Srinivasa Ramanujan

Srinivasa Ramanujan (full name: Srinivasa Ramanujan Iyengar; 22 December 1887 – 26 April 1920) was an Indian mathematician widely regarded as one of the greatest mathematical geniuses in history, renowned for his deep intuitive results in number theory, infinite series, continued fractions, and mathematical analysis.

Born into a Tamil Brahmin family in Erode, in the Madras Presidency of British India, Ramanujan developed a passion for mathematics at an early age and produced thousands of results, many of which were entirely novel. Despite receiving little formal training beyond secondary school, he independently rediscovered a vast body of existing mathematics and went far beyond it. His collaboration with the British mathematician G. H. Hardy at the University of Cambridge brought his work to international attention and produced some of the most celebrated results in modern mathematics.

Early life and education

Ramanujan was born on 22 December 1887 in Erode, Tamil Nadu, to K. Srinivasa Iyengar, a clerk in a cloth merchant's shop, and Komalatammal, a housewife and devotee of the Hindu goddess Namagiri. The family soon moved to Kumbakonam, where Ramanujan grew up and attended local schools.

His mathematical talent became apparent by the age of eleven, when he is said to have surpassed the knowledge of older students who boarded with his family. By the time he was thirteen, he had mastered advanced trigonometry from S. L. Loney's Trigonometry and had begun devising his own theorems. At fifteen or sixteen he obtained a copy of George Shoobridge Carr's A Synopsis of Elementary Results in Pure and Applied Mathematics (1886), a compendium of roughly 5,000 theorems presented without proofs. Ramanujan worked through the book and used it as a launching pad for original research, a method that may partly explain his unconventional style of presenting results without detailed proofs.

In 1903 he earned a scholarship to Government Arts College, Kumbakonam, but lost it the following year because his obsession with mathematics led him to neglect other subjects. He failed his First Arts examination and spent several years in poverty, continuing to pursue mathematical research independently.

Marriage and early career

In 1909, following the custom of the time, Ramanujan's mother arranged his marriage to a ten-year-old girl, Janaki Ammal; the marriage was not consummated until she was older. In search of employment, Ramanujan moved to Madras (now Chennai) and eventually secured a clerical post at the Madras Port Trust in 1912, thanks partly to the support of Indian mathematicians and officials who recognised his unusual abilities. During this period he continued filling notebooks with original results.

Correspondence with G. H. Hardy and move to Cambridge

In January 1913, Ramanujan wrote his now-famous letter to G. H. Hardy, a distinguished number theorist and analyst at Trinity College, Cambridge. The letter contained about 120 statements of theorems, some already known and some entirely new, presented without proofs. Hardy, initially sceptical, showed the results to his colleague J. E. Littlewood; both concluded that the formulae must be the work of a mathematician of the highest calibre.

Hardy arranged for Ramanujan to travel to England, overcoming Ramanujan's initial reluctance on religious grounds (as a Brahmin, he had concerns about crossing the sea). Ramanujan arrived in Cambridge in April 1914 and began one of the most celebrated collaborations in the history of mathematics.

Collaboration with Hardy

Between 1914 and 1919, Ramanujan and Hardy published several joint papers and Ramanujan produced a large body of independent work. Their collaboration on the partition function culminated in the Hardy–Ramanujan asymptotic formula (1918), a landmark result that gave a precise approximation for the number of ways a positive integer can be written as a sum of positive integers.

Hardy later devised his famous genius scale, on which he rated himself at 25 and Ramanujan at 100 — though Hardy himself cautioned against taking such comparisons too literally.

Major mathematical contributions

Ramanujan's notebooks

Ramanujan recorded his findings in a series of notebooks — at least three compiled before his departure for England and a fourth (sometimes called the Lost Notebook, rediscovered by mathematician George Andrews in 1976 at the Wren Library, Cambridge) — containing thousands of results. Many of these took decades for other mathematicians to prove rigorously. The notebooks remain an active area of research.

Taxicab numbers

A celebrated anecdote involves Hardy visiting Ramanujan in a nursing home and remarking that the taxi he had arrived in bore the uninteresting number 1729. Ramanujan immediately replied that 1729 is in fact the smallest number expressible as the sum of two cubes in two different ways (1³ + 12³ = 9³ + 10³ = 1729). This incident gave rise to the concept of taxicab numbers in combinatorics.

Mock theta functions

In his final year, Ramanujan introduced the concept of mock theta functions in a letter to Hardy, written shortly before his death. The full significance of these functions was not understood until the early twenty-first century, when mathematicians connected them to the modern theory of harmonic Maass forms, vindicating Ramanujan's intuition across a gap of nearly a century.

Other results

• Highly composite numbers and superior highly composite numbers
• Ramanujan prime and Ramanujan–Soldner constant
• Rogers–Ramanujan identities (rediscovered independently)
• Ramanujan's master theorem
• Contributions to the theory of elliptic functions and modular forms, which later became important in the proof of Fermat's Last Theorem

Honours and legacy

In 1918, Ramanujan was elected a Fellow of the Royal Society — one of the youngest Fellows ever and among the first Indians to receive the distinction — and a Fellow of Trinity College, Cambridge.

His health had deteriorated seriously during his time in England, likely owing to a combination of nutritional deficiencies (he was a strict vegetarian and struggled to find suitable food during the First World War years), the cold climate, and possibly tuberculosis. He returned to India in 1919 and died on 26 April 1920 in Chetput, Madras, aged 32. The precise cause of his death remains debated by historians of medicine; a 1994 reanalysis suggested hepatic amoebiasis as a possible cause, though tuberculosis has historically been the most commonly cited explanation.

Ramanujan's birthday, 22 December, is celebrated in India as National Mathematics Day. The Government of India declared 2012 the National Mathematics Year to mark the 125th anniversary of his birth. Numerous institutions, prizes, and mathematical objects bear his name, and his life has been depicted in the 2015 biographical film The Man Who Knew Infinity, based on Robert Kanigel's 1991 biography of the same title.

Frequently asked questions

Was Ramanujan self-taught?

Ramanujan was largely self-taught, having had no formal university training in mathematics. His main early resource was Carr's Synopsis, and he developed most of his results independently before gaining access to the wider mathematical community.

Did Ramanujan attend Cambridge University?

Yes. Ramanujan arrived at Cambridge in April 1914 and remained there until 1919, working closely with G. H. Hardy and J. E. Littlewood. He was later elected a Fellow of Trinity College, Cambridge.

What is the Hardy–Ramanujan number?

The Hardy–Ramanujan number is 1729, the smallest positive integer expressible as the sum of two positive cubes in two distinct ways. The name commemorates the famous anecdote involving Hardy's taxi.

How old was Ramanujan when he died?

Ramanujan died on 26 April 1920 at the age of 32, having returned to India the previous year in failing health.

Are Ramanujan's notebooks still studied today?

Yes. Ramanujan's notebooks, including the Lost Notebook rediscovered in 1976, continue to be a rich source of research. Many results they contain have only been proved — or even fully understood — decades after his death.