How Srinivasa Ramanujan transformed mathematics, and left the world wondering how he did it

There is a prize for excellence in mathematics known as the Fields Medal (often considered the equivalent of the Nobel Prize in mathematics), established in 1936. It is awarded once every four years for outstanding work by mathematicians under the age of 40. Ramanujan could have won this prize more than once.

His pre-Cambridge days

We will not recount his biography here, as it has already been covered by science writers many times. He pursued mathematics on his own through books, without any guidance from peers. He was unable to talk or discuss his work with others because his circle did not include anyone who could even appreciate it, let alone understand it.

Also read: How cryptic Ramanujan still inspires, boggles minds of young researchers

However, his letter to G.H. Hardy in 1910 made all the difference. After a brief introduction, the letter contained pages of results. Some of these results were bizarre, and Hardy immediately realised that he was not looking at a letter from a crank but from an extraordinary genius.

This letter, with its remarkable contents, became a talking point in University of Cambridge for a few days after its arrival. Hardy felt that only an exceptional mind, deeply immersed in mathematical thinking, could produce such relations. For example, a couple of these relations are:

Only someone who understood how to handle a divergent series could write such an equation. For Prof. G. H. Hardy, it was a shock that such results could come from a person who openly declared that he had no formal training in mathematics. Hardy received 120 theorems in his first two letters from Srinivasa Ramanujan, many of which were new and unheard of, even to Hardy and J. E. Littlewood.

Also read: When Ramanujan did mathemagic with a taxi number

Hardy quickly arranged for Ramanujan to travel to University of Cambridge. The difficulties he faced in overcoming religious restrictions, and how he eventually reached London in 1914, are well documented. This was during World War I, which affected all the countries in Europe. Britain and its colonies were deeply involved in the war effort.

Amid such an eventful period, Ramanujan’s collaboration with Hardy and his team continued for the next four years. He produced 17 remarkable series for the value of π. In school textbooks, we often use π = 22/7, which is a crude approximation, correct only to two decimal places. Ramanujan’s series converge very rapidly, even the first two terms yield accuracy up to 16 decimal places. One such example is:

Based on his ideas, new series have been developed to compute the value of π to billions of digits using powerful computers.

The Cambridge Days

His time at University of Cambridge was difficult from the beginning, particularly because of the weather and the limited availability of strict vegetarian food. Fortunately, he had the support of his friend P. C. Mahalanobis (later the founder of the Indian Statistical Institute and an initiator of the Planning Commission), who was also a graduate student at the time. His presence seems to have helped Ramanujan adjust socially.

Another challenge was that his way of doing mathematics was very different from the formal approach followed there. For him, results were paramount, with less emphasis on proofs and formal logical derivations. He is believed to have remarked, “An equation means nothing to me unless it expresses a thought of God.” This can perhaps be interpreted as his desire to create equations whose beauty and symmetry convey deep insight.

For Srinivasa Ramanujan, all religions seemed equally true. His faith and beliefs have often been romanticised by Indian biographers, sometimes more than his mathematical contributions. Despite his spiritual outlook and G. H. Hardy’s strong atheism, the two had no difficulty collaborating. Bruce C. Berndt has also pointed out that people often exaggerate divine inspiration in Ramanujan’s work. In reality, he carefully recorded his results in three notebooks, while working out methods on a slate because he could not afford paper for permanent writing.

During his Cambridge years, Ramanujan published 21 papers, five of them in collaboration with Hardy. These works covered number theory, infinite series, and modular functions. The Hardy-Ramanujan formula provides an approximation for the number of ways an integer can be partitioned. The Hardy–Ramanujan–Littlewood circle method was also developed during this period.

Also read: A century later, why the world still needs Ramanujan

He was elected a Fellow of the Royal Society in 1918, just before his return to India in 1919. At the time, he was the second Indian to receive this honour (the first being Ardaseer Cursetjee). He was also the first Indian Fellow of Trinity College, Cambridge, and among the youngest Fellows ever elected.

Health issues

Srinivasa Ramanujan suffered greatly due to the harsh climate, lack of nutritious food, and depression caused by separation from his young wife and family. Even before going to Cambridge, he had serious liver problems that were largely ignored. In London, he was (mis)diagnosed with tuberculosis and vitamin deficiency and was confined to a sanatorium.

Because of his worsening physical and mental condition, he even attempted suicide. Hardy felt it would be best for him to recover in the company of his family and arranged his return to India. He left in March 1919. When he returned to Madras (now Chennai), he was severely emaciated, reduced to skin and bone, as noted by his wife Janaki Ammal. He stayed in Chetpet under the patronage of Namberumal Chetty, a well-known real estate businessman. His residence, known as “Gometra,” is mentioned in accounts such as those by S. Sriram in Madras Musings.

He received medical care from P. Chandrasekar of the Madras Medical College from September 1919 until his death in April 1920. Later assessments suggested that he may have suffered from hepatic amoebiasis rather than tuberculosis, though the exact diagnosis remains debated. American biographer Robert Kanigel, author of The Man Who Knew Infinity: A Life of the Genius Ramanujan (2016), supported tuberculosis as the likely cause while acknowledging the amoebiasis theory.

The final days

Even in extremely poor health, Ramanujan continued working on mathematics in near-complete isolation, assisted only by his wife. Despite pain and fever, he worked while propped up on pillows. His wife would hand him a slate whenever needed and carefully preserved the sheets of paper he produced in a leather suitcase he had brought from London.

A few months before his death, he wrote his last letter to Hardy describing his discovery of a new class of functions called mock theta functions. Remarkably, these functions were much later found to have applications in modern physics, including in counting the states of black holes in gravitational theory.

After crossing the seas, his community had distanced itself from him, and only a few close relatives and friends attended his funeral. He passed away at his residence in Chetpet, Chennai (contrary to some incorrect claims that he died in Kumbakonam).