Srinivasa Ramanujan, The Greatest Mathematical Autodidact
Srinivasa Ramanujan is widely regarded as one of the finest mathematicians of the 20th century. Despite the fact that he was a self-taught mathematician with a short 32-years lifespan, he made major contributions to Pure Mathematics in a variety of domains, including mathematical analysis, number theory, infinite series, and continued fractions, that only a few could overtake in their lifetime. He was a highly devout Indian guy who attributed his talent and capability to his family goddess Namagiri Thayar (a version of the Hindu goddess Laxmi).
Ramanujan was born on December 22, 1887, in Erode, India, a small village in the country’s southern part, at the residence of his maternal grandparents. His father, Kuppuswamy Srinivasa, was a sari shop clerk, while his mother, Komalatammal, was a housewife who also sang in nearby temples. His mother taught him about Brahmin culture, and he followed. From his childhood, he was deeply committed to his faith, culture, and customs.
Ramanujan’s interest in mathematics began at an early age although he did not come from a mathematical background. He first encountered formal mathematics when he was 10 years old. He also learnt mathematics from two college students who were staying as lodgers at his home. Throughout his scholastic career, he showed excellent performance in mathematics.
In 1903, when he was 15 years old, he obtained a copy of a book called “Synopsis of Elementary Results in Pure and Applied Mathematics”. It was written by G. S. Carr in 1886. This book contains a compilation of about 5000 theorems (most of which have only a brief proof or no hint of proof) for the majority of the fundamental mathematics known at the time. It was not a mathematical classic, rather intended to help English mathematics students prepare for the infamously tough Tripos examination. But Ramanujan, on the other hand, became so inspired that he began developing his own theorems and proofs. This book is considered to have awakened the genius. The following year, he independently created and analyzed the Bernoulli numbers as well as computed the Euler– Mascheroni constant up to 15 decimal places. In the same year, 1904, Ramanujan graduated from Town Higher Secondary School with an award for mathematics. He was then granted a scholarship to study at the Government Arts College, Kumbakonam. However, he never paid enough attention to other subjects. He spent most of his time doing maths. As a result, he failed the majority of other subjects, causing him to lose his scholarship. Then he left Government Arts College and enrolled at Pachaiyappa’s College in Madras, but for the same reasons, he dropped out. He began studying himself and carried out his research.
But, Ramanujan suffered from acute poverty as a result of being unemployed. He needed a job to support himself and his family. He was also in need of someone who could comprehend his works. Ramachandra Rao, the secretary of the Indian Mathematical Society at the time, later acquired his works. Ramanujan’s works influenced Rao. He wished to assist him with his research. He provided Ramanujan financial assistance. In 1911, Ramanujan published his first papers in the Journal of the Indian Mathematical Society. It was 17 pages long on Bernoulli Numbers. Ramanujan’s name gradually gained recognition among Indian mathematicians. Later in 1912, Ramanujan got a position as an accounting clerk at Madras Port Trust. He used to finish his duties swiftly and spent his spare time doing maths.
In 1913, with the help of his friends and supporters, Ramanujan wrote to mathematicians at Cambridge, seeking validation of his works. However, he received no recommendation from the first two professors, H. F. Baker and E. W. Hobson. Afterwards, he wrote to G. H. Hardy.
Hardy was an English pure mathematician mostly known for his contributions to number theory and mathematical analysis. He is also widely known for his 1940 essay, A Mathematician’s Apology. When Ramanujan wrote to him, he was a Cambridge professor and was already a leading mathematician in England.
Receiving a 9-page letter from an Indian mathematician, Hardy, like the previous two mathematicians to whom Ramanujan wrote, questioned whether the works in the letter were fraudulent. Some of the mathematics stated by Ramanujan in his letter had already been done by other mathematicians.
“Srinivasa Ramanujan was the strangest man in all of mathematics, probably in the entire history of science. He has been compared to a bursting supernova, illuminating the darkest, most profound corners of mathematics, before being tragically struck down by tuberculosis at the age of 33… Working in total isolation from the main currents of his field, he was able to rederive 100 years’ worth of Western mathematics on his own. The tragedy of his life is that much of his work was wasted rediscovering known mathematics.”
— Michio Kaku on Ramanujan
Ramanujan was unaware of that as he was not exposed much to the works of other mathematicians. On the other hand, some of the maths were unique to Hardy. Ramanujan’s theorems on continuous fractions astounded Hardy. Hardy then provided the paper to his colleague, J. E. Littlewood. Littlewood was also a mathematician and Hardy’s close friend. Hardy and Littlewood collaborated together for a long time. Littlewood was equally awestruck by Ramanujan’s work. Both of them admired his work. Hardy eagerly responded to Ramanujan, expressing his desire to assist him and his accomplishments and also asked for proof of the theorems mentioned in the letter. He also arranged for Ramanujan to visit Cambridge. However, Ramanujan declined to travel owing to cultural differences. On the other hand, he received a monthly research fellowship of 75 rupees at the University of Madras, thanks to the assistance of both Indian and English mathematicians. He continued to publish his works in Journal of the Indian Mathematical Society. Hardy eventually assigned E. H. Neville, an English mathematician, to persuade Ramanujan to visit Cambridge. Ramanujan agreed, but not his mother. People believe she accepted afterwards because she had a vivid dream in which the family goddess told her to stand no longer between her son and the fulfillment of his life’s purpose. In March 1914, Ramanujan finally left for England.
By the time he arrived in England on April 14,1914, his English was excellent. Thus, he didn’t feel much difficult in communicating there. With the help of Hardy, he got a grant from Trinity College, Cambridge. However, Ramanujan struggled to acclimate to the culture and food of England. He was deeply committed to his culture and norms. He had to cook his own meals in order for it to be pure according to his beliefs. The English weather seemed inappropriate for him, and he was subjected to minor prejudice from his peers. In Hardy’s own words, his relationship with Ramanujan was “ the only romantic event” of his life. Hardy intended his statement to imply that they had similar beliefs and viewpoints and admired one other’s findings.
Ramanujan and Hardy collaborated for almost 5 years in Cambridge. He published many of his papers in English and European journals. One remarkable result of the Hardy-Ramanujan collaboration was a asymptotic formula for partition function p(n). p(n) is the number of possible methods to express a positive integer ’n’ as a sum of positive integers. For instance, lets take the integer 4. Now, 4 can be expressed as 1+1+1+1, 1+1+2, 3+1, 2+2, and 4, resulting in five different partitions of 4. Hence, p(4)=5. Hardy and Ramanujan created the circle method to prove the asymptotic formula.
Ramanujan’s years in England were mathematically fruitful, and also he received acclaim that he and Hardy hoped for. In 1916, he received a bachelor of science degree in research from Cambridge. He was elected to the London Mathematical Society the following year. In 1917, he was elected a Fellow of the Royal Society, and in 1918, he became the first Indian Fellow of Trinity College.
Ramanujan suffered tuberculosis in 1917, but his condition recovered enough for him to return to India in 1919. But his health condition deteriorated again, and he died the next year, on April 26, 1920. In addition to his published papers, Ramanujan left three notebooks and a ‘lost’ notebook containing about 3000 to 4000 mathematical claims, mostly identities and equations with no proof. The majority of these have already been confirmed, while only a handful have been proven erroneous.
December 22 (Ramanujan’s birthday) is celebrated as the National Mathematics Day every year in India since 2012 as a tribute to Ramanujan.
Believing that his family goddess provided him all the knowledge, Ramanujan is believed to have said,
”An equation for me has no meaning unless it expresses a thought of God.”
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