Srinivasa Ramanujan: The Man Who Seemed to Understand Infinity

Introduction

In January 1913, the Cambridge mathematician Godfrey H. Hardy received an extraordinary letter from an unknown clerk working in Madras. The letter contained pages of mathematical formulas, many of which appeared without proofs or explanations.

Hardy initially wondered whether the writer was a fraud. However, after examining the formulas with his colleague John E. Littlewood, he realised that some were unlike anything they had previously seen.

The writer was Srinivasa Ramanujan, a largely self-taught Indian mathematician living in poverty. Within a few years, he would become a Fellow of the Royal Society and one of the most remarkable mathematicians of the twentieth century.

A Childhood Dominated by Numbers

Srinivasa Ramanujan was born on 22nd December 1887 in Erode, a town in southern India. He grew up mainly in Kumbakonam, where his father worked as a clerk in a cloth merchant’s shop, and his mother sang at a local temple.

Ramanujan displayed an exceptional memory and an intense interest in mathematics. By his early teenage years, he was working through advanced trigonometry and developing results independently.

His life changed when he obtained a copy of George Shoobridge Carr’s A Synopsis of Elementary Results in Pure and Applied Mathematics. The book contained thousands of mathematical results, but provided very few detailed proofs.

Most students would have used the book as a reference. Ramanujan treated it as a collection of challenges, attempting to prove each result and then discovering new formulas of his own.

The book's style may have influenced his later work. Ramanujan frequently recorded conclusions without explaining how he had reached them, creating serious difficulties for mathematicians trying to verify his discoveries.

Academic Failure and Mathematical Genius

Ramanujan performed brilliantly in mathematics, but he had little interest in other subjects. He received a scholarship to Government Arts College in Kumbakonam, but lost it because he neglected everything except mathematics.

He later enrolled at Pachaiyappa’s College in Madras, where the same problem occurred. His mathematical ability was obvious, but he failed the examinations required for a university degree.

For several years, Ramanujan lived in extremely difficult circumstances. He continued filling notebooks with formulas while struggling to find regular employment.

In 1909, he married Janaki Ammal, who was still a child at the time, as was customary in some Indian communities then. The responsibility of supporting a family increased his need to find paid work.

Ramanujan eventually obtained a position as a clerk at the Madras Port Trust in 1912. His employers recognised his unusual mathematical ability and allowed him some freedom to continue his research.

The Letter That Reached Cambridge

Ramanujan attempted to interest several British mathematicians in his work. Some failed to reply, while others could not understand the unusual collection of formulas he sent them.

On 16th January 1913, he wrote to Godfrey H. Hardy at Trinity College, Cambridge. Ramanujan introduced himself as a poorly paid clerk without a university education and included numerous mathematical results.

Hardy was already a respected number-theory specialist. He could see that some of Ramanujan’s formulas were familiar and that a few were incorrect, but others were entirely new.

The most remarkable results were so original that Hardy believed nobody could have invented them merely to deceive. A fraud would have required a mathematician of exceptional ability.

Hardy later said that he could compare Ramanujan only with great mathematicians such as Leonhard Euler and Carl Jacobi. He recognised that the unknown clerk possessed a level of mathematical intuition that appeared almost impossible to explain.

Coming to Cambridge

Hardy arranged for Ramanujan to travel to England, although religious and family concerns initially made the journey difficult. Ramanujan was an orthodox Hindu, and parts of his community discouraged him from crossing the sea. After receiving family approval, he left India in March 1914 and arrived in England the following month. He began working with Hardy and Littlewood at Cambridge.

The partnership between Hardy and Ramanujan brought together two very different approaches to mathematics. Hardy believed strongly in formal proof, while Ramanujan relied heavily upon intuition, experimentation and his extraordinary ability to recognise numerical patterns.

Hardy tried to teach Ramanujan the importance of rigorous proof without damaging the creative imagination that made him unique. This was not always easy because explaining an established theorem to Ramanujan could immediately inspire him to develop several new ideas. Their collaboration produced important work, particularly in number theory and the study of partitions.

What Are Partitions?

A partition describes the different ways in which a whole number can be written as a sum of positive whole numbers with the order of the numbers not mattering. For example, the number four has five partitions: 4; 3 + 1; 2 + 2; 2 + 1 + 1; and 1 + 1 + 1 + 1. As numbers become larger, the number of possible partitions increases extremely rapidly. The number 100 has more than 190 million partitions.

Ramanujan and Hardy developed a formula to estimate how many partitions a large number has. Their work introduced what became known as the circle method, which later became an important technique in analytic number theory.

Ramanujan also discovered remarkable patterns called congruences. For example, the number of partitions of certain numbers is always divisible by five, seven or eleven. These patterns were not obvious from the definition of partitions. They revealed a hidden structure that mathematicians have continued investigating ever since.

Infinite Series and Calculating π

Ramanujan discovered several rapidly converging infinite series involving π. An infinite series adds together an endless sequence of terms, with each term usually becoming progressively smaller. Some formulas approach their answer very slowly and require many terms to produce an accurate result. Ramanujan’s formulas could produce several correct digits of pi with each additional term.

His series later became important in computer calculations of pi. Modern algorithms inspired by Ramanujan’s work have been used to calculate trillions of decimal places. Ramanujan did not have an electronic computer or even a modern calculator. He reached these results through notebooks, a slate and an astonishing ability to recognise relationships between numbers.

Highly Composite Numbers

Ramanujan also studied numbers possessing more divisors than any smaller positive number. These are called highly composite numbers. The number twelve is highly composite because it has six positive divisors: 1, 2, 3, 4, 6 and 12. No smaller number has as many divisors.

The subject may appear simple, but Ramanujan’s treatment was deep and systematic. His paper on highly composite numbers extended beyond fifty pages and formed the basis of the research degree he received from Cambridge in 1916.

The work demonstrated that Ramanujan could produce detailed mathematical arguments when required, although his natural preference remained the rapid discovery of formulas and relationships.

The Story of 1729

One of the most famous stories about Ramanujan concerns a visit Hardy made while Ramanujan was ill. Hardy mentioned that he had arrived in a taxi numbered 1729 and thought it seemed a rather uninteresting number. Ramanujan immediately replied that 1729 was extremely interesting. It is the smallest positive number that can be expressed as the sum of two positive cubes in two different ways. It can be written as one cubed plus twelve cubed, and it can also be written as nine cubed plus ten cubed.

The story illustrates Ramanujan’s extraordinary familiarity with numbers. Where Hardy saw an ordinary taxi number, Ramanujan immediately recognised a rare mathematical property. The number 1729 is now commonly called the Hardy-Ramanujan number.

Recognition and Illness

Ramanujan was elected a Fellow of the Royal Society in May 1918 for his work on elliptic functions and number theory. At only thirty, he was among the youngest people elected to the Society.

Later that year, he became the first Indian elected to a Fellowship at Trinity College, Cambridge. These honours were exceptional for someone who had arrived in England without a conventional university education.

However, Ramanujan’s health deteriorated badly during his years in Britain. Wartime shortages made it difficult for him to maintain his strict vegetarian diet, while the cold climate, isolation and demanding workload probably added to his difficulties.

He spent long periods in hospitals and nursing homes. His exact illness remains disputed, although several diagnoses have been proposed, including tuberculosis, severe vitamin deficiency and complications from an earlier parasitic infection.

The Mock Theta Functions

Ramanujan returned to India in 1919, but he continued working despite his worsening health. In January 1920, he wrote his final important letter to Hardy. The letter described a new collection of mathematical objects that Ramanujan called mock theta functions. He provided examples and identities, but no complete general theory explaining them.

For decades, mathematicians struggled to understand exactly where these functions belonged. Later research connected them with harmonic Maass forms, representation theory, combinatorics and mathematical physics.

Mock theta functions have even appeared in research connected with black holes and string theory. Ideas recorded by a dying mathematician in 1920 therefore found unexpected applications in twenty-first-century physics.

The Lost Notebook

Ramanujan left behind several notebooks containing thousands of formulas. Mathematicians spent decades proving, correcting and extending the results recorded within them.

In 1976, the American mathematician George Andrews discovered a collection of Ramanujan’s later papers in the library of Trinity College. The collection became known as the lost notebook, although it had not literally been lost and was more accurately a group of unpublished manuscripts.

The pages contained hundreds of results from the final period of Ramanujan’s life, particularly concerning mock theta functions and related series. The discovery created enormous excitement because it opened new areas of research more than fifty years after Ramanujan’s death.

Intuition, Religion and Proof

Ramanujan was deeply religious and associated his mathematical inspiration with the Hindu goddess Namagiri. He reportedly believed that mathematical ideas were revealed to him through dreams and spiritual experiences. This has sometimes encouraged romantic accounts in which Ramanujan simply received complete formulas through divine intervention. The truth was probably more complicated.

Ramanujan worked constantly, performed extensive calculations and developed an exceptional memory for numerical relationships. His intuition was not a substitute for effort, but the result of an extraordinary mind concentrating almost entirely upon mathematics.

Some of his results were incorrect, and others had already been discovered. However, a remarkable number were new, profound, and eventually proved true.

An Exceptionally Short Life

Ramanujan died in Kumbakonam on 26th April 1920, aged only thirty-two. His career at the highest level of mathematics had lasted fewer than ten years. Despite this tragically short life, he produced nearly 4,000 recorded results. His work influenced number theory, infinite series, continued fractions, modular forms, combinatorics and mathematical physics.

Hardy regarded his association with Ramanujan as the most special period of his own life. When Hardy informally rated mathematicians according to natural ability, he gave himself a score of 25, Littlewood 30, the German mathematician David Hilbert 80 and Ramanujan 100.

Conclusion

Srinivasa Ramanujan’s life demonstrates that mathematical genius does not always develop through conventional education. He failed college examinations, lived in poverty and worked as a clerk, yet his notebooks contained discoveries that challenged the finest mathematicians in Cambridge.

His greatest strength was an almost unbelievable ability to recognise hidden patterns. Hardy supplied the discipline of proof, but Ramanujan supplied ideas that nobody else had imagined.

More than a century after his death, mathematicians still study his notebooks and uncover the meaning of his formulas. Some results have reached areas of mathematics and physics that did not exist during his lifetime.

Ramanujan did not merely solve existing problems. He left behind new mathematical worlds for later generations to explore, and that may be the clearest sign of his extraordinary genius.

Editor’s Note

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