1887 – 1920 • The Man Who Knew Infinity
A self-taught genius from colonial India who produced results of extraordinary depth and beauty, filling notebooks with thousands of formulas that mathematicians are still proving a century later.
Srinivasa Ramanujan was born on 22 December 1887 in Erode, Tamil Nadu, into a Tamil Brahmin Iyengar family. His father worked as a clerk in a sari shop, and the family was poor.
By age 12, he had mastered trigonometry. At 15, he obtained a copy of Carr's "Synopsis of Elementary Results in Pure Mathematics" — a bare compendium of 5,000 formulas without proofs. This became his mathematical bible, and he set out to prove every result himself.
His obsession with mathematics cost him: he failed his college exams twice because he neglected all other subjects. He lived in poverty, dependent on the goodwill of patrons, while filling notebooks with extraordinary results.
Ramanujan filled three notebooks with nearly 3,900 results — mostly without proofs. Many were rediscoveries of known theorems; many others were decades ahead of their time.
He grew up in Kumbakonam, a temple town known as "the Cambridge of South India." His devout mother deeply influenced his spiritual outlook on mathematics.
Ramanujan emerged from colonial India, where access to advanced mathematical literature was extremely limited. The Indian mathematical tradition, while ancient (Aryabhata, Brahmagupta, Bhaskara), had been in decline for centuries.
In early 20th-century Cambridge, Hardy and Littlewood were leading a revolution in rigorous analytic number theory. Hardy's emphasis on proof stood in stark contrast to Ramanujan's intuitive, formula-driven approach.
World War I (1914-1918) coincided almost exactly with Ramanujan's time in England, adding hardship: vegetarian food was scarce, the climate was brutal for a South Indian, and loneliness compounded his declining health.
A devout Hindu Brahmin in wartime England, Ramanujan struggled with food, weather, isolation, and racism. He attempted suicide at least once during his time at Cambridge.
Ramanujan's style — intuitive, formula-based, often without proof — clashed with the rigorous Cambridge tradition. Hardy's greatest achievement may have been bridging these two worlds without destroying Ramanujan's genius.
p(n) counts the number of ways to write n as a sum of positive integers. Ramanujan discovered astonishing congruences and, with Hardy, an exact formula.
The 1918 paper with Hardy on p(n) introduced the revolutionary "circle method", one of the most powerful techniques in analytic number theory.
The idea was to express p(n) as a contour integral of the generating function and evaluate it by analysing singularities near roots of unity on the unit circle. The dominant contribution comes from points near e^(2πi/q) for small q.
Rademacher later refined this into an exact convergent series for p(n), one of the most remarkable formulas in all of mathematics.
Ramanujan discovered that p(n) satisfies striking divisibility patterns for n in certain arithmetic progressions modulo 5, 7, and 11. These are not coincidental — they reflect deep connections to modular forms.
Dyson's rank (1944) explained the mod 5 and mod 7 congruences combinatorially. Andrews and Garvan's crank (1988) finally explained all three, fulfilling a prediction Dyson made as a student.
Ono (2000) proved that p(n) satisfies congruences modulo every prime, extending Ramanujan's observation far beyond his original three primes.
Ramanujan produced formulas of breathtaking beauty, many of which were not proven until decades later.
Ramanujan's 1914 series for 1/π converges at a staggering rate: each term adds roughly 8 decimal digits. The Chudnovsky brothers' modification (1989), adding ~14 digits per term, was used to compute billions of digits of π.
These formulas connect to modular equations and singular moduli — special values of elliptic functions at complex multiplication points. The number 9801 = 99² and 396 = 4 × 99 are not arbitrary; they arise from the j-invariant at a specific CM point.
In his final letter to Hardy (January 1920), written on his deathbed, Ramanujan described 17 mysterious functions he called "mock theta functions."
For 80 years, nobody understood their true nature. In 2002, Sander Zwegers showed they are holomorphic parts of harmonic Maass forms, opening an entirely new chapter in modular form theory.
Mock theta functions now appear in string theory (black hole entropy), combinatorics, and algebraic geometry — fields that didn't exist when Ramanujan wrote them down.
Ramanujan studied the discriminant function Δ(q) = q Π(1-q^n)^24 = Σ τ(n) q^n, defining the Ramanujan tau function τ(n).
He conjectured three remarkable properties:
The tau function is the prototypical example of a modular form of weight 12 for the full modular group. Ramanujan's conjectures about it anticipated the entire Langlands program.
Modular forms encode arithmetic information: elliptic curves, L-functions, Galois representations. Ramanujan's intuition about τ(n) presaged the deepest connections in modern number theory, from Shimura-Taniyama to the proof of Fermat's Last Theorem.
τ(1)=1, τ(2)=−24, τ(3)=252, τ(4)=−1472, τ(5)=4830, τ(6)=−6048, ...
Calculate extensively by hand
Perceive deep patterns in numbers
State bold, precise formulas
Fill notebooks (often without proof)
Ramanujan rarely provided proofs in the conventional sense. Hardy estimated that about two-thirds of Ramanujan's results were rediscoveries; the remaining third were new, and of those, many were so deep that proofs came only decades later.
His working method involved extensive numerical computation on a slate (paper was expensive), erasing as he went — leaving only final results in his notebooks.
Ramanujan attributed his mathematical insights to the Hindu goddess Namagiri, saying she would appear in his dreams and present formulas. Whether taken literally or as metaphor for subconscious processing, this reflects a profoundly different relationship to mathematical creativity than the Western tradition.
"An equation for me has no meaning unless it expresses a thought of God."
— RamanujanRamanujan's health deteriorated rapidly in England. He was hospitalised repeatedly from 1917 onwards. The cold, damp English climate, vegetarian diet limitations during wartime, and profound homesickness took a devastating toll.
He was likely suffering from hepatic amoebiasis (an amoebic liver infection contracted in India), though for decades his illness was misdiagnosed as tuberculosis. He returned to India in 1919 but died just a year later, on 26 April 1920, at the age of 32.
Ramanujan faced every possible barrier: poverty, lack of formal training, colonial hierarchies, racial prejudice, religious dietary restrictions, and geographic isolation from the mathematical community.
That he produced work of the highest quality despite these obstacles speaks to an almost incomprehensible natural talent. Hardy rated mathematical ability on a scale of 100, giving himself 25, Littlewood 30, Hilbert 80, and Ramanujan 100.
After his death, a sheaf of papers with 600+ formulas was found — his work from the last year of life. Rediscovered by George Andrews in 1976 in the Trinity College library, it launched decades of new research.
The Ramanujan conjecture, Ramanujan sums, highly composite numbers, and the tau function remain central to analytic and algebraic number theory.
His work on modular equations, theta functions, and mock theta functions anticipated the Langlands program and modern automorphic forms theory.
The partition function, Rogers-Ramanujan identities, and q-series continue to inspire vast research programs in algebraic combinatorics.
Mock theta functions appear in black hole entropy calculations. Ramanujan's mathematics connects to theoretical physics in ways he could never have imagined.
His pi formulas (and their Chudnovsky descendants) have been used to compute trillions of digits of pi, pushing the frontiers of computational mathematics.
Ramanujan Day (Dec 22) is India's National Mathematics Day. His story has inspired books, films ("The Man Who Knew Infinity"), and generations of Indian mathematicians.
Modular forms and elliptic curves, deeply connected to Ramanujan's work, underpin modern elliptic curve cryptography (ECC) used in internet security.
Ramanujan sums and Ramanujan subspaces are used in period estimation and harmonic analysis of signals, with applications in speech and music processing.
The Rogers-Ramanujan identities appear naturally in the hard hexagon model in statistical mechanics, as discovered by Rodney Baxter (1981).
Mock modular forms appear in the computation of quantum black hole entropy (Dabholkar, Murthy, Zagier), connecting Ramanujan's deathbed mathematics to quantum gravity.
Robert Kanigel (1991). The definitive biography, vividly telling Ramanujan's life story and the Hardy-Ramanujan collaboration.
G.H. Hardy (1940). Hardy's own account of Ramanujan's mathematics, written with deep admiration and mathematical precision.
Bruce C. Berndt (1985-1998). Monumental effort to prove every result in Ramanujan's original notebooks. Essential mathematical reference.
George Andrews & Bruce Berndt (2005-2013). Analysis of the stunning "lost notebook" found in 1976.
Bruce C. Berndt (2006). Accessible introduction to Ramanujan's mathematics for advanced undergraduates.
"An equation for me has no meaning unless it expresses a thought of God."
— Srinivasa RamanujanSrinivasa Ramanujan • 1887–1920 • A flame that burned too bright and too brief