Srinivasa Ramanujan: Life and Legacy in Advanced Number Theory

Srinivasa Ramanujan was an Indian mathematical genius whose contributions reshaped twentieth-century number theory, mathematical analysis, infinite series, and continued fractions. Born on December 22, 1887, in Erode in the Madras Presidency, Ramanujan demonstrated extraordinary numerical intuition during childhood in Kumbakonam despite receiving no formal advanced training in pure mathematics. His mathematical development accelerated upon encountering G. S. Carr's Synopsis of Elementary Results in Pure and Applied Mathematics at age sixteen. While working as an accounting clerk at the Madras Port Trust, Ramanujan compiled notebooks filled with thousands of unproved mathematical formulas. In early 1913, his correspondence containing approximately one hundred and twenty theorems reached English mathematician G. H. Hardy at Trinity College, Cambridge, who recognized his original formulations and arranged his travel to England in 1914.

During their celebrated five-year collaboration at Cambridge, Ramanujan and Hardy produced landmark discoveries in analytic number theory. Their most famous joint achievement was the Hardy-Ramanujan asymptotic formula for the partition function, published in 1918. The partition function calculates the number of distinct ways an integer can be expressed as a sum of positive integers. Ramanujan and Hardy devised the circle method, an analytical technique utilizing complex integration on the unit circle, to approximate partition numbers with unprecedented precision. Ramanujan independently identified partition congruences, demonstrating that partitions exhibit periodic divisibility properties modulo five, seven, and eleven. He also formulated the Ramanujan conjecture concerning the coefficients of the tau function in the Fourier expansion of modular forms, a problem that remained unsolved until Belgian mathematician Pierre Deligne proved it in 1974.

Ramanujan's work introduced original mathematical concepts that continue to influence theoretical physics, string theory, and computer algorithms. He investigated highly composite numbers, cataloging integers that possess more divisors than any preceding integer and establishing foundations for modern divisor analysis. During his final illness before passing away on April 26, 1920, at age thirty-two, Ramanujan wrote his last letter to Hardy introducing mock theta functions, which mimicked modular forms without sharing their full transformation properties. In 1976, American mathematician George Andrews rediscovered Ramanujan's Lost Notebook in the Wren Library at Trinity College, containing over six hundred mathematical formulas composed during the final year of his life. Modern supercomputers employ Ramanujan's rapidly converging infinite series to calculate the mathematical constant pi to trillions of decimal digits.

Reviewed by the Master10 Editorial Board for accuracy, clarity and competitive-exam relevance.Editorial Policy

Key Concepts & Self-Assessment20 Key Facts

Review key Srinivasa Ramanujan Number Theory exam facts and rate your mastery to track revision.

Progress: 0/20 Rated 0 Mastered 0 Review Later

#1
Srinivasa Ramanujan was born on December 22, 1887, in Erode, Madras Presidency, a date observed annually as National Mathematics Day in India since 2012.
#2
Ramanujan's mathematical style was deeply influenced by G. S. Carr's text, Synopsis of Elementary Results in Pure and Applied Mathematics, studied during his youth.
#3
While working at the Madras Port Trust under Francis Spring, Ramanujan received support from Indian Mathematical Society founder V. Ramaswamy Aiyer.
#4
In January 1913, Ramanujan initiated correspondence with English mathematician G. H. Hardy at Cambridge University, presenting roughly 120 original theorems.
#5
Ramanujan arrived at Trinity College, Cambridge, in April 1914 to begin an intense five-year mathematical partnership with Hardy and J. E. Littlewood.
#6
In 1918, Ramanujan became the second Indian elected as a Fellow of the Royal Society of London and the first Indian elected a Fellow of Trinity College, Cambridge.
#7
The Hardy-Ramanujan asymptotic formula published in 1918 established the first analytical solution for calculating integer partition function values.
#8
Hardy and Ramanujan invented the circle method to evaluate partition asymptotics, which became a foundational tool in modern analytic number theory.
#9
Ramanujan discovered fundamental partition congruences, demonstrating that the partition function p(n) is divisible by 5, 7, and 11 at regular mathematical intervals.
#10
The integer 1729 is known as the Hardy-Ramanujan number, defined as the smallest positive integer expressible as the sum of two cubes in two distinct ways.
#11
The decomposition of 1729 corresponds to the algebraic identities one cubed plus twelve cubed and nine cubed plus ten cubed.
#12
In 1915, Ramanujan published a pioneering paper on highly composite numbers, analyzing positive integers with more divisors than any smaller integer.
#13
Ramanujan derived rapidly converging infinite series for the inverse of pi, which function as mathematical foundations for modern computational algorithms.
#14
The Ramanujan conjecture regarding the growth rate of the tau function was formulated in 1916 and stood open for nearly six decades.
#15
Belgian mathematician Pierre Deligne proved the Ramanujan conjecture in 1974 as part of his proof of the Weil conjectures, earning the Fields Medal in 1978.
#16
In his final letter to Hardy in January 1920, Ramanujan introduced seventeen mock theta functions, whose algebraic framework was fully classified decades later.
#17
Mathematician George Andrews discovered Ramanujan's Lost Notebook in the Wren Library at Trinity College, Cambridge, in the spring of 1976.
#18
Ramanujan primes represent a sequence of prime numbers that guarantee the existence of at least a specified number of primes between x and x divided by two.
#19
Ramanujan passed away at the young age of thirty-two on April 26, 1920, in Chetput, Madras, due to complications from hepatic amoebiasis.
#20
The SASTRA Ramanujan Prize was established in 2005 to award young mathematicians under the age of thirty-two who make outstanding contributions to mathematics.

Subject Specialist Commentary

Analytical perspective & practical exam advice from the Master10 academic board

Educator's Insight

Srinivasa Ramanujan proved that mathematical genius can discover profound universal truths through pure numerical insight. Working without formal university training, he filled notebooks with thousands of equations on infinite series and partitions. His collaboration with G. H. Hardy united intuitive breakthroughs with rigorous analytical proofs, producing original formulas that continue to guide modern number theory, modular forms, and quantum physics.

In competitive examinations, questions on famous personalities test Ramanujan's landmark dates, honors, and specific mathematical concepts. Be careful not to confuse his Fellow of the Royal Society year (1918) or National Mathematics Day (December 22) with other national commemorations. Remember that 1729 is a taxicab number involving sums of two cubes. To recall his primary mathematical contributions, use the mnemonic PRIME: Partition asymptotic formula, Royal Society election in 1918, Integer 1729 taxicab property, Mock theta functions, and Equations computing pi.

Srinivasa Ramanujan: Life and Legacy in Advanced Number Theory

Key Concepts & Self-Assessment20 Key Facts

Subject Specialist Commentary

Related Knowledge Topics to Discover

Looking for more GK practice?