Srinivasa Ramanujan

 Srinivasa Ramanujan or Srinivasa Ramanujan Aiyangar (22 December 1887 – 26 April 1920) was an Indian mathematician whose intuitive brilliance in number theory, infinite series, and continued fractions reshaped 20th-century mathematics. Largely self-taught, he produced nearly 3,900 results — identities, theorems, and equations — most of them without formal proofs, before dying at the age of 32. His notebooks continued to yield new mathematics for decades after his death.

Early life and self-education

Ramanujan was born in Erode, Tamil Nadu, into a Tamil Iyengar family; his father, K. Srinivasa Iyengar, worked as a clerk in a cloth shop, and his mother, Komalatammal, sang in a local temple^[1]. The family moved to Kumbakonam when he was young, and it was there that Ramanujan attended Town Higher Secondary School. He won academic prizes for mathematics and showed a talent that quickly outgrew his school curriculum^[2].

At age 16, Ramanujan obtained G. S. Carr's A Synopsis of Elementary Results in Pure and Applied Mathematics, a dense volume containing roughly 6,000 theorems with almost no proofs. He did not simply memorise them — he derived the missing proofs himself, often inventing entirely new approaches. This book became the foundation of his mathematical thinking, and by his late teens he was investigating Bernoulli numbers and had calculated the Euler–Mascheroni constant to 15 decimal places^[3].

He won a scholarship to Government Arts College in Kumbakonam in 1904, but his single-minded devotion to mathematics caused him to fail every other subject, and he lost his scholarship by 1906. He left Pachaiyappa's College in Madras without a degree and spent years in financial hardship, supported briefly by a research fellowship from the University of Madras^[4].

First publications and the letter to Hardy

Ramanujan's first paper, a study of Bernoulli numbers, appeared in the Journal of the Indian Mathematical Society in 1911 and startled readers with its originality^[5]. By then he was working as a clerk at the Madras Port Trust, where colleagues encouraged him to reach out to leading British researchers. On 16 January 1913, aged 25, Ramanujan wrote to G. H. Hardy at Trinity College, Cambridge, enclosing 120 theorems^[6].

Hardy initially feared the letter came from a crank, but after hours of careful analysis he and J. E. Littlewood concluded the results were genuine — some were already known, many were not. Hardy later wrote that the theorems "must be true, because if they were not true, no one would have had the imagination to invent them." He invited Ramanujan to Cambridge, and in March 1914 the mathematician sailed for England^[7].

Cambridge and the Hardy collaboration

Ramanujan arrived at Trinity College in 1914, just months before the First World War. Over the next five years he and Hardy produced more than a dozen joint papers, the most influential of which concerned the partition function p(n) — the number of ways a positive integer can be expressed as a sum of smaller integers. Together they invented the circle method, which gave the first accurate approximations for partitions of large numbers and remains one of the central tools of analytic number theory today^[8]. In the same work they discovered three remarkable congruences for p(n) — divisibility patterns by 5, 7, and 11 that resisted proof for years and eventually opened entire research programmes^[9].

In 1916 Ramanujan earned a BA by research from Cambridge for his thesis on highly composite numbers — integers with more divisors than any smaller number. In 1918, at the age of 31, he was elected a Fellow of the Royal Society, one of the youngest ever, and became the first Indian Fellow of Trinity College^[10]. Littlewood observed that every positive integer was one of Ramanujan's "personal friends," a remark that captured his uncanny ability to remember the idiosyncrasies of numbers^[11].

The taxi number 1729

The most famous anecdote from Ramanujan's life occurred when Hardy visited him in a nursing home in Putney, London. Hardy remarked that the taxicab he had ridden in, number 1729, seemed dull and hoped it was not an unlucky omen. Ramanujan replied instantly that it was in fact highly interesting — it was the smallest number expressible as the sum of two cubes in two different ways: 1729 = 1³ + 12³ = 9³ + 10³^[12]. Numbers with this property are now called taxicab numbers, and 1729 is known as the Hardy–Ramanujan number. Ramanujan had already noted this identity in his notebooks years before the hospital visit^[13].

Mock theta functions and the Lost Notebook

Ramanujan's health deteriorated in England's cold climate. A strict vegetarian, he struggled to eat properly and suffered prolonged malnutrition. He returned to India in 1919 and, in his final months, wrote a letter to Hardy on 12 January 1920 introducing 17 functions he called mock theta functions — objects resembling classical theta functions but lacking complete modularity. Mathematicians could not understand them for nearly a century; it was only with Sander Zwegers' 2002 thesis that they were placed within the framework of harmonic weak Maass forms, opening new directions in number theory and mathematical physics^[14].

After his death, a sheaf of 138 pages containing roughly 600 formulas — the Lost Notebook — was discovered in 1976 by George Andrews among G. N. Watson's papers at Trinity College, a find Andrews compared to discovering Beethoven's tenth symphony^[15]. Ramanujan also published rapidly converging series for π in a 1914 paper, formulas that would not power actual computation until the mid-1980s and underpin modern π-algorithms^[16]. He developed a theory of divergent series based on the Euler–Maclaurin formula and assigned 1 + 2 + 3 + ⋯ the value −1/12, later formalised as Ramanujan summation^[17].

Death and legacy

Ramanujan died on 26 April 1920 in Kumbakonam at the age of 32^[18]. On 26 December 2011, Prime Minister Manmohan Singh declared 22 December — Ramanujan's birthday — National Mathematics Day during the 125th birth anniversary celebrations at the University of Madras, observed every year since 2012^[19]. The Ramanujan Institute for Advanced Study in Mathematics was founded at the University of Madras in 1950 as a memorial to his work, and the SASTRA Ramanujan Prize, awarded annually since 2005 to mathematicians aged 32 or younger, keeps his name at the centre of contemporary research^[20]. His ideas continue to influence cryptography, string theory, black hole physics, and algorithm design — a lifetime of insight compressed into 32 years^[21].