Srinivasa Ramanujan was one of India’s greatest mathematicians. His work on the analytical theory of numbers, continued fractions, elliptic functions, and the infinite series was considered as a substantial commitment to the history of mathematics. Ramanujan was born into a Tamil Brahmin family on 22nd December 1887 in Madras (which is now known as Tamil Nadu) during the British rule in India. His father was a clerk at a clothing shop whereas his mother was a housewife. Due to his father’s busy schedule and the hardship of life back in the days of British rule in India Ramanujan grew closer to his mother over time. She taught Ramanujan about all the religious traditions to all the prayers of their culture.
Ramanujan had always been a bright student during his years of middle school. After successfully completing his middle school with a distinction, Ramanujan was introduced to Formal Mathematics in his High School for the very first time in his life. By the age of 11, he had a mathematical knowledge of 2 undergraduate students who were guests at his house. On turning 13 he had mastered the art of advanced trigonometry while discovering sophisticated formulas on his own. He started receiving merit certificates and scholarly awards
at the age of 14. He showed passion towards geometry and infinite series due to which he developed his own method to solve quartic equations and tried solving quantic equations using radicals. One of the key elements in awakening his mathematical genius was when he was introduced to a book written by G.S.Carr called “Synopsis of Elementary Results in Pure and Applied Mathematics” which was a collection of nearly 5000 theorems. Ramanujan studied all the theorems present in the book in a very brief manner. Later Ramanujan independently developed and investigated the Bernoulli numbers and calculated Euler’s constant (Gamma) up to 15 decimal places. Upon graduating high school because of his intelligence and knowledge he was awarded a scholarship to college.
Ramanujan was a man of indefinite genius in himself. Due to his deep passion for mathematics, he lost his scholarship because of that he decided to drop out of college and pursue his independent research in the field of mathematics. He struggled for years with no money in hand and no food to eat but a mind full of unimaginable concepts and theorems. In the year 1910, V. Ramaswamy Aiyer who was the founder of The Indian Mathematical Society saw Ramanujan’s potential and offered him a research position at the University of Madras. Aiyer saw Ramanujan’s exceptional mathematical skills and was taken by surprise so he decided to refer him to one of his mathematician friend named Ramachandra Rao who was the secretary of The Indian Mathematical Society. Rao doubted Ramanujan’s work and thought of him as a phony. Rao after speaking to a couple of his other sources decided to give Ramanujan a chance
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and listened to him talk on his theory of divergent series, hypergeometric series and elliptic integrals which forced Rao to believe in extraordinary skills and knowledge of Ramanujan.
Ramanujan published his work in the Journal of the Indian Mathematical Society where one of the first problems he proposed was to find the value of:
He waited for a solution for about 6 months but failed to receive any. So he supplied the solution in his first book, where he formulated the equation that could be used to solve the infinitely nested radicals problem.
Ramanujan’s first formal paper in the journal was on the properties of Bernoulli numbers. One property he discovered was that the fractions were Bernoulli numbers were always divisible by 6. He also managed to develop a method of calculation Bn based on previous Bernoulli numbers like:
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In spring of 1913 Rao & Aiyer tried to present Ramanujan’s work to British mathematicians M.J.M Hill from University College London, H.F Baker & E.W Hobson from Cambridge University but none of them offered to take Ramanujan as one of their students. Later on 16th January 1913, Ramanujan wrote 9 pages of mathematics proposing his theories to G.H Hardy, coming from an unknown mathematician made hardy believe those manuscripts as a possible fraud. Later Hardy recognized some of the Ramanujan’s formula but others seem scarcely possible to believe. One of the theorems Hardy found amazing was
Hardy was also impressed by Ramanujan’s work relating to infinite series:
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Hardy knew the first result as Bauer determined it in 1859 but the second was new to Hardy as it was derived from a class of functions called Hyper Geometric Series. Hardy found these results much more intriguing than Gauss’s work on
integrals. After seeing Ramanujan’s theorems on continued fractions Hardy stated that he had never seen anything in the least like them before. He figured Ramanujan’s theorems must be true because if they were not true then no one would have the imagination to invent them. Later discussing Ramanujan’s paper with his colleague J.E Littlewood Hardy concluded that those papers were the most remarkable he has received. He believed Ramanujan as a mathematician of the highest quality and exceptional originality and power. Hardy expressed his interest in Ramanujan’s work and invited him to study to England. At first, Ramanujan refused to go to a foreign land due to his religious belief but late he agreed and on March 1914 he decided to leave for England to begin his work with Littlewood and Hardy.
After spending about 5 years in Cambridge collaborating with Hardy and Littlewood Ramanujan published part of his findings. He was awarded a Bachelor of Science degree by Research (Renamed as Ph.D.) for his work on highly composite numbers. Hardy remarked that it was one of the most unusual papers seen in mathematical research at that time. In the year 1917 Ramanujan was elected to the London Mathematical Society. In the year 1918, Hardy and Ramanujan studied the partition function. They gave a non-convergent asymptotic series that permits exact computation of an integer. The same year he was elected as a Fellow of the Royal Society. At the age 31, Ramanujan was one of the youngest fellows in the history of the Royal Society. He was the 1st Indian to be elected as a Fellow of Trinity College Cambridge in the year 1918.
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Ramanujan often said, “An equation for me has no meaning unless it represents a thought of God”. Throughout his life, Ramanujan suffered from health problems. He was diagnosed with Tuberculosis and severe vitamin deficiency due to which he decided to return to Indian and spent his last few years with his family. In the year 1920, Ramanujan died at a very young age of 32. Hardy believed that Ramanujan’s discoveries are unusually rich and there is more to them than initially meets the eye. Hardy also quoted that “Every positive integer was one of Ramanujan’s personal friend.”