The self-taught Indian genius whose extraordinary intuition for numbers produced results that continue to astonish and inspire mathematicians a century later.
Srinivasa Ramanujan Iyengar was born on 22 December 1887 in Erode, a small town in the Madras Presidency of British India (now in the state of Tamil Nadu). He was born into a Tamil Brahmin Iyengar family of modest means. His father, Kuppuswamy Srinivasa Iyengar, worked as a clerk in a sari shop in Kumbakonam, earning a meager salary. His mother, Komalatammal, was a devout woman who sang devotional songs at a local temple. Ramanujan grew up primarily in Kumbakonam, a town known for its numerous temples and as a center of traditional learning in South India.
From a very young age, Ramanujan displayed an exceptional aptitude for mathematics. He was a quiet, contemplative child who excelled at school without apparent effort. By the age of eleven, he had exhausted the mathematical knowledge of two college students who were lodging at his family's home. By twelve, he had mastered advanced trigonometry, and he began developing his own results and investigating mathematical problems on his own. His classmates and teachers recognized him as a prodigy, and he regularly won academic prizes.
A pivotal moment in Ramanujan's intellectual development came in 1903, when he obtained a copy of A Synopsis of Elementary Results in Pure and Applied Mathematics by George Shoobridge Carr. This book, a collection of approximately 5,000 mathematical formulas and results presented largely without proofs, became Ramanujan's primary mathematical companion. Working through the book, Ramanujan not only verified the results but began extending them, developing his own formulas and discovering new relationships. Carr's book, while unremarkable in itself, served as the springboard for Ramanujan's extraordinary mathematical explorations.
Ramanujan's obsession with mathematics came at a significant personal cost. After receiving a scholarship to attend the Government Arts College in Kumbakonam in 1904, he became so absorbed in his mathematical work that he neglected his other subjects and failed his examinations. He lost his scholarship and left the college without a degree. A second attempt at formal education at Pachaiyappa's College in Madras also ended in failure for the same reason: Ramanujan could not bring himself to study subjects other than mathematics.
Without a degree and with no formal credentials, Ramanujan struggled to find employment and support for his mathematical work. He spent several years in near poverty, living on the charity of friends and well-wishers while continuing to fill notebooks with mathematical results. These notebooks, which would later become famous, contained thousands of formulas, identities, and theorems that Ramanujan had discovered on his own, often without any knowledge of the existing mathematical literature. Many of the results were completely new; others were rediscoveries of known theorems; and some were so novel and unexpected that they would not be fully understood for decades.
During this difficult period, Ramanujan sought recognition from the mathematical community by writing to prominent Indian mathematicians and, later, to mathematicians in England. Some of his Indian contacts recognized his talent and helped him secure a modest position as a clerk in the Madras Port Trust in 1912. This position provided a small but stable income and gave Ramanujan time to continue his mathematical work. More importantly, it brought him into contact with several educated individuals who encouraged him to reach out to mathematicians in England.
"An equation for me has no meaning unless it expresses a thought of God." — Srinivasa Ramanujan, expressing his deeply spiritual approach to mathematics.
On 16 January 1913, Ramanujan wrote a letter to G. H. Hardy, a leading mathematician at Trinity College, Cambridge. The letter, which would become one of the most famous documents in the history of mathematics, contained a brief introduction of Ramanujan's circumstances and a selection of approximately 120 mathematical results. Ramanujan explained that he was a clerk in Madras with no university education and that he had been conducting mathematical research on his own. He asked Hardy to evaluate his work and, if it had merit, to help him gain recognition.
Hardy received the letter and initially set it aside, but something about the formulas caught his attention. He showed them to his colleague and collaborator J. E. Littlewood, and the two men spent the evening examining Ramanujan's results. Some of the formulas were already known, confirming that Ramanujan had independently rediscovered significant results. Others were completely new and astonishing in their depth and beauty. Hardy later recalled that the formulas had to be true because, if they were not, no one would have had the imagination to invent them. He concluded that Ramanujan was a mathematician of the highest caliber.
Hardy wrote back to Ramanujan, asking for proofs of some of the results and encouraging him to come to Cambridge. Arranging the trip was not straightforward. Ramanujan, as a devout Brahmin, initially had religious reservations about crossing the ocean. There were also practical obstacles, including the need for financial support and official permissions. With the help of several supporters in India, including the mathematician Seshu Aiyar and the philanthropist Narayana Iyer, as well as a scholarship from the University of Madras, these obstacles were eventually overcome. In March 1914, Ramanujan sailed for England.
Ramanujan arrived in London in April 1914 and soon settled into life at Trinity College, Cambridge. His collaboration with Hardy proved to be one of the most remarkable and productive partnerships in the history of mathematics. Hardy provided the rigorous training in modern mathematical methods that Ramanujan lacked, while Ramanujan brought an extraordinary intuition and an apparently inexhaustible supply of new ideas. The two men complemented each other perfectly, and their joint work produced some of the most important mathematical results of the early twentieth century.
The adjustment to life in England was challenging for Ramanujan. The climate was cold and damp, a stark contrast to the tropical heat of South India. As a strict vegetarian and devout Hindu, he had difficulty finding suitable food, particularly during the wartime rationing that began shortly after his arrival, as World War I broke out in August 1914. Ramanujan was socially isolated, with few Indians in Cambridge and limited opportunities for the kind of community life he had known in India. Despite these hardships, he threw himself into his mathematical work with extraordinary energy.
During his five years at Cambridge, Ramanujan published numerous papers, both independently and in collaboration with Hardy. He was elected a Fellow of the Royal Society in 1918, one of the youngest men and the first Indian to receive this honor for mathematical research. He was also elected a Fellow of Trinity College, the first Indian to achieve this distinction. These recognitions were remarkable achievements for a man who had arrived in England only four years earlier without a university degree.
Among the most celebrated results from the Hardy-Ramanujan collaboration was their work on the partition function, which counts the number of ways a positive integer can be expressed as a sum of positive integers. Their asymptotic formula for the partition function, published in 1918, was a tour de force of analytic number theory that combined deep insight with technical virtuosity. The formula provided remarkably accurate estimates for the number of partitions and introduced a new method, later known as the circle method, that became one of the most powerful tools in analytic number theory.
Ramanujan had an extraordinary gift for discovering formulas involving infinite series and continued fractions. Many of his results in these areas were completely unexpected and seemed to come from nowhere. He discovered rapidly converging series for computing the mathematical constant pi that are far more efficient than classical formulas. One of his series, rediscovered and refined in the 1980s by the Chudnovsky brothers, became the basis for algorithms that have been used to compute trillions of digits of pi, setting multiple world records.
His work on continued fractions was equally remarkable. Ramanujan discovered beautiful and deep identities involving continued fractions, some of which are connected to deep results in number theory and the theory of modular forms. His Rogers-Ramanujan identities, which relate certain continued fractions to infinite products, have become fundamental results with connections to statistical mechanics, representation theory, and the theory of vertex algebras.
Ramanujan's contributions to the theory of partitions were among his most important and enduring. Beyond the asymptotic formula developed with Hardy, Ramanujan discovered remarkable congruence properties of the partition function. He showed that the number of partitions of certain classes of integers is always divisible by 5, 7, or 11, depending on the residue of the number modulo these primes. These Ramanujan congruences were later proved rigorously and have led to deep developments in the theory of modular forms and algebraic number theory.
In his final letter to Hardy, written in January 1920 from India shortly before his death, Ramanujan described a new class of functions that he called mock theta functions. He provided a list of seventeen examples and described some of their properties, but he did not provide a rigorous definition or a general theory. For decades, mock theta functions remained mysterious and poorly understood. It was not until the early twenty-first century that the mathematician Sander Zwegers, building on work by others, developed a comprehensive theory that placed Ramanujan's mock theta functions within the framework of harmonic Maass forms. This breakthrough opened entirely new areas of research in number theory, combinatorics, and mathematical physics.
Ramanujan also made significant contributions to the study of highly composite numbers, positive integers that have more divisors than any smaller positive integer. His 1915 paper on this subject, one of the longest he published, provided a detailed analysis of the structure and distribution of these numbers. The work demonstrated Ramanujan's ability to combine computational skill with theoretical insight and has remained a standard reference on the subject.
Throughout his time in England, Ramanujan's health had been fragile. The cold climate, the difficulties of maintaining his vegetarian diet during wartime, and the relentless intensity of his mathematical work all took their toll. In 1917, Ramanujan was diagnosed with tuberculosis, though some modern researchers have suggested that he may actually have suffered from hepatic amoebiasis, a parasitic infection of the liver contracted in India before his departure. Whatever the precise diagnosis, his condition deteriorated steadily, and he spent much of 1917 and 1918 in various sanatoriums and nursing homes in England.
Despite his illness, Ramanujan continued to work on mathematics with remarkable determination. Some of his most profound results, including the work on mock theta functions, date from this period of declining health. Hardy visited him regularly, and it was during one such visit that the famous anecdote about the number 1729 occurred. Hardy mentioned that he had arrived in a taxi with the number 1729 and remarked that it seemed a rather dull number. Ramanujan immediately replied that on the contrary, 1729 is a very interesting number, as it is the smallest number expressible as the sum of two cubes in two different ways (1729 = 1 cubed + 12 cubed = 9 cubed + 10 cubed). The number 1729 has since been known as the Hardy-Ramanujan number.
In March 1919, Ramanujan returned to India, hoping that the warmer climate and familiar surroundings would restore his health. He settled in Madras (now Chennai), where he continued to work on mathematics despite his worsening condition. His final letters to Hardy, written in January 1920, contained the remarkable descriptions of mock theta functions that would occupy mathematicians for the next century. Ramanujan died on 26 April 1920, at the age of 32, leaving behind a body of work that, despite its relatively small volume, was of extraordinary depth and originality.
Ramanujan's mathematical output is preserved in several notebooks that he kept throughout his life, as well as in the papers he published during his short career. The first two notebooks, compiled before his departure for England, contain thousands of results, most presented without proof. A third notebook, sometimes called the "lost notebook," was discovered in 1976 by the mathematician George Andrews among the papers of G. N. Watson at the Wren Library of Trinity College, Cambridge. This notebook contained results from the last year of Ramanujan's life, including his work on mock theta functions.
The systematic study and verification of Ramanujan's notebooks has been a major undertaking in twentieth and twenty-first century mathematics. Bruce Berndt of the University of Illinois spent decades working through the first two notebooks, eventually publishing five volumes of detailed commentary that verified and proved virtually all of Ramanujan's results. Andrews and Berndt have similarly worked through the lost notebook, publishing multiple volumes. This massive scholarly effort has confirmed the extraordinary accuracy and depth of Ramanujan's work and has led to numerous new mathematical discoveries.
In India, Ramanujan is celebrated as a national hero. His birthday, 22 December, is observed as National Mathematics Day, and the Ramanujan Mathematical Society, founded in 1985, promotes mathematical research and education. The SASTRA Ramanujan Prize, awarded annually since 2005 to a mathematician under the age of 32 for outstanding contributions to areas influenced by Ramanujan, has become one of the most prestigious awards for young mathematicians worldwide.
Ramanujan's legacy in mathematics is immense and continues to grow. His results have found applications in areas far beyond anything he could have imagined, including string theory, black hole physics, computer science, and cryptography. The theory of modular forms, to which Ramanujan made fundamental contributions, has become one of the central areas of modern number theory and was essential to Andrew Wiles's 1995 proof of Fermat's Last Theorem. Ramanujan's conjectures and open problems continue to drive mathematical research, and new discoveries inspired by his work appear regularly in the mathematical literature.
The story of Ramanujan's life has captured the imagination of people far beyond the mathematical community. His journey from poverty and obscurity in colonial India to the highest levels of mathematical achievement in Cambridge is one of the most inspiring stories in the history of science. The 2015 biographical film The Man Who Knew Infinity, based on Robert Kanigel's 1991 biography, brought Ramanujan's story to a wide global audience. The film dramatized his relationship with Hardy and the cultural challenges he faced in England, introducing millions of viewers to the human side of mathematical genius.
Hardy himself, who was not given to overstatement, ranked Ramanujan alongside Euler and Jacobi in terms of pure mathematical talent. He described his collaboration with Ramanujan as the one truly romantic incident in his life. For Hardy, Ramanujan's ability to arrive at correct and profound mathematical results through intuition and a process that seemed to bypass conventional reasoning was both a source of wonder and a challenge to his own philosophy of mathematics, which emphasized rigor and proof above all else.
Perhaps the most remarkable aspect of Ramanujan's legacy is that much of his work remains only partially understood. Mathematicians continue to discover new depths in his formulas and identities, finding connections to areas of mathematics and physics that did not exist during his lifetime. The mock theta functions, which Ramanujan described in his final letter to Hardy, are now known to be connected to deep structures in algebraic geometry, representation theory, and quantum physics. As the mathematician Freeman Dyson observed, Ramanujan's formulas sometimes seem to arrive from the future, waiting for the mathematical theories that would eventually explain them.
Ramanujan's life and work stand as a powerful testament to the universality of mathematical talent. Born into circumstances that offered almost no opportunity for advanced education, working without access to libraries, mentors, or a mathematical community, Ramanujan nonetheless produced results that placed him among the greatest mathematicians in history. His story challenges assumptions about who can contribute to the highest levels of intellectual achievement and serves as a reminder that mathematical genius can emerge in the most unlikely of circumstances.