Srinivasa Ramanujan

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Srinivasa Ramanujan


Born 22 December 1887(1887-12-22)
Erode, Tamil Nadu, India
Died 26 April 1920 (aged 32)
Chetput, (Madras), Tamil Nadu, India

Residence Tamil Nadu, India
Fields Mathematician
Alma mater Trinity College, Cambridge
Academic advisors G. H. Hardy and J. E. Littlewood
Known for Landau–Ramanujan constant

Mock theta functions
Ramanujan conjecture
Ramanujan prime
Ramanujan–Soldner constant
Ramanujan theta function
Ramanujan's sum

Rogers–Ramanujan identities

Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan (Tamil: சீனிவாச இராமானுஜன் or ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (22 December 1887 – 26 April 1920) was an Indian mathematician and self taught genius who, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions.

Born and raised in Erode, Tamil Nadu, India, Ramanujan first encountered formal mathematics at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S L Loney.[1] He had mastered them by age 12, and even discovered theorems of his own. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant. He received a scholarship to study at Government College in Kumbakonam, but lost it when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself.[2] In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge. Only G. H. Hardy recognized the brilliance of his work, subsequently inviting Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge, dying of illness, malnutrition and possibly liver infection in 1920 at the age of 32.

During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identities and equations).[3] Although a small number of these results were actually false and some were already known, most of his claims have now been proven correct.[4] He stated results that were both original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research.[5] However, some of his major discoveries have been rather slow to enter the mathematical mainstream. Recently, Ramanujan's formulae have found applications in crystallography and string theory. The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.[6]

Early life

Ramanujan was born on 22 December 1887 in Erode, Tamil Nadu, India, at the residence of his maternal grandparents.[7] His father, K. Srinivasa Iyengar worked as a clerk in a sari shop and hailed from the district of Thanjavur.[8] His mother, Komalatammal or Komal Ammal (Ammal in Tamil is equivalent of Madam in English) was a housewife and also sang at a local temple.[9] They lived in Sarangapani Street in a traditional home in the town of Kumbakonam. The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son named Sadagopan, who died less than three months later. In December 1889, Ramanujan had smallpox and recovered, unlike thousands in the Thanjavur district who succumbed to the disease that year.[10] 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, but both children died before their first birthdays.

On 1 October 1892, Ramanujan was enrolled at the local school.[11] In March 1894, he was moved to a Telugu medium school. After his maternal grandfather lost his job as a court official in Kanchipuram,[12] Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School.[13] After his paternal grandfather died, he was sent back to his maternal grandparents, who were now living in Madras. He did not like school in Madras, and he tried to avoid going to school. His family enlisted a local constable to make sure he attended school. Within six months, Ramanujan was back in Kumbakonam.[13]

Since Ramanujan's father was at work most of the day, his mother took care of him as a child. He had a close relationship with her. From her, he learned about tradition and puranas. He learned to sing religious songs, to attend pujas at the temple and particular eating habits – all of which are part of Brahmin culture.[14] 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 finished first in the district.[15] That year, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the first time.[15]

By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book on advanced trigonometry written by S. L. Loney.[16][17] He completely mastered this book by the age of 13 and discovered sophisticated theorems on his own. By 14, he was receiving merit certificates and academic awards which continued throughout his school career and also assisted the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers.[18] He completed mathematical exams in half the allotted time, and showed a familiarity with infinite series. When he was 16, Ramanujan came across the book A Synopsis of Elementary Results in Pure and Applied Mathematics by George S. Carr.[19] This book was a collection of 5000 theorems, and it introduced Ramanujan to the world of mathematics. The next year, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal places.[20] His peers of the time commented that they "rarely understood him" and "stood in respectful awe" of him.[18]

When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than the maximum possible marks.[18] He received a scholarship to study at Government College in Kumbakonam,[21][22] However, Ramanujan was so intent on studying mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process.[23] In August 1905, he ran away from home, heading towards Visakhapatnam.[24] 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 Fine Arts degree exam in December 1906 and again a year later. Without a degree, he left college and continued to pursue independent research in mathematics. At this point in his life, he lived in extreme poverty and was often near the point of starvation.[25]

Adulthood in India

On 14 July 1909, Ramanujan was married to a nine-year old bride, Janaki Ammal.[26] – in the branch of Hinduism to which Ramanujan belonged,^[which?] marriage was a formal engagement that was consummated only after the bride turned 17 or 18, as per the traditional calendar. After the marriage, Ramanujan developed a hydrocele testis, an abnormal swelling of the tunica vaginalis, an internal membrane in the testicle.[27] The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac. His family did not have the money for the operation, but in January 1910, a doctor volunteered to do the surgery for free.[28] After his successful surgery, Ramanujan searched for a job. He stayed at friends' houses while he went door to door around the city of Madras (now Chennai) looking for a clerical position. To make some money, he tutored some students at Presidency College who were preparing for their F.A. exam.[29] In late 1910, Ramanujan was sick again, possibly as a result of the surgery earlier in the year. He feared for his health, and even told his friend, R. Radakrishna Iyer, to "hand these [my mathematical notebooks] over to Professor Singaravelu Mudaliar [mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the Madras Christian College."[30] After Ramanujan recovered and got back his notebooks from Iyer, he took a northbound train from Kumbakonam to Villupuram, a coastal city under French control.[31][32]

Attention from mathematicians

He met deputy collector V. Ramaswami Iyer, who had recently founded the Indian Mathematical Society.[33] Ramanujan, wishing for a job at the revenue department where Iyer worked, showed him his mathematics notebooks. As Iyer later recalled:

I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.[34]

Iyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras.[33] Some of these friends looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Mathematical Society.[35][36][37] Ramachandra Rao was impressed by Ramanujan's work, but doubted that it was actually his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understanding for his work, but concluded that he was not a phony.[38] Ramanujan's friend, C. V. Rajagopalachari, persisted with Ramachandra Rao and tried to quell any doubts over Ramanujan's academic integrity. Rao agreed to give him another chance, and he listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, which Rao said ultimately "converted" him to believe in Ramanujan's mathematical brilliance.[38] When Rao asked him what he wanted, Ramanujan replied that he needed some work and financial support. Rao consented and sent him to Madras. He continued his mathematical research with Rao's financial aid taking care of his daily needs. Ramanujan, with the help of Ramaswami Iyer, had his work published in the Journal of Indian Mathematical Society.[39]