Indian mathematician (1887โ1920) who, with almost no formal training in advanced mathematics, independently produced thousands of original results in number theory, infinite series, continued fractions, and modular forms. His 1913 letter to Cambridge mathematician G.H. Hardy launched one of the most remarkable collaborations in mathematical history. He died at age 32, leaving notebooks that mathematicians are still fully deciphering over a century later.
In January 1913, a 25-year-old clerk in Madras (now Chennai), India, with no university degree, mailed a letter to G.H. Hardy, one of the most respected mathematicians at Cambridge University. The letter contained about 120 mathematical theorems and formulas, without any proofs โ just the results themselves, some of which looked bizarre or even wrong at first glance.
Ramanujan grew up in poverty in Tamil Nadu, India. He taught himself advanced mathematics largely from a single book โ a compendium of results called A Synopsis of Elementary Results in Pure and Applied Mathematics by George Carr โ working through and extending it entirely on his own, without formal guidance, developing an extraordinary and unconventional mathematical intuition.
Hardy arranged for Ramanujan to come to Cambridge in 1914, where the two collaborated intensely for about five years, producing groundbreaking papers together. However, Ramanujan struggled with the cold English climate, wartime food shortages, and cultural isolation, and fell seriously ill (likely tuberculosis, though some modern medical historians suggest other possibilities). He returned to India in 1919 and died the following year, aged just 32.
A famous story: while Ramanujan was ill in hospital, Hardy visited and mentioned his taxi's number, 1729, calling it "rather a dull number." Ramanujan immediately replied that it was actually fascinating โ it's the smallest number expressible as the sum of two cubes in two different ways: 1729 = 1ยณ + 12ยณ = 9ยณ + 10ยณ. Numbers with this property are now called "taxicab numbers" in his honour.
| Born | 22 December 1887, Erode, Tamil Nadu, India |
| Died | 26 April 1920, Kumbakonam, India (aged 32) |
| Key collaborator | G.H. Hardy, Cambridge University |
| Notebooks | 3 notebooks + "Lost Notebook" (rediscovered 1976), containing ~3,900 results |
Ramanujan and Hardy developed the Hardy-Ramanujan asymptotic formula for p(n), the partition function (the number of ways to write n as a sum of positive integers, ignoring order). Their formula gives a remarkably accurate approximation for p(n) even for large n, using an unusual "circle method" they developed together โ a technique that became foundational to later analytic number theory.
Ramanujan introduced a number-theoretic function ฯ(n) (the Ramanujan tau function) arising from the coefficients of a specific modular form (the discriminant function). He conjectured several deep properties of ฯ(n), some of which were only fully proved decades later using sophisticated tools, including Pierre Deligne's proof (1974) of the Ramanujan conjecture on the growth rate of ฯ(n), which followed from Deligne's proof of the Weil conjectures (earning Deligne a Fields Medal).
Ramanujan discovered remarkably rapid-converging infinite series for computing ฯ, including:
This series converges so quickly that each additional term adds roughly 8 correct decimal digits โ these series are the basis for some modern computer algorithms used to calculate ฯ to trillions of digits.
In his final letter to Hardy (1920), written while dying, Ramanujan introduced entirely new mathematical objects he called "mock theta functions" โ without a rigorous theoretical framework explaining what they were or why they worked. These functions remained mysterious for nearly 90 years until Sander Zwegers (2002, doctoral thesis) finally provided a rigorous theoretical explanation, connecting them to the modern theory of harmonic Maass forms.
In 1976, mathematician George Andrews discovered a sheaf of approximately 100 pages of Ramanujan's mathematical work in the Trinity College Library archive, previously overlooked and uncatalogued. Dubbed the "Lost Notebook," it contained roughly 600 additional formulas and results from the final year of Ramanujan's life, including the mock theta functions and numerous other results whose significance is still being explored by researchers over four decades after its rediscovery. Bruce Berndt and George Andrews have spent much of their careers systematically working through and proving Ramanujan's results across his notebooks.
A distinctive and much-discussed feature of Ramanujan's work is that his notebooks record results with essentially no accompanying proofs โ a striking departure from standard mathematical practice. Historians and mathematicians (Berndt, Hardy) attribute this partly to Ramanujan's largely self-directed education (learning primarily from Carr's results-focused compendium, which itself provided minimal proofs) and partly to his own described experience of mathematical insight, which he sometimes attributed to intuition or even, in his own religious framework, to the goddess Namagiri โ a claim that reflects his personal spiritual worldview and is reported respectfully by historians without either dismissing or mathematically endorsing the framework itself.
A substantial and ongoing project in 20th and 21st century number theory has been systematically verifying and proving Ramanujan's unproved claims. The overwhelming majority have been confirmed as correct once proper proofs were constructed โ a track record considered remarkable given the informal, intuition-driven manner in which the results were originally recorded. A small number of claims have been found to be incorrect or imprecisely stated, which is itself notable given the sheer volume and difficulty of the results involved.
Ramanujan's work on modular forms and mock theta functions has found unexpected applications well beyond pure number theory, including in string theory and the physics of black holes (through connections to modular forms discovered by physicists including work connecting to the Monstrous Moonshine phenomena), demonstrating how abstract number-theoretic structures can resurface in seemingly unrelated areas of theoretical physics decades after their original discovery.
Ramanujan was elected a Fellow of the Royal Society in 1918 (one of the youngest Fellows in the Society's history, and the first Indian to be elected) and a Fellow of Trinity College, Cambridge, the same year โ remarkable achievements for someone who had arrived in England only four years earlier with no formal university degree.