Srinivasa Ramanujan

Ramanujan's birthplace on 18 Alahiri Street, Erode, now in Tamil Nadu

Ramanujan's home on Sarangapani Sannidhi Street, Kumbakonam

Ramanujan (literally, "younger brother of Rama", a Hindu deity[14] was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode during the British Colonial Rule (present-day Tamil Nadu, India), at the residence of his maternal grandparents.[15] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a clerk in a sari shop.[16][2] His mother, Komalatammal, was a housewife and sang at a local temple.[17] They lived in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam.[18] The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later. In December 1889, Ramanujan contracted smallpox, but recovered, unlike the 4,000 others who died in a bad year in the Thanjavur district around this time. He moved with his mother to her parents' house in Kanchipuram, near Madras (now Chennai). His mother gave birth to two more children, in 1891 and 1894, both of whom died before their first birthdays.[14]

On 1 October 1892, Ramanujan was enrolled at the local school.[19] After his maternal grandfather lost his job as a court official in Kanchipuram,[20] Ramanujan and his mother moved back to Kumbakonam, and he was enrolled in Kangayan Primary School.[21] When his paternal grandfather died, he was sent back to his maternal grandparents, then living in Madras. He did not like school in Madras, and tried to avoid attending. His family enlisted a local constable to make sure he attended school. Within six months, Ramanujan was back in Kumbakonam.[21]

Since Ramanujan's father was at work most of the day, his mother took care of the boy, and they had a close relationship. From her, he learned about tradition and puranas, to sing religious songs, to attend pujas at the temple, and to maintain particular eating habits—all part of Brahmin culture.[22] At Kangayan Primary School, Ramanujan performed well. Just before turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, and arithmetic with the best scores in the district.[23] That year, Ramanujan entered Town Higher Secondary School, where he encountered formal mathematics for the first time.[23]

A child prodigy by age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book written by S. L. Loney on advanced trigonometry.[24][25] He mastered this by the age of 13 while discovering sophisticated theorems on his own. By 14, he received merit certificates and academic awards that continued throughout his school career, and he assisted the school in the logistics of assigning its 1,200 students (each with differing needs) to its approximately 35 teachers.[26] He completed mathematical exams in half the allotted time, and showed a familiarity with geometry and infinite series. Ramanujan was shown how to solve cubic equations in 1902. He would later develop his own method to solve the quartic. In 1903, he tried to solve the quintic, not knowing that it was impossible to solve with radicals.[27]

In 1903, when he was 16, Ramanujan obtained from a friend a library copy of A Synopsis of Elementary Results in Pure and Applied Mathematics, G. S. Carr's collection of 5,000 theorems.[28][29] Ramanujan reportedly studied the contents of the book in detail.[30] The next year, Ramanujan independently developed and investigated the Bernoulli numbers and calculated the Euler–Mascheroni constant up to 15 decimal places.[31] His peers at the time said they "rarely understood him" and "stood in respectful awe" of him.[26]

When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than the maximum.[32] He received a scholarship to study at Government Arts College, Kumbakonam,[33][34] but was so intent on mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process.[35] In August 1905, Ramanujan ran away from home, heading towards Visakhapatnam, and stayed in Rajahmundry[36] for about a month.[35] He later enrolled at Pachaiyappa's College in Madras. There, he passed in mathematics, choosing only to attempt questions that appealed to him and leaving the rest unanswered, but performed poorly in other subjects, such as English, physiology, and Sanskrit.[37] Ramanujan failed his Fellow of Arts exam in December 1906 and again a year later. Without an FA degree, he left college and continued to pursue independent research in mathematics, living in extreme poverty and often on the brink of starvation.[38]

In 1910, after a meeting between the 23-year-old Ramanujan and the founder of the Indian Mathematical Society, V. Ramaswamy Aiyer, Ramanujan began to get recognition in Madras's mathematical circles, leading to his inclusion as a researcher at the University of Madras.[39]

On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 April 1994),[40] a girl his mother had selected for him a year earlier and who was ten years old when they married.[41][42][43] It was not unusual then for marriages to be arranged with girls at a young age. Janaki was from Rajendram, a village close to Marudur (Karur district) Railway Station. Ramanujan's father did not participate in the marriage ceremony.[44] As was common at that time, Janaki continued to stay at her maternal home for three years after marriage, until she reached puberty. In 1912, she and Ramanujan's mother joined Ramanujan in Madras.[45]

After the marriage, Ramanujan developed a hydrocele testis.[46] The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac, but his family could not afford the operation. In January 1910, a doctor volunteered to do the surgery at no cost.[47]

After his successful surgery, Ramanujan searched for a job. He stayed at a friend's house while he went from door to door around Madras looking for a clerical position. To make money, he tutored students at Presidency College who were preparing for their Fellow of Arts exam.[48]

In late 1910, Ramanujan was sick again. He feared for his health, and told his friend R. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the Madras Christian College."[49] After Ramanujan recovered and retrieved his notebooks from Iyer, he took a train from Kumbakonam to Villupuram, a city under French control.[50][51] In 1912, Ramanujan moved with his wife and mother to a house in Saiva Muthaiah Mudali street, George Town, Madras, where they lived for a few months.[52] In May 1913, upon securing a research position at Madras University, Ramanujan moved with his family to Triplicane.[53]

Ramanujan (centre) and his colleague G. H. Hardy (rightmost), with other scientists, outside the Senate House, Cambridge, c.1914–19

Whewell's Court, Trinity College, Cambridge

Ramanujan departed from Madras aboard the S.S. Nevasa on 17 March 1914.[90] When he disembarked in London on 14 April, Neville was waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took up residence on Whewell's Court, a five-minute walk from Hardy's room.[91]

Hardy and Littlewood began to look at Ramanujan's notebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems in the notebooks. Hardy saw that some were wrong, others had already been discovered, and the rest were new breakthroughs.[92] Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a Jacobi",[93] while Hardy said he "can compare him only with Euler or Jacobi."[94]

Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood, and published part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs, and working styles. In the previous few decades, the foundations of mathematics had come into question and the need for mathematically rigorous proofs recognised. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on his intuition and insights. Hardy tried his best to fill the gaps in Ramanujan's education and to mentor him in the need for formal proofs to support his results, without hindering his inspiration—a conflict that neither found easy.

Ramanujan was awarded a Bachelor of Arts by Research degree[95][96] (the predecessor of the PhD degree) in March 1916 for his work on highly composite numbers, sections of the first part of which had been published the preceding year in the Proceedings of the London Mathematical Society. The paper was more than 50 pages long and proved various properties of such numbers. Hardy disliked this topic area but remarked that though it engaged with what he called the 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra of inequalities'.[97]

On 6 December 1917, Ramanujan was elected to the London Mathematical Society. On 2 May 1918, he was elected a Fellow of the Royal Society,[98] the second Indian admitted, after Ardaseer Cursetjee in 1841. At age 31, Ramanujan was one of the youngest Fellows in the Royal Society's history. He was elected "for his investigation in elliptic functions and the Theory of Numbers." On 13 October 1918, he was the first Indian to be elected a Fellow of Trinity College, Cambridge.[99]

Ramanujan had numerous health problems throughout his life. His health worsened in England; possibly he was also less resilient due to the difficulty of keeping to the strict dietary requirements of his religion there and because of wartime rationing in 1914–18. He was diagnosed with tuberculosis and a severe vitamin deficiency, and confined to a sanatorium. In 1919, he returned to Kumbakonam, Madras Presidency, and in 1920 he died at the age of 32. After his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting of formulae on singular moduli, hypergeometric series and continued fractions.[45]

Ramanujan's widow, Smt. Janaki Ammal, moved to Bombay. In 1931, she returned to Madras and settled in Triplicane, where she supported herself on a pension from Madras University and income from tailoring. In 1950, she adopted a son, W. Narayanan, who eventually became an officer of the State Bank of India and raised a family. In her later years, she was granted a lifetime pension from Ramanujan's former employer, the Madras Port Trust, and pensions from, among others, the Indian National Science Academy and the state governments of Tamil Nadu, Andhra Pradesh and West Bengal. She continued to cherish Ramanujan's memory, and was active in efforts to increase his public recognition; prominent mathematicians, including George Andrews, Bruce C. Berndt and Béla Bollobás made it a point to visit her while in India. She died at her Triplicane residence in 1994.[44][45]

A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D. A. B. Young[100] concluded that his medical symptoms—including his past relapses, fevers, and hepatic conditions—were much closer to those resulting from hepatic amoebiasis, an illness then widespread in Madras, than tuberculosis. He had two episodes of dysentery before he left India. When not properly treated, amoebic dysentery can lie dormant for years and lead to hepatic amoebiasis, whose diagnosis was not then well established.[101] At the time, if properly diagnosed, amoebiasis was a treatable and often curable disease;[101][102] British soldiers who contracted it during the First World War were being successfully cured of amoebiasis around the time Ramanujan left England.[103]

Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners.[104] He lived a simple life at Cambridge.[105] Ramanujan's first Indian biographers describe him as a rigorously orthodox Hindu. He credited his acumen to his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked to her for inspiration in his work[106] and said he dreamed of blood drops that symbolised her consort, Narasimha. Later he had visions of scrolls of complex mathematical content unfolding before his eyes.[107] He often said, "An equation for me has no meaning unless it expresses a thought of God."[108]

Hardy cites Ramanujan as remarking that all religions seemed equally true to him.[109] Hardy further argued that Ramanujan's religious belief had been romanticised by Westerners and overstated—in reference to his belief, not practice—by Indian biographers. At the same time, he remarked on Ramanujan's strict vegetarianism.[110]

Similarly, in an interview with Frontline, Berndt said, "Many people falsely promulgate mystical powers to Ramanujan's mathematical thinking. It is not true. He has meticulously recorded every result in his three notebooks," further speculating that Ramanujan worked out intermediate results on slate that he could not afford the paper to record more permanently.[8]

In mathematics, there is a distinction between insight and formulating or working through a proof. Ramanujan proposed an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a byproduct of his work, new directions of research were opened up. Examples of the most intriguing of these formulae include infinite series for π, one of which is given below:

${\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.}$

This result is based on the negative fundamental discriminant d = −4 × 58 = −232 with class number h(d) = 2. Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 396², which is related to the fact that

${\textstyle e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .}$

This might be compared to Heegner numbers, which have class number 1 and yield similar formulae.

Ramanujan's series for π converges extraordinarily rapidly and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation 9801√2/4412 for π, which is correct to six decimal places; truncating it to the first two terms gives a value correct to 14 decimal places. See also the more general Ramanujan–Sato series.

One of Ramanujan's remarkable capabilities was the rapid solution of problems, illustrated by the following anecdote about an incident in which P. C. Mahalanobis posed a problem:

Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x?' This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. 'It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind', Ramanujan replied."[111][112]

His intuition also led him to derive some previously unknown identities, such as

${\displaystyle {\begin{aligned}&\left(1+2\sum _{n=1}^{\infty }{\frac {\cos(n\theta )}{\cosh(n\pi )}}\right)^{-2}+\left(1+2\sum _{n=1}^{\infty }{\frac {\cosh(n\theta )}{\cosh(n\pi )}}\right)^{-2}\\[6pt]={}&{\frac {2\Gamma ^{4}\left({\frac {3}{4}}\right)}{\pi }}={\frac {8\pi ^{3}}{\Gamma ^{4}\left({\frac {1}{4}}\right)}}\end{aligned}}}$

for all θ such that ${\displaystyle |\Re (\theta )|<\pi }$ and ${\displaystyle |\Im (\theta )|<\pi }$, where Γ(z) is the gamma function, and related to a special value of the Dedekind eta function. Expanding into series of powers and equating coefficients of θ⁰, θ⁴, and θ⁸ gives some deep identities for the hyperbolic secant.

In 1918, Hardy and Ramanujan studied the partition function P(n) extensively. They gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. In 1937, Hans Rademacher refined their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae called the circle method.[113]

In the last year of his life, Ramanujan discovered mock theta functions.[114] For many years, these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms.

The number 1729 is known as the Hardy–Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital. In Hardy's words:[118]

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

Immediately before this anecdote, Hardy quoted Littlewood as saying, "Every positive integer was one of [Ramanujan's] personal friends."[119]

The two different ways are:

${\displaystyle 1729=1^{3}+12^{3}=9^{3}+10^{3}.}$

Generalisations of this idea have created the notion of "taxicab numbers".

In his obituary of Ramanujan, written for Nature in 1920, Hardy observed that Ramanujan's work primarily involved fields less known even among other pure mathematicians, concluding:

His insight into formulae was quite amazing, and altogether beyond anything I have met with in any European mathematician. It is perhaps useless to speculate as to his history had he been introduced to modern ideas and methods at sixteen instead of at twenty-six. It is not extravagant to suppose that he might have become the greatest mathematician of his time. What he actually did is wonderful enough… when the researches which his work has suggested have been completed, it will probably seem a good deal more wonderful than it does to-day.[76]

Hardy further said:[120]

He combined a power of generalisation, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day. The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was..."

When asked about the methods Ramanujan employed to arrive at his solutions, Hardy said they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account."[121] He also said that he had "never met his equal, and can compare him only with Euler or Jacobi".[121]

K. Srinivasa Rao has said,[122] "As for his place in the world of Mathematics, we quote Bruce C. Berndt: 'Paul Erdős has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100. Hardy gave himself a score of 25, J. E. Littlewood 30, David Hilbert 80 and Ramanujan 100.'" During a May 2011 lecture at IIT Madras, Berndt said that over the last 40 years, as nearly all of Ramanujan's conjectures had been proven, there had been greater appreciation of Ramanujan's work and brilliance, and that Ramanujan's work was now pervading many areas of modern mathematics and physics.[114][123]

Bust of Ramanujan in the garden of Birla Industrial & Technological Museum in Kolkata, India

The year after his death, Nature listed Ramanujan among other distinguished scientists and mathematicians on a "Calendar of Scientific Pioneers" who had achieved eminence.[124] Ramanujan's home state of Tamil Nadu celebrates 22 December (Ramanujan's birthday) as 'State IT Day'. Stamps picturing Ramanujan were issued by the government of India in 1962, 2011, 2012 and 2016.[125]

Since Ramanujan's centennial year, his birthday, 22 December, has been annually celebrated as Ramanujan Day by the Government Arts College, Kumbakonam, where he studied, and at the IIT Madras in Chennai. The International Centre for Theoretical Physics (ICTP) has created a prize in Ramanujan's name for young mathematicians from developing countries in cooperation with the International Mathematical Union, which nominates members of the prize committee. SASTRA University, a private university based in Tamil Nadu, has instituted the SASTRA Ramanujan Prize of US$10,000 to be given annually to a mathematician not exceeding age 32 for outstanding contributions in an area of mathematics influenced by Ramanujan.[126]

Based on the recommendations of a committee appointed by the University Grants Commission (UGC), Government of India, the Srinivasa Ramanujan Centre, established by SASTRA, has been declared an off-campus centre under the ambit of SASTRA University. House of Ramanujan Mathematics, a museum of Ramanujan's life and work, is also on this campus. SASTRA purchased and renovated the house where Ramanujan lived at Kumabakonam.[126]

In 2011, on the 125th anniversary of his birth, the Indian government declared that 22 December will be celebrated every year as National Mathematics Day.[127] Then Indian Prime Minister Manmohan Singh also declared that 2012 would be celebrated as National Mathematics Year.[128]

Ramanujan IT City is an information technology (IT) special economic zone (SEZ) in Chennai that was built in 2011. Situated next to the Tidel Park, it includes 25 acres (10 ha) with two zones, with a total area of 5.7 million square feet (530,000 m²), including 4.5 million square feet (420,000 m²) of office space.[129]

Commemorative stamps released by India Post (by year):

1962

2011

2012

2016

• The Man Who Loved Numbers is a 1988 PBS NOVA documentary about Ramanujan (S15, E9).[130]
• The Man Who Knew Infinity is a 2015 film based on Kanigel's book. British actor Dev Patel portrays Ramanujan.[131][132][133]
• Ramanujan, an Indo-British collaboration film chronicling Ramanujan's life, was released in 2014 by the independent film company Camphor Cinema.[134] The cast and crew include director Gnana Rajasekaran, cinematographer Sunny Joseph and editor B. Lenin.[135][136] Indian and English stars Abhinay Vaddi, Suhasini Maniratnam, Bhama, Kevin McGowan and Michael Lieber star in pivotal roles.[137]
• Nandan Kudhyadi directed the Indian documentary films The Genius of Srinivasa Ramanujan (2013) and Srinivasa Ramanujan: The Mathematician and His Legacy (2016) about the mathematician.[138]
• Ramanujan (The Man Who Reshaped 20th Century Mathematics), an Indian docudrama film directed by Akashdeep released in 2018.[139]
• M. N. Krish's thriller novel The Steradian Trail weaves Ramanujan and his accidental discovery into its plot connecting religion, mathematics, finance and economics.[140][141]
• Partition, a play by Ira Hauptman about Hardy and Ramanujan, was first performed in 2013.[142][143][144][145]
• The play First Class Man by Alter Ego Productions[146] was based on David Freeman's First Class Man. The play centres around Ramanujan and his complex and dysfunctional relationship with Hardy. On 16 October 2011 it was announced that Roger Spottiswoode, best known for his James Bond film Tomorrow Never Dies, is working on the film version, starring Siddharth.[147]
• A Disappearing Number is a British stage production by the company Complicite that explores the relationship between Hardy and Ramanujan.[148]
• David Leavitt's novel The Indian Clerk explores the events following Ramanujan's letter to Hardy.[149][150]
• Google honoured Ramanujan on his 125th birth anniversary by replacing its logo with a doodle on its home page.[151][152]
• Ramanujan was mentioned in the 1997 film Good Will Hunting, in a scene where professor Gerald Lambeau (Stellan Skarsgård) explains to Sean Maguire (Robin Williams) the genius of Will Hunting (Matt Damon) by comparing him to Ramanujan.[153]

• George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part I (Springer, 2005, ISBN 0-387-25529-X)[154]
• George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part II, (Springer, 2008, ISBN 978-0-387-77765-8)
• George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part III, (Springer, 2012, ISBN 978-1-4614-3809-0)
• George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part IV, (Springer, 2013, ISBN 978-1-4614-4080-2)
• George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part V, (Springer, 2018, ISBN 978-3-319-77832-7)
• M. P. Chaudhary, A simple solution of some integrals given by Srinivasa Ramanujan, (Resonance: J. Sci. Education – publication of Indian Academy of Science, 2008)[155]
• M.P. Chaudhary, Mock theta functions to mock theta conjectures, SCIENTIA, Series A : Math. Sci., (22)(2012) 33–46.
• M.P. Chaudhary, On modular relations for the Roger-Ramanujan type identities, Pacific J. Appl. Math., 7(3)(2016) 177–184.