Srinivasa Ramanujan

Srinivasa Ramanujan (1887–1920) was an extraordinary Indian mathematician whose intuitive grasp of numbers and deep insight into mathematical patterns revolutionised number theory, infinite series, and mathematical analysis. Despite minimal formal training, his contributions transformed modern mathematics, earning him international acclaim and a place among the greatest mathematical geniuses in history. His life story, marked by brilliance, hardship, and an insatiable curiosity for numbers, continues to inspire mathematicians and scholars worldwide.

Early Life and Education

Ramanujan was born on 22 December 1887 in Erode, Tamil Nadu, and grew up in Kumbakonam, a temple town renowned for its cultural and intellectual traditions. His father, K. Srinivasa Iyengar, worked as a clerk in a cloth shop, and his mother, Komalatammal, was a devout housewife who influenced his spiritual outlook.
From an early age, Ramanujan displayed an exceptional talent for mathematics. At the age of 10, he topped his class at Town High School, and by 13, he had mastered advanced trigonometry from Loney’s textbook. He independently developed theorems on infinite series and prime numbers, astonishing his teachers with his originality and speed.
Despite his mathematical brilliance, Ramanujan’s focus on numbers led him to neglect other subjects, resulting in his failure in college examinations. This setback deprived him of a formal degree, but his passion for mathematics remained undiminished. Working with old notebooks and limited resources, he continued to produce remarkable theorems that would later astound professional mathematicians.

Mathematical Work and Discoveries

Ramanujan’s work encompassed several fields of pure mathematics, including number theory, continued fractions, infinite series, and modular functions. His style was characterised by intuition and originality rather than formal proof, a feature that made his results both mysterious and fascinating.
Some of his most celebrated contributions include:

• Ramanujan’s Prime Number Formula: An elegant approximation for the number of primes less than a given number, contributing to analytic number theory.
• Partition Function: His collaboration with G. H. Hardy produced an asymptotic formula for partition numbers, revealing deep insights into combinatorics.
• Ramanujan’s Tau Function (τ(n)): A highly complex function connected with modular forms, which later influenced developments in algebraic geometry and string theory.
• Mock Theta Functions: Introduced during the last year of his life, these functions became central to 20th-century research in q-series and modular forms.
• Ramanujan’s Constant and Continued Fractions: He derived numerous identities and approximations, including near-exact results for π (pi), some of which are used in modern computer algorithms.

His notebooks, comprising thousands of theorems without proofs, remain a treasure trove for mathematicians. Many of his formulas were later proved correct, highlighting his extraordinary intuition that seemed to transcend conventional logic.

Correspondence with G. H. Hardy

In 1913, Ramanujan sent a letter containing over a hundred theorems to Professor G. H. Hardy at the University of Cambridge. Initially sceptical, Hardy soon realised that the results displayed genius. Struck by their originality, he described them as “profound and entirely novel,” unlike anything seen before in Western mathematics.
Recognising Ramanujan’s potential, Hardy arranged for him to come to Cambridge University in 1914. Despite initial cultural and health challenges, Ramanujan thrived intellectually, collaborating closely with Hardy and producing some of his most influential work. Their partnership led to several landmark papers, including studies on:

• The partition function and asymptotic formulae.
• The circle method for estimating integer partitions.
• Highly composite numbers, a field that Ramanujan developed in great depth.

Hardy later described their collaboration as one of the most extraordinary in mathematical history and rated Ramanujan’s mathematical ability as being “a genius of the highest order.”

Life at Cambridge and Later Years

During his years at Cambridge (1914–1919), Ramanujan was elected a Fellow of the Royal Society (FRS) in 1918, one of the youngest in history and the first Indian to receive this distinction in mathematics. The same year, he became a Fellow of Trinity College, Cambridge.
However, his stay in England was fraught with difficulties. The harsh climate, unfamiliar diet, and the isolation of wartime Britain affected his health severely. He was a strict vegetarian, which made finding suitable food difficult, and he suffered from malnutrition and depression.
Diagnosed with tuberculosis and liver infection, Ramanujan returned to India in 1919. Despite declining health, he continued to work tirelessly, producing his “Lost Notebook”, which contained groundbreaking material on mock theta functions and q-series. He passed away on 26 April 1920, at the age of 32, in Kumbakonam.

The Lost Notebook and Posthumous Recognition

Decades after his death, Ramanujan’s “Lost Notebook” was rediscovered in 1976 by mathematician George Andrews at the Wren Library, Cambridge. The manuscript contained hundreds of formulas that have since influenced diverse mathematical fields, including string theory, black hole physics, and computer algorithms.
Modern mathematicians continue to explore his ideas, confirming the accuracy and depth of his work. The Ramanujan Journal, established in 1997, publishes research inspired by his discoveries, highlighting the ongoing relevance of his mathematical insights.

Style and Philosophy of Mathematics

Ramanujan’s approach to mathematics was profoundly intuitive. He often described his insights as divine revelations from Namagiri Thayar, a Hindu goddess he worshipped devotedly. To him, mathematics was not merely logical reasoning but a form of spiritual truth. He famously remarked, “An equation for me has no meaning unless it expresses a thought of God.”
His notebooks show minimal formal derivations, reflecting an instinctive grasp of numerical relationships that defied conventional methodologies. This blend of spirituality and mathematical creativity remains one of the most remarkable aspects of his personality.

Recognition and Legacy

Srinivasa Ramanujan’s legacy is celebrated worldwide. His life and work symbolise the triumph of intellectual brilliance over adversity and the universality of scientific knowledge. Among his many honours and recognitions are:

• Fellow of the Royal Society (1918) – First Indian mathematician to receive the honour.
• Fellow of Trinity College, Cambridge (1918).
• The Ramanujan Institute for Advanced Study in Mathematics, established in Chennai.
• The SASTRA Ramanujan Prize, awarded annually to young mathematicians for outstanding contributions in fields influenced by his work.
• National Mathematics Day, celebrated on 22 December each year in India, commemorating his birth anniversary.