SRINIVAS RAMANUJAN

An equation for me has no meaning unless if expresses a thought of God.                                                  

                                                                     -- Srinivas Ramanujan

Srinivasa Ramanujan Iyengar, known to most simply as Ramanujan as the other components of his name are surnames borne of various traditions, was born on 22 December, 1887 in the small village of Erode, India. Legend tells Ramanujan’s mother became pregnant several years after marriages only after her father prayed to the Goddess Namagiri to bless his daughter with offspring. The son of an accountant, Ramanujan was born into the Brahmin caste - the highest, and most orthodox Hindu, level of the Indian caste system. While he was deeply influenced by the traditions of this caste, his family, like most others in Southern India around this time, survived in relative poverty. Ramanujan was one of six children, three of whom died before their first birthdays. At age two Ramanujan contracted smallpox. Ramanujan survived. He would go on to live a short, often sickly, life. But Ramanujan's life was a life of extraordinary genius which would see him became India's foremost mathematician and a legend for the ages.

Ramanujan started school at age five. While “quiet and meditative”, his mathematical abilities were recognized quite early. He had a great ability to repeat all of the formulas and theorems he had been taught. He could recite the digits of pi and the square root of two to as many places as listeners could bear to hear. The apparently critical moment in Ramanujan’s mathematical development was at age 15 when he finally came into possession of a significant book of mathematics - A Synopsis of Elementary Results in Pure and Applied Mathematics by George S. Carr. This was a lengthy, two volume compendium that contained many of the important mathematical results through the middle part of the nineteenth century.

It was hardly a text from which one could learn - it was very terse, almost dictionary like. Yet Ramanujan set out to understand and establish all of the results in this text on his own. His obsession with mathematics kept him from fulfilling the scholarships that he had won to government colleges. He was inseparable from the two large notebooks that he filled with his mathematical ideas. Ramanujan would later claim that his inspirations came in the form of dreams from the Goddess Namakkal.

In the early part of 1909 Ramanjuan became quite ill. Later that year Ramanujan was married in an arranged marriage. He would not live with his wife, age nine, until she turned twelve. Throughout this time Ramanujan unwillingly accepted financial support from local benefactors so he could pursue his mathematics ruminations. One such benefactor described him as “Miserably poor... A short uncouth figure, stout, unshaved, not overclean, with one conspicuous feature - shining eyes... He never craved for any distinction. He wanted ... that simple food should be provided for him without exertion on his part and that he should be allowed to dream on.” When Ramanujan tired of this support he became an office clerk. But at each opportunity he sought to share his mathematical work with others who might be in a position to judge or appreciate it. 

On the heals of several fortuitous introductions, Ramanujan was encouraged enough to write to G.H. Hardy - then a Fellow at Trinity College, Cambridge and who would become one of the twentieth century’s most famous mathematicians. 

Ramanujan’s letter, dated 16 January, 1913, is a picture of modesty:

I beg to introduce myself to you as a clerk in the Accounts Department... I have no university education... but I am striking out a new path for myself... The results I get are termed by the local mathematicians as “startling”... [Yet] the local mathematicians are not able to understand me in my higher flights... I would request

you to go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published... Requesting to be excused for the trouble I give you, I remain, Dear Sir, Yours truly, S. Ramanujan.

The letter included more than 100 theorems that Ramanujan had discovered. They are fabulously intricate wonders such as:

Prof Hardy: [These formulas ] defeated me completely. I had never seen anything in the least like this before. A single look at them is enough to show they could only be written down by a mathematician of the highest class. They must be true because no one would have the imagination to invent them

Hardy immediately arranged for Ramanujan to come to England. Yet the prejudices of India’s caste system made Ramanujan feel that he could not accept Hardy’s invitation to come to Cambridge. Moreover, Ramanujan’s mother would not give her consent. Hardy

made every effort to encourage him to come to Cambridge, even enlisting his friends as allies.

While he was able to sway Ramanujan, it was not until Ramanujan’s mother announced that “she had a dream on the previous night, in which she saw her son seated in a big hall amidst a group of Europeans, and that the goddess Namagiri had commanded her not to stand in the way of her son fulfilling his life’s purpose.” On 17 March, 1914 Ramanujan sailed to England. In April he was admitted to Trinity College - a remarkable honor

Yet outside of mathematics Ramanujan did not prosper as well. He had a very difficult time adjusting to life in England. He was a strict vegetarian. He cooked all of his food himself and had difficulty obtaining food that was compatible with his usual diet. The climate of England was totally different from his native India. The stone buildings of Trinity College were cold and damp. Never having encountered such cold, Ramanujan slept in his overcoat, wrapped in a shawl. It was not until a fellow Indian student at Trinity realized that Ramanujan did not understand the purpose of the many blankets spread neatly on his bed that Ramanujan learned to lift up the blankets and slide under them to keep warm. England joined World War I. Trinity College housed open air hospitals for wounded soldiers and adequate vegetarian food became harder to come by. Ramanujan was often sick. In the spring of 1917 he became particularly ill. He was diagnosed with tuberculosis and placed in a sanatorium.

Despite his illness, his impact spread and his reputation grew. In May of 1918 he was elected as a Fellow to the Royal Society - one of the highest academic honors of the time. His election was all the more remarkable as it came on a first ballot, was the first time an Indian had been so honored, and it came at the remarkably young age of 30. Several other major honors were also bestowed on him during this year. These honors seemed to have buoyed his health for a short time and he continued to develop beautiful and important mathematical discoveries. 

Early in 1919 his health worsened again. He returned to India where he died on 26 April, 1920 at the age of 32. He had no children. He was survived by his wife and his parents.

The richness of Ramanujan’s mathematical legacy is in sharp contrast to trials of his brief and difficult life. Ramanujan’s own published works are of sufficient importance to consider him one of the elite mathematicians of the twentieth century. But his impact did not stop there. 

He left many notebooks full of unpublished theorems, results, and ideas. These notebooks have been intensely studied by mathematicians and have resulted in hundreds of papers whose contributions are direct results of the work laid out by Ramanujan. 

Indeed, almost eighty years later, the notebooks of Ramanujan served as the impetus of the major new discovery by Ken Ono that you investigated in Topic 5. Ono says that while he "was familiar with a lot of what he had done through the writings of more modern mathematicians, I didn't suspect that I would learn anything from studying Ramanujan's notes." 

However, one mathematical identity, written in a particularly obtuse fashion, even for Ramanujan, struck Ono. "This can't be right." Yet it was. This one identity helped Ono establish "spectacular" results that are "the most important work on partition congruences since the epic work of Ramanujan" and among the most notable of the past decade. Ono "learned a valuable lesson.

It sometimes really pays to read the original." Who knows how many other mathematical gems are still unearthed in Ramanujan's notebooks.