seemed equally true to him.[84]

Hardyfurther argued that Ramanujan's religiousness had been romanticised by Westerners andoverstatedin reference to his belief, not practiceby Indian biographers. At the same time, he

remarked 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 the
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most interesting of these formulae include the intriguing infinite series for, one of which isgiven below

This result is based on the negative fundamental discriminant d= 458 with class numberh(d)= 2 (note that 571358 = 26390 and that 9801=9999; 396=499) and is related to the factthat

Compare to Heegner numbers, which have class number 1 and yield similar formulae.Ramanujan's series for converges extraordinarily rapidly (exponentially) and forms the basis ofsome of the fastest algorithms currently used to calculate . Truncating the sum to the first term

also gives the approximation for , which is correct to six decimal places.

One of his remarkable capabilities was the rapid solution for problems. He was sharing a roomwith P. C. Mahalanobis who had a problem, "Imagine that you are on a street with housesmarked 1 through n. There is a house in between (x) such that the sum of the house numbers toleft of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are nand x?" This is a bivariate problem with multiple solutions. Ramanujan thought about it and gavethe answer with a twist: He gave a continued fraction. The unusual part was that it was thesolution to the whole class of problems. Mahalanobis was astounded and asked how he did it. "Itis 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," Ramanujanreplied.[85][86]

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

for all , where is the gamma function. Expanding into series of powers and equating

coefficients of , , and gives some deep identities for the hyperbolic secant.

In 1918, Hardy and Ramanujan studied the partition function P(n) extensively and gave a non-convergent asymptotic series that permits exact computation of the number of partitions of aninteger. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergentseries solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerfulnew method for finding asymptotic formulae, called the circle method.

[87]
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He discovered mock theta functions in the last year of his life.[88]For many years these functionswere a mystery, but they are now known to be the holomorphic parts of harmonic weakMaassforms.

The Ramanujan conjecture

Main article: RamanujanPetersson conjecture

Although there are numerous statements that could bear the nameRamanujan conjecture, there isone statement that was very influential on later work. In particular, the connection of thisconjecture with conjectures ofAndr Weil in algebraic geometry opened up new areas ofresearch. That Ramanujan conjecture is an assertion on the size of the Tau-function, which has asgenerating function the discriminant modular form (q), a typical cusp form in the theory ofmodular forms. It was finally proven in 1973, as a consequence ofPierre Deligne's proof of theWeil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal in1978 for his work on Weil conjectures.[89]

Ramanujan's notebooksFurther information: Ramanujan's lost notebook

While still in Madras, Ramanujan recorded the bulk of his results in four notebooks ofloose leafpaper. These results were mostly written up without any derivations. This is probably the originof the misperception that Ramanujan was unable to prove his results and simply thought up thefinal result directly. Mathematician Bruce C. Berndt, in his review of these notebooks andRamanujan's work, says that Ramanujan most certainly was able to make the proofs of most ofhis results, but chose not to.

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, and then transfer just theresults to paper. Using a slate was common for mathematics students in the Madras Presidency atthe time. He was also quite likely to have been influenced by the style ofG. S. Carr's bookstudied in his youth, which stated results without proofs. Finally, it is possible that Ramanujanconsidered his workings to be for his personal interest alone; and therefore recorded only theresults.[90]

The first notebook has 351 pages with 16 somewhat organized chapters and some unorganizedmaterial. The second notebook has 256 pages in 21 chapters and 100 unorganised pages, with thethird notebook containing 33 unorganised pages. The results in his notebooks inspired numerouspapers 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 BruceBerndt.[90]A fourth notebook with 87 unorganised pages, the so-called "lost notebook", wasrediscovered in 1976 by George Andrews.

[3]

Notebooks 1, 2 and 3 were published as a two-volume set in 1957 by the Tata Institute ofFundamental Research (TIFR), Mumbai, India. This was a photocopy edition of the originalmanuscripts, 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 collector's edition. These wereproduced from scanned and microfilmed images of the original manuscripts by expert archivistsofRoja Muthiah Research Library, Chennai.

Ramanujan

Hardy number 1729

The number 1729 is known as the HardyRamanujan number after a famous anecdote of theBritish mathematician G. H. Hardy regarding a visit to the hospital to see Ramanujan. In Hardy'swords:[91]

I remember once going to see him when he was ill at Putney. I had ridden in taxi cabnumber 1729 and remarked that the number seemed to me rather a dull one, and that Ihoped 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."

The two different ways are

1729 = 13

+ 123

= 93

+ 103.

Generalizations of this idea have created the notion of"taxicab numbers". Coincidentally, 1729is also a Carmichael number.

Other mathematicians' views of Ramanujan

Hardy said : "The limitations of his knowledge were as startling as its profundity. Here was aman who could work out modular equations and theorems... to orders unheard of, whose masteryof continued fractions was... beyond that of any mathematician in the world, who had found forhimself the functional equation of the zeta function and the dominant terms of many of the mostfamous problems in the analytic theory of numbers; and yet he had never heard of a doublyperiodic function or ofCauchy's theorem, and had indeed but the vaguest idea of what a functionof a complex variable was...".

[92]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."[93]Healso stated that he had "never met his equal, and can compare him only with Euler or Jacobi."[93]

Quoting K. Srinivasa Rao,[94]"As for his place in the world of Mathematics, we quote Bruce C.Berndt: 'Paul Erdshas passed on to us Hardy's personal ratings of mathematicians. Suppose thatwe rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself ascore of 25, J.E. Littlewood 30, David Hilbert 80 and Ramanujan 100.'"

Professor Bruce C. Berndt of the University of Illinois, during a lecture at IIT Madras in May2011, stated that over the last 40 years, as nearly all of Ramanujan's theorems have been proven
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right, there had been a greater appreciation of Ramanujan's work and brilliance. Further, hestated Ramanujan's work was now pervading many areas of modern mathematics andphysics.

[95][88]

In his bookScientific Edge, noted physicist Jayant Narlikar spoke of "Srinivasa Ramanujan,

discovered by the Cambridge mathematician Hardy, whose great mathematical findings werebeginning to be appreciated from 1915 to 1919. His achievements were to be fully understoodmuch later, well after his untimely death in 1920. For example, his work on the highly compositenumbers (numbers with a large number of factors) started a whole new line of investigations inthe theory of such numbers."

During his lifelong mission in educating and propagating mathematics among the school childrenin India, Nigeria and elsewhere, P.K. Srinivasan has continually introduced Ramanujan'smathematical works.

Recognition

Ramanujan's home state ofTamil Nadu celebrates 22 December (Ramanujan's birthday) as 'StateIT Day', memorializing both the man and his achievements, as a native of Tamil Nadu. A stamppicturing Ramanujan was released by the Government of India in 1962the 75th anniversary ofRamanujan's birthcommemorating his achievements in the field of number theory,

[96]and a

new design was issued on December 26, 2011, by the India Post.[97][98]

Since the Centennial year of Ramanujan, every year 22 Dec, is celebrated as Ramanujan Day bythe Government Arts College, Kumbakonam where he had studied and later dropped out. It iscelebrated by the Department of Mathematics by organising one-, two-, or three-day seminars byinviting eminent scholars from universities/colleges, and participants are mainly students of

Mathematics, research scholars, and professors from local colleges. It has been planned tocelebrate the 125th birthday in a grand manner by inviting the foreign Eminent Mathematicalscholars of this century viz., G E Andrews. and Bruce C Berndt, who are very familiar with thecontributions and works of Ramanujan.

Ramanujan's work and life are celebrated on 22 December at the Indian Institute of Technology(IIT), Madras in Chennai. The Department of Mathematics celebrates this day by organising aNational Symposium On Mathematical Methods and Applications (NSMMA) for one day byinviting Eminent Indian and foreign scholars.

A prize for young mathematicians from developing countries has been created in the name ofRamanujan by the International Centre for Theoretical Physics (ICTP), in cooperation with theInternational Mathematical Union, who nominate members of the prize committee. TheShanmugha Arts, Science, Technology & Research Academy (SASTRA), based in the state ofTamil Nadu in South India, has instituted the SASTRA Ramanujan Prize of $10,000 to be givenannually to a mathematician not exceeding the age of 32 for outstanding contributions in an areaof 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
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since 2005, at an international conference conducted by SASTRA in Kumbakonam, Ramanujan'shometown, around Ramanujan's birthday, 22 December.

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.

[99]

Dr Manmohan Singh also declared that the year 2012 would becelebrated as the National Mathematics Year.
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Documents

Srinivasa Ramanujan FRS