During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations).[6] Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research.[7] Nearly all his claims have now been proven correct.[8] The Ramanujan Journal, a scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan,[9] and his notebooks—containing summaries of his published and unpublished results—have been analysed and studied for decades since his death as a source of new mathematical ideas. As late as 2011 and again in 2012, researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death.[10][11] He became one of the youngest Fellows of the Royal Society and only the second Indian member, and the first Indian to be elected a Fellow of Trinity College, Cambridge. Of his original letters, Hardy stated that a single look was enough to show they could have been written only by a mathematician of the highest calibre, comparing Ramanujan to mathematical geniuses such as Euler and Jacobi.
In 1919, ill health—now believed to have been hepatic amoebiasis (a complication from episodes of dysentery many years previously)—compelled Ramanujan's return to India, where he died in 1920 at the age of 32. His last letters to Hardy, written in January 1920, show that he was still continuing to produce new mathematical ideas and theorems. His "lost notebook", containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976.
A deeply religious Hindu,[12] Ramanujan credited his substantial mathematical capacities to divinity, and said the mathematical knowledge he displayed was revealed to him by his family goddess Namagiri Thayar. He once said, "An equation for me has no meaning unless it expresses a thought of God."[13]
Early life
Ramanujan (literally, "younger brother of Rama", a Hindu deity[14]:12) was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode, Madras Presidency (now Tamil Nadu, India), at the residence of his maternal grandparents.[14]:11 His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a clerk in a sari shop.[14]:17–18[15] His mother, Komalatammal, was a housewife and sang at a local temple.[16] They lived in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam.[17] The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later. In December 1889 Ramanujan contracted smallpox, but recovered, unlike the 4,000 others who died in a bad year in the Thanjavur district around this time. He moved with his mother to her parents' house in Kanchipuram, near Madras (now Chennai). His mother gave birth to two more children, in 1891 and 1894, both of whom died before their first birthdays.[14]:12
On 1 October 1892 Ramanujan was enrolled at the local school.[14]:13 After his maternal grandfather lost his job as a court official in Kanchipuram,[14]:19 Ramanujan and his mother moved back to Kumbakonam and he was enrolled in Kangayan Primary School.[14]:14 When his paternal grandfather died, he was sent back to his maternal grandparents, then living in Madras. He did not like school in Madras, and tried to avoid attending. His family enlisted a local constable to make sure he attended school. Within six months, Ramanujan was back in Kumbakonam.[14]:14
Since Ramanujan's father was at work most of the day, his mother took care of the boy, and they had a close relationship. From her he learned about tradition and puranas, to sing religious songs, to attend pujas at the temple, and to maintain particular eating habits—all part of Brahmin culture.[14]:20 At Kangayan Primary School Ramanujan performed well. Just before turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography and arithmetic with the best scores in the district.[14]:25 That year Ramanujan entered Town Higher Secondary School, where he encountered formal mathematics for the first time.[14]:25
A child prodigy 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 written by S. L. Loney on advanced trigonometry.[18][19] He mastered this by the age of 13 while discovering sophisticated theorems on his own. By 14 he received merit certificates and academic awards that continued throughout his school career, and he assisted the school in the logistics of assigning its 1,200 students (each with differing needs) to its approximately 35 teachers.[14]:27 He completed mathematical exams in half the allotted time, and showed a familiarity with geometry and infinite series. Ramanujan was shown how to solve cubic equations in 1902; he developed his own method to solve the quartic. The following year he tried to solve the quintic, not knowing that it could not be solved by radicals.
In 1903, when he was 16, Ramanujan obtained from a friend a library copy of A Synopsis of Elementary Results in Pure and Applied Mathematics, G. S. Carr's collection of 5,000 theorems.[14]:39[20] Ramanujan reportedly studied the contents of the book in detail.[21] The book is generally acknowledged as a key element in awakening his genius.[21] The next year Ramanujan independently developed and investigated the Bernoulli numbers and calculated the Euler–Mascheroni constant up to 15 decimal places.[14]:90 His peers at the time said they "rarely understood him" and "stood in respectful awe" of him.[14]:27
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.[14] He received a scholarship to study at Government Arts College, Kumbakonam,[14]:28[14]:45 but was so intent on mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process.[14]:47 In August 1905 Ramanujan ran away from home, heading towards Visakhapatnam, and stayed in Rajahmundry[22] for about a month.[14]:47–48 He later enrolled at Pachaiyappa's College in Madras. There he passed in mathematics, choosing only to attempt questions that appealed to him and leaving the rest unanswered, but performed poorly in other subjects, such as English, physiology and Sanskrit.[23] Ramanujan failed his Fellow of Arts exam in December 1906 and again a year later. Without an FA degree, he left college and continued to pursue independent research in mathematics, living in extreme poverty and often on the brink of starvation.[14]:55–56
In 1910, after a meeting between the 23-year-old Ramanujan and the founder of the Indian Mathematical Society, V. Ramaswamy Aiyer, Ramanujan began to get recognition in Madras's mathematical circles, leading to his inclusion as a researcher at the University of Madras.[24]
