Srinivasa Ramanujan: Greatest Mathematical Genius of the Twentieth Century

In January 1913, a ten-page letter arrived at the office of G.H. Hardy, one of the most eminent mathematicians in Britain, at Trinity College, Cambridge. It had been sent by an unknown clerk at the Madras Port Trust in India, a man named Srinivasa Ramanujan who had no university degree, no institutional affiliation, no academic credentials of any kind.

Hardy almost dismissed it. Every prominent mathematician received letters from cranks — people who claimed to have squared the circle or solved the unsolvable. But something stopped him. He showed the letter to his colleague J.E. Littlewood, and the two of them spent the evening going through the 120 theorems it contained. By the end of the night, they had reached a conclusion that Hardy would later describe as self-evident: they were reading the work of a genius of the first rank.

“They must be true,” Hardy said of the formulas, “because if they were not true, no one would have had the imagination to invent them.”

That letter changed both their lives. It also changed mathematics — not just in the five years Ramanujan spent at Cambridge, but for the century that followed, and into the present day, where formulas he wrote while dying in a rented room in Madras are turning out to describe the physics of black holes and the mathematics of string theory.

Ramanujan died at 32. He left behind three notebooks, a lost fourth discovered half a century after his death, and a body of work so far ahead of its time that mathematicians are still catching up.

A Childhood in Kumbakonam: The Making of a Mathematical Mind

Srinivasa Ramanujan Iyengar was born on December 22, 1887, in Erode, a small town in Tamil Nadu, southern India. His family moved shortly after to Kumbakonam, where his father worked as a clerk in a cloth merchant’s shop. The family was Brahmin — caste Hindu — and devoutly religious. Ramanujan would remain deeply religious throughout his life, attributing many of his mathematical insights to the goddess Namagiri of Namakkal, whom his family worshipped. He described her as appearing to him in dreams, writing equations on his tongue. Whether this was metaphor, vision, or something else entirely, no one can say.

His mathematical gifts were apparent before he was a teenager. By eleven he had mastered the mathematics of the college students who boarded in his family home, exhausting their knowledge and asking them questions they could not answer. At sixteen he encountered a book — George Carr’s A Synopsis of Elementary Results in Pure and Applied Mathematics — that would shape his entire approach to mathematics.

It was an unusual book: a dense catalogue of thousands of mathematical theorems listed without proofs, designed as a reference tool for students preparing for Cambridge examinations. For most students it would have been a guide. For Ramanujan it was a challenge. He worked through every theorem, reconstructing the proofs himself, and then went beyond — filling the margins and extra pages with results of his own that Carr had never imagined.

The problem was that this approach — working alone, reconstructing foundations, generating results without proofs — was to define and also to limit him. He proved extraordinary things. But he proved them in ways that were often invisible even to himself, proceeding by intuition and pattern recognition that he could not always translate into the formal language of mathematical argument that the Western tradition required.

Years of Struggle: Poverty, Failure, and Notebooks

Ramanujan’s gifts did not translate into academic success. He won a scholarship to the Government College in Kumbakonam at fifteen, but failed his examinations the following year — not in mathematics, where he was untouchable, but in English and physiology, subjects he could not make himself care about. He lost his scholarship. He tried college again, failed again, and dropped out entirely.

For several years he lived in poverty in Madras, dependent on the charity of friends and the modest support of occasional patrons. He filled notebooks with mathematics — results, conjectures, identities, formulas — without income, without recognition, without anyone around him who could evaluate what he was producing. He married in 1909, on his mother’s arrangement, a girl named Janaki who was then about ten years old — a traditional Brahmin marriage of the time, in which the wife would join the household when old enough. He continued to fill his notebooks.

He applied for government jobs and support, writing to officials and mathematicians in India. Most did not know what to make of him. A few recognised that something unusual was present. Ramachandra Rao, a mathematician and government official in Madras, gave him a stipend for a period after hearing him explain his work — and later recalled that Ramanujan had shown him results on hypergeometric series that Rao had no framework to evaluate but could see were extraordinary. Eventually Ramanujan secured a clerkship at the Madras Port Trust, where he worked from 1912. He was a competent clerk. The notebooks continued to fill.

The Letter That Changed Everything

Ramanujan had been writing to mathematicians. Most did not reply. Two British mathematicians had written back in 1912, discouraging him without engaging seriously with his work. In January 1913, with the encouragement of colleagues at the Port Trust, he wrote to Hardy.

The letter that Hardy received was methodical in its presentation — pages of theorems, organised by topic, with no proofs and no explanation of method. Hardy and Littlewood recognised results that were known and correctly stated, results that were new and appeared to be true, and results that seemed initially impossible but turned out, on careful analysis, to be correct. There were also some that were simply wrong — Ramanujan’s intuition occasionally outpaced even his own remarkable accuracy. But the overall impression was overwhelming: this was not a crank. This was someone operating at a level Hardy had not encountered before.

Hardy immediately arranged support. The University of Madras granted Ramanujan a scholarship. E.H. Neville, a colleague of Hardy’s who was lecturing in India, visited Ramanujan and worked to overcome his hesitation about travelling to England — a significant obstacle, since orthodox Brahmin tradition discouraged crossing the sea. After prolonged persuasion, partly through the claimed intervention of his goddess Namagiri in a dream, Ramanujan agreed. He sailed from India on March 17, 1914. Hardy later wrote that the discovery of Ramanujan was the most romantic event of his mathematical life. “I have never done anything,” he said, “half as wonderful as discovering him.”

Cambridge: The Greatest Mathematical Collaboration of the Twentieth Century

Ramanujan arrived at Trinity College in April 1914. He was 26. England was cold, grey, and culturally alien. He was a strict vegetarian in a country where vegetarian food was barely understood. He cooked his own meals in his rooms, struggled with the climate, and missed India with a homesickness that those around him could see but could not fully address. He was not unhappy — his letters home were warm and sometimes funny — but he was out of his element in every way except the one that mattered most.

The collaboration with Hardy was immediate and productive. Hardy possessed exactly what Ramanujan lacked: a rigorous command of formal mathematical proof, an understanding of the contemporary landscape of European mathematics, and the analytical discipline to translate Ramanujan’s intuitions into publishable work. Ramanujan possessed exactly what Hardy lacked: an almost supernatural capacity for seeing mathematical patterns and relationships that no amount of formal training produced.

Hardy later attempted a famous assessment of mathematical talent on a scale of 0 to 100. He gave himself 25. He gave his collaborator Littlewood 30. He gave David Hilbert, the greatest mathematician of the previous generation, 80. He gave Ramanujan 100.

Between 1914 and 1919, Ramanujan produced over thirty research papers. Some were joint work with Hardy. The most celebrated was their 1918 paper on the partition function — the function that counts how many ways a positive integer can be expressed as a sum of positive integers. The number of partitions of 5, for example, is 7: the number grows rapidly and had no known formula. Hardy and Ramanujan developed what became known as the circle method — an elegant technique using complex analysis — to produce an asymptotic formula for the partition function of extraordinary accuracy. The result was immediately recognised as a masterwork. It founded a branch of mathematics, analytic number theory, that remains active today.

In 1918, Ramanujan was elected a Fellow of the Royal Society — one of the youngest people ever to receive the honour. A few months later he became a Fellow of Trinity College. He had arrived in England with no degree. He left as one of the most distinguished mathematicians in the world.

The Illness That Could Not Be Named

By 1917, Ramanujan was seriously ill. The diagnosis changed over the years — tuberculosis was the primary one, though some doctors suggested other conditions — and the cold, damp English climate was clearly not helping. He was admitted to multiple sanatoriums. He lost significant weight. There were periods when those around him feared he would not survive.

Hardy visited him regularly in hospital. On one famous occasion, Hardy arrived in a taxi whose number was 1729 — an unremarkable number, he commented. Ramanujan disagreed at once. It was the smallest number expressible as the sum of two positive cubes in two different ways: 1³ + 12³ = 9³ + 10³. This exchange — Ramanujan’s instant recognition of the property of a cab number, in a hospital bed while gravely ill — has become the most told story in the history of mathematics. It is told because it perfectly illustrates something true: Ramanujan did not do mathematics. He lived in it. Numbers were not objects he examined. They were a landscape he inhabited.

He returned to India in 1919 as his health improved slightly. He was thirty-one. He continued working. Even bedridden and weakening, he filled pages with new mathematics — including the results that would become, after his death, the contents of the lost notebook. He died on April 26, 1920, in Kumbakonam, at the age of thirty-two. He had been working until near the end.

The Three Notebooks and the Lost One

Ramanujan left behind three notebooks — physical books in which he had recorded his mathematical results from around 1903 onward. They contained approximately 3,900 results, most without proofs. The task of verifying and proving these results occupied mathematicians for decades. Bruce Berndt of the University of Illinois spent over twenty years working through them systematically, publishing his findings in five volumes. The majority of the results turned out to be correct. A small number were wrong or required significant modification. The process of proving them generated new mathematics — results in areas that Ramanujan had intuited but that required entirely new theoretical frameworks to rigorously establish.

But the most dramatic story belongs to the fourth notebook — what became known as the lost notebook. When Ramanujan returned to India in 1919, he brought with him a manuscript of approximately 138 pages containing work from his final months — results on mock theta functions, q-series, and other topics he had been developing in his last year. After his death the manuscript passed through several hands, was associated with the papers of the mathematician G.N. Watson, was nearly destroyed when Watson’s papers were due to be incinerated after his death in 1965, survived by chance, and was sent to the Wren Library at Trinity College Cambridge.

There it sat, unsorted and unidentified, until the spring of 1976, when the American mathematician George Andrews visited Cambridge for a conference and, on a suggestion from a colleague, examined the Watson papers. Andrews had written his doctoral dissertation on mock theta functions — precisely the subject of Ramanujan’s final work. When he began leafing through the manuscript, he recognised what he was looking at immediately. It was, he would later say, one of the most extraordinary moments of his professional life. The discovery was subsequently described as the mathematical equivalent of finding Beethoven’s tenth symphony.

The lost notebook contained over 600 formulas. Most had no proofs. None of the results had been published. And they were, in the judgment of every mathematician who studied them, among the most profound things Ramanujan had ever written — produced in the last year of his life, while dying, in a rented house in Madras.

What Ramanujan Actually Discovered: A Guide for Non-Mathematicians

Ramanujan worked across several areas of mathematics. A brief account of the most important gives a sense of what his achievement actually was.

The partition function. As described above, Ramanujan and Hardy’s work on counting the ways of partitioning numbers was foundational. But Ramanujan went further — he identified specific arithmetic patterns in the partition numbers, noticing that p(5n+4) is always divisible by 5, p(7n+5) is always divisible by 7, and p(11n+6) is always divisible by 11. These are called Ramanujan’s congruences, and they were initially simply observed — he saw the pattern in the numbers and wrote it down. Proving why they were true took decades. The deepest proof, using a theoretical framework that did not exist in Ramanujan’s time, arrived only in 2000.

Highly composite numbers. Ramanujan defined and characterised a class of numbers — numbers with more divisors than any smaller number — that had not been systematically studied before. His paper on this subject, published in 1915, founded a new area of number theory.

Infinite series for pi. Ramanujan discovered several rapidly converging infinite series that compute the value of pi. One of his formulas, discovered around 1910, converges so quickly that each additional term adds roughly eight more correct decimal places. When it was first published in the West it seemed impossible — and indeed it could not be proved rigorously until 1987. His series are now used in the fastest known algorithms for computing pi to billions of decimal places.

Mock theta functions. Perhaps Ramanujan’s deepest work, produced in his final year. Theta functions are a class of mathematical objects with beautiful symmetry properties, important in number theory, complex analysis, and physics. Ramanujan found functions that almost, but not quite, had these symmetry properties — functions that behaved like theta functions in some regions but broke the symmetry in others. He called them mock theta functions and described them in his last letter to Hardy.

What they actually were — what mathematical structure they represented — was not understood until 2002, when the mathematician Sander Zwegers proved they were the holomorphic parts of mathematical objects called harmonic weak Maass forms. This breakthrough revealed connections to modular forms that have since been found to describe the entropy of black holes and calculations in string theory. Ramanujan had found these objects by intuition eighty years before the theoretical framework needed to understand them existed.

The Source of Ramanujan’s Genius: What We Know and What We Don’t

Hardy wrestled throughout his life with the question of how Ramanujan did what he did. The standard framework for mathematical discovery — deep study of existing results, formal training in proof techniques, gradual extension of known methods — did not describe Ramanujan at all. He had not had access to most of the mathematics he was rediscovering and extending. He often produced results that were already known in Europe — but arrived at them independently, by routes that turned out to be entirely new. He produced results that were not yet known anywhere, that no one had even been looking for.

Hardy’s conclusion, reached reluctantly by a man who distrusted mystical explanations, was that Ramanujan possessed a form of mathematical intuition that had no parallel in his experience — an ability to perceive mathematical truths directly, without the scaffolding of proof, in a way that was unlike anything Hardy had encountered in a lifetime of working with gifted mathematicians. Whether this was neurological, cultural, the product of his specific isolated mode of study, or something else, Hardy could not determine.

What is clear is that the isolation was both Ramanujan’s limitation and, in a profound sense, his gift. Working alone in India, without access to contemporary European mathematics, he was forced to rediscover and reinvent from first principles. This meant he sometimes wasted effort on results that were already known. But it also meant that he approached problems from directions that no formally trained mathematician would have attempted — and occasionally those directions led somewhere new.

Ramanujan himself attributed his insights to the goddess Namagiri, to dreams, to visions. Hardy did not believe in supernatural intervention. But he also did not know how else to describe what he had witnessed. “I have sometimes wondered,” he wrote years after Ramanujan’s death, “whether there is something in the idea that, in some unexaminable way, he was receiving answers from beyond.”

The Legacy: Why Ramanujan Still Matters in 2026

A century after his death, Ramanujan’s work is not a historical curiosity. It is active mathematics, generating new research, opening new connections, and showing up in places that would have astonished even him.

His mock theta functions, understood through Zwegers’ 2002 framework, are now central to the physics of black holes. The entropy of certain black holes — the quantity that measures the disorder of matter as it falls past the event horizon — can be calculated using formulas that turn out to be expressible in terms of Ramanujan’s mock theta functions. In string theory, the mathematical structures Ramanujan identified appear in calculations of the quantum properties of strings. He was doing physics he did not know he was doing, a century before the physics existed.

His congruences for partition numbers have generated an entire field — the study of partition theory and its connections to modular forms — that continues to produce new results. A 2024 paper in the Annals of Mathematics by mathematicians building on Ramanujan’s circle method opened new directions in the distribution of prime numbers. The Rogers-Ramanujan identities — two beautiful formulas connecting infinite series that Ramanujan found independently in India and that had been proved by L.J. Rogers in 1894 without anyone reading the proof — turned out in the 1980s to describe exactly solvable models in statistical mechanics. They showed up in physics for reasons that are still not fully understood.

As the mathematician Ken Ono, who has spent decades working on Ramanujan’s legacy, puts it: “Ramanujan’s work is like a garden. Every time you think you understand a corner of it, you find another path leading somewhere new.”

The story of Ramanujan connects to broader questions about the nature of mathematical talent and where it comes from — questions that the history of science is full of, from Newton to Einstein to the teenage Galois writing out group theory the night before his fatal duel. For a look at another self-taught mathematical mind whose work transformed physics through a different kind of solitary intensity, see our article on Paul Dirac: the strangest man who predicted antimatter and built the foundation of modern physics. And for the story of the physicist whose joy in discovery most closely resembles Ramanujan’s relationship with mathematics, see our article on Richard Feynman: the Nobel Prize physicist who called curiosity his greatest scientific instrument.

Hardy’s Grief and the Question He Could Never Answer

G.H. Hardy outlived Ramanujan by twenty-seven years. He never fully recovered from the loss. In his 1940 book A Mathematician’s Apology — written late in his life, after illness had ended his research career — he returned repeatedly to Ramanujan as the measure of genuine mathematical greatness. The book is partly a meditation on what it means to have a mathematical gift and what it means to lose it.

“I can never forget the letter,” Hardy wrote. “It changed my life.” He described their collaboration as “the one romantic incident” of his career — and he meant this in the deepest sense, not the trivial one: an encounter with something that could not be explained, that exceeded the categories available for understanding it, that left a permanent impression on everything that came after.

Ramanujan’s wife Janaki survived until 1994, living quietly in Madras. She gave occasional interviews in her later years, speaking of a husband she had barely known — they had lived together for only a few years in total — but who she understood had been extraordinary. “He was not of this world,” she said in one late interview. “He belonged to mathematics.”

Frequently Asked Questions

Who was Srinivasa Ramanujan?

Srinivasa Ramanujan (1887–1920) was a self-taught Indian mathematician from Tamil Nadu who made extraordinary contributions to number theory, infinite series, partition theory, and continued fractions. He had no formal university training in mathematics but produced results of such depth and originality that G.H. Hardy, one of Britain’s leading mathematicians, brought him to Cambridge in 1914. He was elected a Fellow of the Royal Society in 1918 and died at 32, leaving behind thousands of results that mathematicians spent the following century proving and extending.

What is Ramanujan most famous for?

Ramanujan is most famous for his collaboration with G.H. Hardy on the partition function, his congruences for partition numbers, his infinite series for pi, his taxi-cab number story (1729), and his mock theta functions — the last of which turned out, decades after his death, to describe the entropy of black holes and calculations in string theory.

What was Ramanujan’s lost notebook?

Ramanujan’s lost notebook is a 138-page manuscript of mathematical results written in the last year of his life, while he was dying in India. After his death the manuscript passed through several hands and was nearly destroyed before being identified in 1976 by the American mathematician George Andrews at Trinity College Cambridge. It contains over 600 formulas, most without proofs, and is considered perhaps Ramanujan’s deepest work. Its discovery has been called the mathematical equivalent of finding Beethoven’s tenth symphony.

Did Ramanujan have any formal training in mathematics?

No. Ramanujan was almost entirely self-taught. He failed his university examinations because of his neglect of subjects other than mathematics, dropped out of college twice, and had no formal degree. His mathematical knowledge came from working through books independently and filling notebooks with his own results. When he arrived at Cambridge, Hardy had to work with him to develop the formal proof techniques that European mathematics required.

Why are Ramanujan’s formulas still being discovered today?

Ramanujan left behind thousands of results without proofs. Many required entirely new mathematical frameworks to understand — frameworks that did not exist in his time. The mock theta functions he described in his final year were not fully understood until 2002. Results from his notebooks continue to generate new mathematics because they often point toward connections between different areas of the subject that were not previously suspected.

How is Ramanujan’s work connected to modern physics?

Ramanujan’s mock theta functions, understood through modern modular form theory, appear in calculations of black hole entropy in string theory — a connection discovered decades after his death. His Rogers-Ramanujan identities describe exactly solvable models in statistical mechanics. His partition function work underlies calculations in quantum field theory. He was doing mathematics that turned out to describe physical reality in ways he could not have anticipated.

Sources

• Encyclopaedia Britannica — Srinivasa Ramanujan
• MacTutor History of Mathematics — Ramanujan Biography
• Quanta Magazine — Ramanujan Was a Genius. Math Is Still Catching Up (2024)
• Wikipedia — Ramanujan’s Lost Notebook
• Wikipedia — Srinivasa Ramanujan
• Andrews, George — Uncovering Ramanujan’s Lost Notebook: An Oral History
• Springer — Ramanujan’s Lost Notebook, Part I (Andrews & Berndt)
• Nobel Prize — context for Fellows of the Royal Society

About the Author

Baryon is the founder and editor of Web News For Us. Driven by a deep fascination with the biggest unanswered questions in science — from quantum physics and cosmology to the nature of consciousness and the genetic code written into every living cell — he has spent years studying modern physics, biology, and the history of scientific thought. He covers Science & AI, Space, Genetics & Research, and the timeless wisdom of history’s greatest thinkers and mystics.

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