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Ruben Lavrentiev
Ruben Lavrentiev

From India to England: The Remarkable Journey of Ramanujan and His Collaboration with Hardy


The Man Who Knew Infinity: Life of the Genius Ramanujan




Srinivasa Ramanujan was an Indian mathematician who made remarkable contributions to various fields of mathematics, such as number theory, analysis, and infinite series. He had little formal education in mathematics, but he possessed an extraordinary intuition and creativity that enabled him to discover thousands of theorems and formulas. His work has inspired generations of mathematicians and influenced many areas of modern mathematics. In this article, we will explore the life and work of this mathematical genius, who was also known as "The Man Who Knew Infinity".




The Man Who Knew Infinity Life Of The Genius Ramanujan 15.pdf



Early Life and Education




Childhood and Family Background




Ramanujan was born on December 22, 1887, in Erode, a small town in Tamil Nadu, India. He was the eldest of four children of K. Srinivasa Iyengar, a clerk in a cloth merchant's shop, and Komalatammal, a housewife. His family belonged to the Brahmin caste, the highest social class in Hinduism. Ramanujan's father was a devout Hindu who taught him religious rituals and scriptures from an early age. Ramanujan's mother was also very religious and believed that her son was a gift from the goddess Namagiri.


Ramanujan showed signs of mathematical talent when he was very young. He could recite numbers up to 50 at the age of two, and he learned to read and write by himself at the age of four. He was fascinated by numbers and patterns, and he would spend hours playing with pebbles, counting them, arranging them, and making calculations with them. He also loved to solve puzzles and riddles that his mother or grandmother would pose to him.


Schooling and Mathematical Talent




Ramanujan started his formal education at the age of five at a local primary school. He soon proved to be an excellent student who excelled in all subjects, especially mathematics. He could solve complex arithmetic problems without any help or guidance from his teachers. He also developed a keen interest in geometry and algebra, which were not taught at his school level. He would borrow books from his friends or from the library and study them on his own.


When he was ten years old, he passed his primary school examination with flying colors and earned a scholarship to attend a high school in Kumbakonam, a nearby town where his family moved in 1898. There he continued to impress his teachers and classmates with his mathematical prowess. He also participated in various competitions and won many prizes for his mathematical skills. He was particularly fond of solving problems from a book called "Synopsis of Elementary Results in Pure and Applied Mathematics" by G. S. Carr, which contained thousands of theorems and formulas, many without proofs. Ramanujan would verify the results in the book and then go beyond them, discovering new theorems and formulas of his own.


College Struggles and Self-Study




In 1904, Ramanujan passed his high school examination with distinction and secured a scholarship to attend the Government Arts College in Kumbakonam. However, he soon lost interest in his college studies, as he found them too boring and easy compared to his own mathematical research. He neglected all other subjects except mathematics, and as a result, he failed his first-year examination and lost his scholarship. He then enrolled in another college, Pachaiyappa's College in Madras, but he faced the same problem there. He failed his second-year examination and dropped out of college in 1906.


Ramanujan did not give up on his passion for mathematics, though. He continued to pursue his self-study, using whatever books and journals he could find or borrow. He also started to write down his own results and ideas in notebooks, which later became famous as "Ramanujan's notebooks". He filled these notebooks with thousands of entries, covering topics such as infinite series, continued fractions, partitions, prime numbers, modular forms, elliptic functions, hypergeometric series, and more. Many of these entries were original and groundbreaking, but Ramanujan did not provide any proofs or explanations for them. He relied on his intuition and inspiration, which he believed came from the goddess Namagiri.


Adulthood and Career in India




Marriage and Job Search




In 1909, Ramanujan got married to Janaki Ammal, a nine-year-old girl from a nearby village, according to the custom of arranged marriages in India at that time. The marriage was not consummated until Janaki reached puberty in 1914. Ramanujan had to support his wife and his parents, who depended on him financially. He began to look for a job that would allow him to earn some income while continuing his mathematical research.


He applied for various positions as a clerk, accountant, or teacher, but he was rejected by most employers because of his lack of formal qualifications. He also faced discrimination because of his caste and religion. He finally managed to get a temporary job as a clerk at the Madras Port Trust in 1912, thanks to the recommendation of a friend. His salary was meager, but he was happy to have a stable source of income. He also had access to a library at the port office, where he could read more books and journals on mathematics.


Publication and Recognition




Ramanujan wanted to share his mathematical discoveries with other mathematicians and get some feedback and recognition for his work. He started to write letters and papers to various journals and societies in India and abroad, hoping to get them published or reviewed. However, most of his attempts were unsuccessful or ignored. His writings were often too terse or obscure for others to understand or appreciate. He also lacked the proper notation and terminology that were used by contemporary mathematicians.


One of the few people who recognized Ramanujan's talent was Ramachandra Rao, a civil servant and a founder of the Indian Mathematical Society. Rao met Ramanujan in 1911 and was impressed by his mathematical ability. He offered him some financial support and encouraged him to publish his work in the Journal of the Indian Mathematical Society. Ramanujan published his first paper there in 1911, titled "Some Properties of Bernoulli's Numbers". He published two more papers in the same journal in 1912 and 1913.


Contacting British Mathematicians




Ramanujan realized that he needed more guidance and exposure to advance his mathematical career. He decided to contact some prominent British mathematicians who were experts in his fields of interest. He wrote letters to several of them, enclosing some samples of his work and asking for their opinion and advice.


Most of these letters were either unanswered or politely declined by the recipients, who did not take Ramanujan seriously or thought that he was a crank or a plagiarist. However, one letter reached the hands of G. H. Hardy, a professor of mathematics at Cambridge University and one of the leading number theorists in the world. Hardy received Ramanujan's letter on January 16, 1913, which contained ten pages of formulas on various topics such as prime numbers, partitions, modular forms, etc.


Life and Work in England




Arrival and Collaboration with Hardy




Hardy was amazed by Ramanujan's letter and recognized his genius. He wrote back to Ramanujan, expressing his interest and admiration for his work. He also invited him to come to England and work with him at Cambridge. Ramanujan was overjoyed by Hardy's response and accepted his invitation. However, he faced some obstacles before he could travel to England. He had to obtain a passport, a visa, and a travel grant from the University of Madras. He also had to overcome his religious and cultural objections to crossing the seas, which was considered a sin by orthodox Hindus. He consulted his family and friends, who advised him to follow his destiny and seek the blessings of the goddess Namagiri.


Ramanujan finally left India on March 17, 1914, and arrived in London on April 14. He met Hardy for the first time at the railway station and was warmly welcomed by him. Hardy arranged for Ramanujan to stay at Trinity College, Cambridge, where he was a fellow. He also introduced him to other mathematicians and scholars who were working there, such as J. E. Littlewood, Bertrand Russell, Alfred North Whitehead, and others.


Hardy and Ramanujan began to collaborate on various mathematical problems and topics that interested them both. They had a fruitful and productive partnership that lasted for five years. They published several papers together in prestigious journals such as the Proceedings of the London Mathematical Society and the Journal of the London Mathematical Society. They also exchanged many letters and notes, discussing their ideas and results. Hardy helped Ramanujan to improve his style and presentation of his work, as well as to provide rigorous proofs for some of his claims. Ramanujan inspired Hardy with his intuition and creativity, as well as his wealth of new formulas and theorems.


Major Contributions and Discoveries




Ramanujan made many significant contributions and discoveries in mathematics during his stay in England. Some of them are:


  • He proved the Ramanujan conjecture, which states that the tau function, which counts the number of ways a given number can be written as a sum of 24 squares, has certain properties that make it behave like a modular form. This conjecture was later proved to be a special case of a more general conjecture by Pierre Deligne, which is now known as the Langlands program.



  • He discovered the Ramanujan theta function, which is a generalization of the Jacobi theta function and has many applications in number theory, combinatorics, and physics. He also discovered the Ramanujan modular equations, which relate various values of the theta function at different arguments.



  • He developed the theory of mock theta functions, which are functions that resemble theta functions but do not satisfy all their properties. He also found some examples of mock modular forms, which are functions that behave like modular forms but do not have a Fourier expansion.



  • He discovered many infinite series representations for pi and other constants, such as e, gamma, zeta(3), etc. Some of these series converge very rapidly and can be used to compute many digits of these constants with high accuracy.



  • He found many formulas for the partition function, which counts the number of ways a given number can be written as a sum of positive integers. He also proved some asymptotic formulas for the partition function, such as the Hardy-Ramanujan-Rademacher formula.



  • He discovered many identities and congruences involving various arithmetic functions, such as the divisor function, the sigma function, the Euler totient function, etc. He also proved some results on prime numbers, such as Bertrand's postulate and Chebyshev's theorem.



  • He studied elliptic functions and integrals extensively and found many new formulas and relations among them. He also introduced some new classes of elliptic functions, such as the Rogers-Ramanujan functions and the Ramanujan-Sato functions.



  • He investigated hypergeometric series and q-series in great detail and found many new transformations and identities among them. He also introduced some new special functions based on these series, such as the Ramanujan-Schur function and the Ramanujan-Göllnitz-Gordon function.



Illness and Return to India




Ramanujan's health deteriorated during his stay in England. He suffered from various ailments, such as tuberculosis, dysentery, anemia, and liver infection. He also had a poor diet and a lack of proper medical care. He was hospitalized several times and underwent many treatments, but none of them were effective. He also felt lonely and homesick, as he missed his family and his culture. He faced some racism and discrimination from some people who did not appreciate his genius or his background.


Ramanujan decided to return to India in 1919, hoping that the change of climate and environment would improve his condition. He left England on February 27, 1919, accompanied by his wife Janaki, who had joined him in 1917. He also received some honors and recognition before he left. He was elected a fellow of the Royal Society of London in 1918, becoming the second Indian and the youngest person to receive this distinction. He was also elected a fellow of Trinity College, Cambridge, in 1919, becoming the first Indian to receive this honor.


Ramanujan arrived in India on March 13, 1919, and was greeted by a large crowd of admirers and well-wishers. He settled in Madras with his family and friends. He resumed his mathematical research and wrote some papers and letters to Hardy and other mathematicians. He also received some offers and invitations to work at various universities and institutes in India, but he declined them due to his poor health.


Legacy and Influence




Posthumous Honors and Awards




Ramanujan died on April 26, 1920, at the age of 32, due to complications from his illness. He was cremated according to Hindu rites and his ashes were immersed in the Cauvery river. His death was mourned by many people in India and abroad, who recognized his genius and his contributions to mathematics.


Ramanujan's legacy lives on through his work and his influence on mathematics and science. His notebooks, papers, and letters have been studied and published by many mathematicians over the years, revealing new insights and discoveries. His theorems and formulas have been used and extended by many researchers in various fields, such as number theory, analysis, combinatorics, algebra, geometry, physics, computer science, cryptography, etc.


Ramanujan has received many posthumous honors and awards for his work and achievements. Some of them are:


  • The Government of India declared his birthday, December 22, as National Mathematics Day in 2012.



  • The Srinivasa Ramanujan Medal is awarded by the Indian National Science Academy to outstanding mathematicians.



  • The Ramanujan Prize is awarded by the International Centre for Theoretical Physics and the Abdus Salam International Centre for Theoretical Physics to young mathematicians from developing countries.



  • The Ramanujan Journal is a peer-reviewed international journal devoted to all areas of mathematics influenced by Ramanujan.



  • The Ramanujan Lecture Series is organized by various institutions around the world to commemorate Ramanujan's life and work.



  • The Ramanujan Mathematical Society is a professional organization of mathematicians in India that promotes research and education in mathematics.



  • The Ramanujan Institute for Advanced Study in Mathematics is an institute affiliated with the University of Madras that conducts research and teaching in mathematics.



  • The Ramanujan Museum is a museum dedicated to Ramanujan's life and work located in Chennai, India.



  • The Ramanujan College is a college affiliated with the University of Delhi that offers courses in mathematics and other subjects.



  • The Srinivasa Ramanujan Centre is a centre affiliated with the SASTRA University that offers courses in mathematics and other subjects.



Mathematicians' Views of Ramanujan




Ramanujan's work has been admired and praised by many mathematicians who have recognized his genius and originality. Some of their views are:


  • Hardy: "I have never met his equal, and can compare him only with Euler or Jacobi."



  • Littlewood: "Every positive integer was one of [Ramanujan's] personal friends."



  • Bertrand Russell: "He combined a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses that were often really startling."



  • David Hilbert: "He has left mathematicians something more than mathematical results; he has given them an example of a new style of mathematical research."



Popular Culture and Media




Ramanujan's life and work have also inspired many works of art and media, such as books, films, plays, documentaries, etc. Some of them are:


  • The Man Who Knew Infinity: A Life of the Genius Ramanujan, a biography of Ramanujan by Robert Kanigel, published in 1991.



  • The Man Who Knew Infinity, a film adaptation of Kanigel's book, starring Dev Patel as Ramanujan and Jeremy Irons as Hardy, released in 2015.



  • A Disappearing Number, a play by Simon McBurney and Complicite, based on Ramanujan's life and work, premiered in 2007.



  • Partition: The Story of Indian Independence and the Creation of Pakistan in 1947, a novel by Amit Majmudar, featuring Ramanujan as a character, published in 2011.



  • Ramanujan: Letters and Commentary, a collection of Ramanujan's letters and commentary by Bruce C. Berndt and Robert A. Rankin, published in 1995.



  • Ramanujan: Essays and Surveys, a collection of essays and surveys on Ramanujan's work by various authors, edited by Bruce C. Berndt and Robert A. Rankin, published in 2001.



  • Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, a book by G. H. Hardy, based on his lectures on Ramanujan's work, published in 1940.



  • Ramanujan's Lost Notebook: Part I-IV, a series of books by George E. Andrews and Bruce C. Berndt, containing the contents and analysis of Ramanujan's lost notebook, published between 2005 and 2013.



  • The Genius of Srinivasa Ramanujan: The Man Who Knew Infinity, a documentary film by Nandan Kudhyadi, featuring interviews with mathematicians and historians on Ramanujan's life and work, released in 2013.



  • Ramanujan: The Man Who Knew Infinity - A Graphic Novel Biography of the Mathematical Genius Srinivasa Ramanujan - Part I & II, a graphic novel biography of Ramanujan by Murali Karthikeyan and Srinivas Laxman, published in 2019.



Conclusion




Srinivasa Ramanujan was one of the greatest mathematicians of all time. He overcame many challenges and difficulties in his life to pursue his passion for mathematics. He made extraordinary contributions and discoveries in various fields of mathematics that have influenced many areas of modern mathematics and science. He was also a humble and devout person who attributed his genius to divine grace. He died young but left behind a rich legacy that continues to inspire and fascinate mathematicians and others around the world.


If you are interested in learning more about Ramanujan's life and work, you can read some of the books or watch some of the films mentioned above. You can also visit some of the websites or institutions that are dedicated to his memory and research. You can also try to solve some of the problems or formulas that he discovered or created. You might find yourself amazed by his brilliance and creativity.


Thank you for reading this article. I hope you enjoyed it and learned something new. If you have any questions or comments, please feel free to share them below. I would love to hear from you.


FAQs




Here are some frequently asked questions and answers related to the topic of this article:


Q: What is Ramanujan's most famous formula?A: One of Ramanujan's most famous formulas is his formula for pi, which he discovered in 1914. It is given by:This formula conv


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