Quick Info
Born
22 December 1887
Erode, Tamil Nadu state,
India
Died
26 April 1920
Kumbakonam, Tamil Nadu state,
India
Summary
Ramanujan made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.
Srinivasa Ramanujan, a brilliant Indian mathematician, is renowned for his profound contributions to the analytical theory of numbers, elliptic functions, continued fractions, and infinite series.
Ramanujan was born in Erode, a small village around 400 km southwest of Madras (now Chennai), in his grandmother’s house. When he was one year old, his family moved to Kumbakonam, approximately 160 km closer to Madras. His father worked as a clerk in a cloth merchant’s shop in Kumbakonam. Unfortunately, in December 1889, young Ramanujan contracted smallpox.
Around the age of five, Ramanujan began attending primary school in Kumbakonam, although he switched between several primary schools before entering the Town High School in Kumbakonam in January 1898. At Town High School, he excelled in all subjects and proved himself as a well-rounded scholar. In 1900, he started exploring mathematics independently, particularly focusing on summing geometric and arithmetic series.
In 1902, Ramanujan was introduced to solving cubic equations, and he later developed his own method for solving quartic equations. The following year, he attempted to solve the quintic equation, unaware that it couldn’t be solved by radicals.
During his time at Town High School, Ramanujan discovered a mathematics book by G. S. Carr titled “Synopsis of Elementary Results in Pure Mathematics.” While this book allowed Ramanujan to teach himself mathematics due to its concise style, it also influenced his own mathematical writing style. Unfortunately, the book, published in 1886, had become outdated by the time Ramanujan used it.
By 1904, Ramanujan delved into deep mathematical research. He explored series like ∑(1/n) and accurately calculated Euler’s constant to 15 decimal places. He also started studying Bernoulli numbers independently.
Ramanujan’s excellent school performance earned him a scholarship to Government College in Kumbakonam in 1904. However, the following year, his scholarship was not renewed due to his increasing focus on mathematics at the expense of his other subjects. With no financial support and without informing his parents, Ramanujan ran away to Vizagapatnam, around 650 km north of Madras. There, he continued his mathematical work, exploring hypergeometric series and establishing connections between integrals and series, ultimately discovering that he had been studying elliptic functions.
In 1906, Ramanujan moved to Madras, enrolling in Pachaiyappa’s College with the goal of passing the First Arts examination, which would grant him admission to the University of Madras. However, after three months of study, he fell seriously ill, took the First Arts examination, and passed in mathematics but failed all other subjects, resulting in his overall failure. This setback prevented him from entering the University of Madras. Nevertheless, Ramanujan continued his mathematical work independently, with his main reference being Carr’s book.
In 1908, Ramanujan extended his mathematical explorations to include continued fractions and divergent series. Despite his dedication to mathematics, his health deteriorated, leading to an operation in April 1909. It took him a considerable amount of time to recover. During this period, he married a ten-year-old girl named S. Janaki Ammal, arranged by his mother. However, Ramanujan did not live with his wife until she reached the age of twelve.
Continuing his mathematical journey, Ramanujan began posing and solving problems in the Journal of the Indian Mathematical Society. In 1910, he established relationships between elliptic modular equations. His brilliant research paper on Bernoulli numbers, published in 1911 in the Journal of the Indian Mathematical Society, earned him recognition for his mathematical prowess. Despite his lack of a university education, Ramanujan was becoming well-known in the Madras region as a mathematical genius.
In 1911, Ramanujan approached the founder of the Indian Mathematical Society for advice on securing a job. Consequently, he received his first job, a temporary position in the Accountant General’s Office in Madras. Subsequently, it was suggested that he approach Ramachandra Rao, a Collector in Nellore and a founding member of the Indian Mathematical Society. Ramachandra Rao tried to help Ramanujan secure a scholarship, but this attempt was unsuccessful. In 1912, Ramanujan applied for the position of clerk in the accounts section of the Madras Port Trust. In his application, he expressed his passion for mathematics and his continuous self-study of the subject. Despite his lack of a university education, he impressed the hiring authorities, partly thanks to a recommendation from E. W. Middlemast, the Professor of Mathematics at The Presidency College in Madras.
Ramanujan was appointed as a clerk and commenced his duties on March 1, 1912. He was fortunate to work with several individuals trained in mathematics. S. N. Aiyar, the Chief Accountant for the Madras Port Trust, had a mathematical background and even published a paper on Ramanujan’s work regarding the distribution of primes in 1913. C. L. T. Griffith, the professor of civil engineering at the Madras Engineering College, recognized Ramanujan’s abilities and wrote to M. J. M. Hill, the professor of mathematics at University College London, about Ramanujan’s work.
In January 1913, Ramanujan wrote a letter to G. H. Hardy after seeing a copy of Hardy’s 1910 book “Orders of Infinity.” In his letter, Ramanujan introduced himself and shared some of his work, emphasizing his passion for mathematics and his pursuit of new ideas. Hardy, together with Littlewood, examined Ramanujan’s work, which included a long list of unproven theorems. Hardy replied to Ramanujan on February 8, 1913, expressing his interest but requesting proofs for many of Ramanujan’s assertions. He categorized Ramanujan’s results into three groups: known results, new but seemingly less important results, and results that appeared to be both new and significant. Despite his eagerness, Ramanujan faced challenges in providing the rigorous proofs Hardy sought.
The outbreak of World War I disrupted Littlewood’s teaching efforts, leaving Hardy to collaborate with Ramanujan more intensively. However, their collaboration was unique because every time Ramanujan was introduced to new mathematical concepts, he would flood Hardy with original ideas, making it challenging to stick to a structured teaching plan.
In 1916, Ramanujan completed his graduation from Cambridge with a Bachelor of Arts by Research, which was later recognized as a Ph.D. from 1920 onwards. His dissertation centered on highly composite numbers and consisted of seven papers he had published in England.
Ramanujan’s health began to decline significantly in 1917, with doctors fearing for his life. Although he showed some improvement by September, he spent most of his time in various nursing homes. In February 1918, he was elected as a fellow of the Cambridge Philosophical Society, followed
by his election as a Fellow of the Royal Society of London on May 2, 1918, a monumental honor. Additionally, he was elected a Fellow of Trinity College Cambridge, which would run for six years starting on October 10, 1918.
Despite his fragile health, Ramanujan’s honors seemed to have a positive impact, leading to a slight recovery. He resumed his mathematical endeavors and continued producing groundbreaking mathematics. He returned to India on February 27, 1919, but his health remained precarious, and he tragically passed away the following year.
Ramanujan’s letters to Hardy in 1913 contained numerous remarkable results. He independently rediscovered results of Gauss, Kummer, and others related to hypergeometric series. One of his most famous achievements was related to the partition function p(n), which counts the ways an integer n can be expressed as a sum of positive integers. Ramanujan’s work on p(n) led to profound discoveries and conjectures. Although some of his theorems about prime numbers were incorrect, his work in this area inspired further research.
Ramanujan’s legacy lives on through his unpublished notebooks, which contain theorems and results that mathematicians continue to explore. Mathematicians like G. N. Watson published numerous papers inspired by Ramanujan’s work. Although Ramanujan’s life was tragically cut short, his contributions to mathematics remain an enduring testament to his extraordinary talent.
