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### Ramanujan

By Sean Whitear

In 1918 a sickly Indian mathematician was lying in a hospital bed in Putney, when fellow researcher G.H. Hardy paid him a visit. Knowing this unwell mathematician’s fascination for numbers, Hardy brought up the number of the cab he took on the way to the hospital, 1729, and claimed that it wasn’t particularly interesting. The man immediately piped up with conjecture, stating that the number WAS interesting, being the smallest number that can be expressed by the sum of two cubes in two different ways, being 123+13 and 93+103. While not a particular moment of realised genius, with the man making great strides in number theory, infinite series and continued fractions despite no formal training, it is an excellent example of the lifelong genius of Srinivasa Ramanujan.

Ramanujan was born on the 22nd of December 1887 in Erode, Madras Presidency to K. Srinivasa, a clerk, and Komalatammal, a housewife. At the age of two, Ramanujan contracted smallpox, beginning an infinite series of illnesses that were to plague him for his entire life. In this period Ramanujan spent large periods of time with his mother, who was heavily involved in the local Brahmin culture, sowing the seeds for Ramanujan’s deep spirituality as he grew into adulthood. Ramanujan enrolled in a local Primary school in 1892 and graduated in 1897, with high scores in mathematics.

In 1898 Ramanujan attended Town Higher Secondary School, where a true fascination in mathematics first began to form. By the age of 13 he was studying advanced trigonometry, having received material from two local college borders. In 1902 he independently found methods of solving cubic and quartic equations, and spent a good deal of time attempting to solve the quintic. In 1903 Ramanujan received what was to be his cornerstone text: “A Synopsis of Elementary Results in Pure and Applied Mathematics” by G.S. Carr. The book was vital in igniting Ramanujan’s consistent self-discovery, and formed the style in which Ramanujan published his theorems. Next year the budding mathematician graduated, receiving the R. Ranganatha Rao prize and received a scholarship to study at a nearby college.

Unfortunately, college life did not suit Ramanujan. While he excelled in mathematics in his first year, all his other courses were neglected, and consequently he soon failed and dropped out. The same thing happened the next year at another college. Determined to continue in his pursuit of mathematical discovery, he dropped out again in order to study independently. This was a local minimum for Ramanujan, as during this period he lived in extreme poverty. Despite these hazardous conditions, at this point Ramanujan began recording his theorems in what would eventually amount to four notebooks containing over 5000 theorems.

Returning home in 1910, Ramanujan began searching for clerical work, but by the end of the year he was ill again and required surgery. He recovered in 1911 and, despite relapsing, continued looking for work. In this endeavour, he demonstrated some of his theorems to a potential employer. These theorems were quickly sent to R. Ramachandra Rao of the Indian Mathematical Society. The papers were initially thought to be fraud, due to the astounding nature of the theorems, which were rarely provided with full proofs. However, after Rao met Ramanujan in person, the now clerk received funding to conduct research and began publishing problems in Mathematical Journals.

In early 1913 Ramanujan began writing to English mathematicians in order to share his theorems and research. While ignored by some, G.H. Hardy immediately noticed some extraordinary insight in the 100 theorems he was sent. While some had already been discovered and there were some errors, many theorems had never before been identified. While initially thought to be fraudulent again, Hardy was soon won over and requested Ramanujan travel to England in order to continue research. Ramanujan initially refused, with his travel being in conflict with his Brahmin beliefs and against his parents’ wishes, however by the end of the year, he resolved to go in the pursuit of enlightenment and with his parents’ wishes, travelled to England in March 1914.

Ramanujan arrived in April in 1914 and immediately began mathematical research with Hardy. Upon Ramanujan’s arrival, Hardy was again astonished upon gaining full access to Ramanujan’s notebooks. While both were accomplished mathematicians, the two had very different working methods. While Hardy’s methods focused heavily on logic and proofs, Ramanujan relied mainly on intuition and spontaneity. Ramanujan’s method of thought was very heavily spiritually influenced, believing that fully formed theorems were given to him divinely. However, despite these differences, the two managed to work efficiently and productively throughout Ramanujan’s stay in Europe, producing 21 papers in the four years. During these last few years, Ramanujan’s genius finally began to be realised, with the mathematician being awarded the BA Degree for Research from Cambridge University in 1916 and a Fellow of the Royal Society in February 1918.

Unfortunately, England itself was not so accommodating. The weather and lack of a sustainable vegetarian diet due to World War One rationing had a startling impact on the Indian’s health. He soon contracted tuberculosis and spent much of his remaining stay in and out of hospitals, consequently being the biggest obstacle in Ramanujan’s endeavours. In 1918 he began to recover and the next year moved back to India for his health. Unfortunately, he fell ill again and passed away in Madras on 26 April 1920 at the age of 32.

This could have been the end of Ramanujan’s story. However, due to the continued work of Hardy and his London associates, Ramanujan’s theorems continued to be investigated and proven, and with 600 theorems he wrote in the last year of his life, the genius even gifted the world of mathematics posthumously. Even now, ‘Ramanujan’s name is currently at the cutting edge of mathematics, which is pretty impressive when you consider that he’s been dead for 95 years’ (Heyer, Marie, 2015).

*Sean is in a double degree of civil and structural engineering and maths.*