The historical setting
Srinivasa Ramanujan Iyengar (22 December 1887 – 26 April 1920) was an Indian mathematician who worked during the early 20th century. He made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable. Throughout his life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations). 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 further research. Of his thousands of results, most have been proven correct.
The Ramanujan Journal, a scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan, and his notebooks—containing summaries of his published and unpublished results—have been analysed and studied for decades since his death as a source of new mathematical ideas.
Early life
Ramanujan (literally "younger brother of Rama", a Hindu deity) was born in British-ruled India on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode, in modern Tamil Nadu. His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a clerk in a sari shop. His mother, Komalatammal, was a housewife and sang at a local temple. They lived in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam. The family home later became 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.
He moved with his mother to her parents' house in Kanchipuram, near Madras (modern Chennai). His mother gave birth to two more children, in 1891 and 1894, both of whom died before their first birthdays. On 1 October 1892, Ramanujan was enrolled at the local school. After his maternal grandfather lost his job as a court official in Kanchipuram, Ramanujan and his mother moved back to Kumbakonam, and he was enrolled in Kangayan Primary School. When his paternal grandfather died, he was sent back to his maternal grandparents, then living in Madras.
Contacting British mathematicians
In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. M. J. M. Hill of University College London commented that Ramanujan's papers were riddled with holes. He said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the necessary educational background and foundation to be accepted by mathematicians. Although Hill did not offer to take Ramanujan on as a student, he gave thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.
On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest in his work, adding that it was "essential that I should see proofs of some of your assertions". Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to plan for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip. In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land", and his parents were also opposed for the same reason.
Ramanujan's notebooks
While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of looseleaf paper. They were mostly written up without any derivations. This is probably the origin of the misapprehension that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to prove most of his results, but chose not to record the proofs in his notes. This may have been for any number of reasons. Since paper was very expensive, Ramanujan did most of his work and perhaps his proofs on slate, after which he transferred the final results to paper.
At the time, slates were commonly used by mathematics students in the Madras Presidency. He was also quite likely to have been influenced by the style of G. S. Carr's book, which stated results without proofs. It is also possible that Ramanujan considered his work to be for his personal interest alone and therefore recorded only the results. The first notebook has 351 pages with 16 somewhat organised chapters and some unorganised material.