Srinivasa Aiyangar Ramanujan
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Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
. Though he had almost no formal training in
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
, he made substantial contributions to
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
,
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
, and
continued fraction In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
s, including solutions to mathematical problems then considered unsolvable. Ramanujan initially developed his own mathematical research in isolation: according to
Hans Eysenck Hans Jürgen Eysenck (; 4 March 1916 – 4 September 1997) was a German-born British psychologist who spent his professional career in Great Britain. He is best remembered for his work on intelligence and personality, although he worked on other ...
: "He tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered". Seeking mathematicians who could better understand his work, in 1913 he began a postal correspondence with the English mathematician
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
at the
University of Cambridge , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Schola ...
, England. Recognising Ramanujan's work as extraordinary, Hardy arranged for him to travel to Cambridge. In his notes, Hardy commented that Ramanujan had produced groundbreaking new
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
s, including some that "defeated me completely; I had never seen anything in the least like them before", and some recently proven but highly advanced results. During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
s). Many were completely novel; his original and highly unconventional results, such as the
Ramanujan prime In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function. Origins and definition In 1919, Ramanujan published a new proof of Bertrand's postulate which, as ...
, the
Ramanujan theta function In mathematics, particularly -analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant for ...
,
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
formulae and
mock theta function In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight . The first examples of mock theta functions were described by Srinivasa Ramanu ...
s, have opened entire new areas of work and inspired a vast amount of further research. Of his thousands of results, all but a dozen or two have now been proven correct. ''
The Ramanujan Journal ''The Ramanujan Journal'' is a peer-reviewed scientific journal covering all areas of mathematics, especially those influenced by the Indian mathematician Srinivasa Ramanujan. The journal was established in 1997 and is published by Springer Science ...
'', a
scientific journal In academic publishing, a scientific journal is a periodical publication intended to further the progress of science, usually by reporting new research. Content Articles in scientific journals are mostly written by active scientists such as s ...
, was established to publish work in all areas of mathematics influenced by Ramanujan, and his notebooks—containing summaries of his published and unpublished results—have been analysed and studied for decades since his death as a source of new mathematical ideas. As late as 2012, researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death.Deep meaning in Ramanujan's 'simple' pattern
"Mathematical proof reveals magic of Ramanujan's genius"
. ''
New Scientist ''New Scientist'' is a magazine covering all aspects of science and technology. Based in London, it publishes weekly English-language editions in the United Kingdom, the United States and Australia. An editorially separate organisation publishe ...
''.
He became one of the youngest
Fellows of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural knowledge, including mathemati ...
and only the second Indian member, and the first Indian to be elected a Fellow of Trinity College, Cambridge. Of his original letters, Hardy stated that a single look was enough to show they could have been written only by a mathematician of the highest calibre, comparing Ramanujan to mathematical geniuses such as
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and
Jacobi Jacobi may refer to: * People with the surname Jacobi (surname), Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenva ...
. In 1919, ill health—now believed to have been hepatic
amoebiasis Amoebiasis, or amoebic dysentery, is an infection of the intestines caused by a parasitic amoeba ''Entamoeba histolytica''. Amoebiasis can be present with no, mild, or severe symptoms. Symptoms may include lethargy, loss of weight, colonic ulce ...
(a complication from episodes of
dysentery Dysentery (UK pronunciation: , US: ), historically known as the bloody flux, is a type of gastroenteritis that results in bloody diarrhea. Other symptoms may include fever, abdominal pain, and a feeling of incomplete defecation. Complications ...
many years previously)—compelled Ramanujan's return to India, where he died in 1920 at the age of 32. His last letters to Hardy, written in January 1920, show that he was still continuing to produce new mathematical ideas and theorems. His "
lost notebook Ramanujan's lost notebook is the manuscript in which the Indian mathematician Srinivasa Ramanujan recorded the mathematical discoveries of the last year (1919–1920) of his life. Its whereabouts were unknown to all but a few mathematicians until i ...
", containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976. A deeply religious
Hindu Hindus (; ) are people who religiously adhere to Hinduism.Jeffery D. Long (2007), A Vision for Hinduism, IB Tauris, , pages 35–37 Historically, the term has also been used as a geographical, cultural, and later religious identifier for ...
, Ramanujan credited his substantial mathematical capacities to
divinity Divinity or the divine are things that are either related to, devoted to, or proceeding from a deity.divine
, and said his family goddess, Namagiri Thayar, revealed his mathematical knowledge to him. He once said, "An equation for me has no meaning unless it expresses a thought of
God In monotheism, monotheistic thought, God is usually viewed as the supreme being, creator deity, creator, and principal object of Faith#Religious views, faith.Richard Swinburne, Swinburne, R.G. "God" in Ted Honderich, Honderich, Ted. (ed)''The Ox ...
."


Early life

Ramanujan (literally, "younger brother of
Rama Rama (; ), Ram, Raman or Ramar, also known as Ramachandra (; , ), is a major deity in Hinduism. He is the seventh and one of the most popular '' avatars'' of Vishnu. In Rama-centric traditions of Hinduism, he is considered the Supreme Bein ...
", a Hindu deity) was born on 22 December 1887 into a
Tamil Brahmin Tamil Brahmins are an ethnoreligious community of Tamil-speaking Hindu Brahmins, predominantly living in Tamil Nadu, though they number significantly in Telangana, Andhra Pradesh, Kerala, Karnataka, in addition to other regions of India, as wel ...
Iyengar Iyengar (also spelt Ayyangar or Aiyengar, pronounced ) refers to the name of an ethnoreligious community of Tamil-speaking Hindu Brahmins, whose members follow Sri Vaishnavism and the Visishtadvaita philosophy propounded by Ramanuja. Found mos ...
family in
Erode Erode () is a city in the Indian state of Tamil Nadu. Erode is the seventh largest urban agglomeration in the state, after Chennai, Coimbatore, Madurai, Tiruchirapalli, Tiruppur and Salem. It is also the administrative headquarters of the E ...
, in present-day
Tamil Nadu Tamil Nadu (; , TN) is a States and union territories of India, state in southern India. It is the List of states and union territories of India by area, tenth largest Indian state by area and the List of states and union territories of India ...
. His father, Kuppuswamy Srinivasa Iyengar, originally from
Thanjavur district Thanjavur District is one of the Districts of Tamil Nadu, 38 districts of the States and territories of India, state of Tamil Nadu, in southeastern India. Its headquarters is Thanjavur. The district is located in the delta of the Cauvery River ...
, worked as a clerk in a
sari A sari (sometimes also saree or shari)The name of the garment in various regional languages include: * as, শাৰী, xÄrÄ«, translit-std=ISO * bn, শাড়ি, Å›Äá¹›i, translit-std=ISO * gu, સાડી, sÄá¸Ä«, translit-std= ...
shop. His mother, Komalatammal, was a
housewife A housewife (also known as a homemaker or a stay-at-home mother/mom/mum) is a woman whose role is running or managing her family's home—housekeeping, which includes caring for her children; cleaning and maintaining the home; making, buying an ...
and sang at a local temple. They lived in a small traditional home on Sarangapani Sannidhi Street in the town of
Kumbakonam Kumbakonam (formerly spelt as Coombaconum or Combaconum) or Kudanthai is a city municipal corporation in the Thanjavur district in the States of India, Indian state of Tamil Nadu. It is located from Thanjavur and from Chennai and is the headq ...
. 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 Smallpox was an infectious disease caused by variola virus (often called smallpox virus) which belongs to the genus Orthopoxvirus. The last naturally occurring case was diagnosed in October 1977, and the World Health Organization (WHO) c ...
, 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 Kanchipuram ('; ) also known as ''Conjeevaram,'' is a city in the Indian state of Tamil Nadu in the Tondaimandalam region, from Chennaithe capital of Tamil Nadu. Known as the ''City of Thousand Temples'', Kanchipuram is known for its temple ...
, near Madras (now
Chennai Chennai (, ), formerly known as Madras ( the official name until 1996), is the capital city of Tamil Nadu, the southernmost Indian state. The largest city of the state in area and population, Chennai is located on the Coromandel Coast of th ...
). His mother gave birth to two more children, in 1891 and 1894, both of whom died before their first birthdays. On 1 October 1892, Ramanujan was enrolled at the local school. After his maternal grandfather lost his job as a court official in Kanchipuram, Ramanujan and his mother moved back to
Kumbakonam Kumbakonam (formerly spelt as Coombaconum or Combaconum) or Kudanthai is a city municipal corporation in the Thanjavur district in the States of India, Indian state of Tamil Nadu. It is located from Thanjavur and from Chennai and is the headq ...
, and he was enrolled in Kangayan Primary School. 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. 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 Purana (; sa, , '; literally meaning "ancient, old"Merriam-Webster's Encyclopedia of Literature (1995 Edition), Article on Puranas, , page 915) is a vast genre of Indian literature about a wide range of topics, particularly about legends an ...
, to sing religious songs, to attend pujas at the temple, and to maintain particular eating habits—all part of
Brahmin Brahmin (; sa, बà¥à¤°à¤¾à¤¹à¥à¤®à¤£, brÄhmaṇa) is a varna as well as a caste within Hindu society. The Brahmins are designated as the priestly class as they serve as priests (purohit, pandit, or pujari) and religious teachers (guru ...
culture. At Kangayan Primary School, Ramanujan performed well. Just before turning 10, in November 1897, he passed his primary examinations in English,
Tamil Tamil may refer to: * Tamils, an ethnic group native to India and some other parts of Asia **Sri Lankan Tamils, Tamil people native to Sri Lanka also called ilankai tamils **Tamil Malaysians, Tamil people native to Malaysia * Tamil language, nativ ...
, geography, and arithmetic with the best scores in the district. That year, Ramanujan entered Town Higher Secondary School, where he encountered formal mathematics for the first time. A
child prodigy A child prodigy is defined in psychology research literature as a person under the age of ten who produces meaningful output in some domain at the level of an adult expert. The term is also applied more broadly to young people who are extraor ...
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. 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. He completed mathematical exams in half the allotted time, and showed a familiarity with
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
. 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 In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
, not knowing that it was impossible to solve with radicals. 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 George Shoobridge Carr (1837–1914) was a British mathematician. He wrote ''Synopsis of Pure Mathematics'' (1886). This book, first published in England in 1880, was read and studied closely by mathematician Srinivasa Ramanujan when he was a t ...
's collection of 5,000 theorems. Ramanujan reportedly studied the contents of the book in detail. The next year, Ramanujan independently developed and investigated the
Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s and calculated the
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural l ...
up to 15 decimal places. His peers at the time said they "rarely understood him" and "stood in respectful awe" of him. 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. He received a scholarship to study at
Government Arts College, Kumbakonam The Government Arts College, previously known as the Government Arts College for Men, is an autonomous college based in the town of Kumbakonam in Tamil Nadu, India. This college is functioning under Bharathidasan University. The Vice-Chancellor ...
, 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. In August 1905, Ramanujan ran away from home, heading towards
Visakhapatnam , image_alt = , image_caption = From top, left to right: Visakhapatnam aerial view, Vizag seaport, Simhachalam Temple, Aerial view of Rushikonda Beach, Beach road, Novotel, Novotel Visakhapatnam, INS Kursura (S20), INS ...
, and stayed in
Rajahmundry Rajahmundry, officially known as Rajamahendravaram, is a city in the Indian state of Andhra Pradesh and District headquarters of East Godavari district. It is the sixth most populated city in the state. During British rule, the District of Rajah ...
for about a month. He later enrolled at
Pachaiyappa's College Pachaiyappa's College is one of the oldest educational institutions in Chennai, in the South Indian state of Tamil Nadu. In addition, it is the first sole Indian college in Madras Presidency. History Pachaiyappa's College, Chennai is the resul ...
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. 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. In 1910, after a meeting between the 23-year-old Ramanujan and the founder of the
Indian Mathematical Society Indian Mathematical Society (IMS) is the oldest organization in India devoted to the promotion of study and research in mathematics. The Society was founded in April 1907 by V. Ramaswami Aiyar with its headquarters at Pune. The Society started ...
,
V. Ramaswamy Aiyer V. Ramaswamy Aiyer (4 August 1871 – 22 January 1936) was a civil servant The civil service is a collective term for a sector of government composed mainly of career civil servants hired on professional merit rather than appointed or elect ...
, Ramanujan began to get recognition in Madras's mathematical circles, leading to his inclusion as a researcher at the
University of Madras The University of Madras (informally known as Madras University) is a public university, public State university (India), state university in Chennai, Tamil Nadu, India. Established in 1857, it is one of the oldest and among the most prestigiou ...
.


Adulthood in India

On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 April 1994), a girl his mother had selected for him a year earlier and who was ten years old when they married.> 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 Karur District is one of the 38 districts (a district located centrally along the Kaveri and Amaravathi River, Amaravathi rivers) in the Indian state of Tamil Nadu. The main town in Karur District is the city of Karur, which is also the district ...
) Railway Station. Ramanujan's father did not participate in the marriage ceremony. 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. After the marriage, Ramanujan developed a
hydrocele testis A hydrocele testis is an accumulation of clear fluid within the cavum vaginale, the potential space between the layers of the tunica vaginalis of the testicle. It is the most common form of hydrocele and is often referred to simply as a "hydrocele ...
.> 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. 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. In late 1910, Ramanujan was sick again. He feared for his health, and told his friend R. Radakrishna Iyer to "hand is notebooksover to Professor Singaravelu Mudaliar
he mathematics professor at Pachaiyappa's College He or HE may refer to: Language * He (pronoun), an English pronoun * He (kana), the romanization of the Japanese kana 㸠* He (letter), the fifth letter of many Semitic alphabets * He (Cyrillic), a letter of the Cyrillic script called ''He'' ...
or to the British professor Edward B. Ross, of the
Madras Christian College Madras Christian College (MCC) is a liberal arts and sciences college in Chennai, India. Founded in 1837, MCC is one of Asia's oldest extant colleges. The college is affiliated to the University of Madras but functions as an autonomous institu ...
." After Ramanujan recovered and retrieved his notebooks from Iyer, he took a train from Kumbakonam to
Villupuram Viluppuram, Villupuram, or Vizhuppuram () is a Municipality and the administrative headquarters of Viluppuram district. Located south west of a Tiruvannamalai and north west of Cuddalore null The town serves as a major railway junction, ...
, a city under French control. In 1912, Ramanujan moved with his wife and mother to a house in Saiva Muthaiah Mudali street, George Town,
Madras Chennai (, ), formerly known as Madras ( the official name until 1996), is the capital city of Tamil Nadu, the southernmost Indian state. The largest city of the state in area and population, Chennai is located on the Coromandel Coast of th ...
, where they lived for a few months. In May 1913, upon securing a research position at Madras University, Ramanujan moved with his family to
Triplicane Triplicane, known in the vernacular as Thiruvallikeni, is one of the oldest neighbourhoods of Chennai, India. It is situated on the Bay of Bengal coast and about from Fort St George. The average elevation of the neighbourhood is 14 metres ...
.


Pursuit of career in mathematics

In 1910, Ramanujan met deputy collector
V. Ramaswamy Aiyer V. Ramaswamy Aiyer (4 August 1871 – 22 January 1936) was a civil servant The civil service is a collective term for a sector of government composed mainly of career civil servants hired on professional merit rather than appointed or elect ...
, who founded the Indian Mathematical Society. Wishing for a job at the revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled:
I was struck by the extraordinary mathematical results contained in
he notebooks He or HE may refer to: Language * He (pronoun), an English pronoun * He (kana), the romanization of the Japanese kana 㸠* He (letter), the fifth letter of many Semitic alphabets * He (Cyrillic), a letter of the Cyrillic script called ''He'' ...
I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.
Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras. Some of them looked at his work and gave him letters of introduction to
R. Ramachandra Rao Diwan Bahadur Raghunatha Rao Ramachandra Rao (c. 1871 – July 1936) was an Indian civil servant, mathematician and social and political activist who served as District Collector in British India. Early life and education Ramachandra Rao w ...
, the district collector for
Nellore Nellore is a city located on the banks of Penna River, in Nellore district of the Indian state of Andhra Pradesh. It serves as the headquarters of the district, as well as Nellore mandal and Nellore revenue division. It is the List of cities in ...
and the secretary of the Indian Mathematical Society. Rao was impressed by Ramanujan's research but doubted that it was his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable
Bombay Mumbai (, ; also known as Bombay — the official name until 1995) is the capital city of the Indian state of Maharashtra and the ''de facto'' financial centre of India. According to the United Nations, as of 2018, Mumbai is the second- ...
mathematician, in which Saldhana expressed a lack of understanding of his work but concluded that he was not a fraud. Ramanujan's friend C. V. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic integrity. Rao agreed to give him another chance, and listened as Ramanujan discussed
elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s, hypergeometric series, and his theory of
divergent series In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must ...
, which Rao said ultimately convinced him of Ramanujan's brilliance. When Rao asked him what he wanted, Ramanujan replied that he needed work and financial support. Rao consented and sent him to Madras. He continued his research with Rao's financial aid. With Aiyer's help, Ramanujan had his work published in the ''Journal of the Indian Mathematical Society. One of the first problems he posed in the journal was to find the value of: : \sqrt. He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied an incomplete solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely
nested radical In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression. Examples include :\sqrt, which arises in discussing the regular pentagon, and more co ...
s problem. : x+n+a = \sqrt Using this equation, the answer to the question posed in the ''Journal'' was simply 3, obtained by setting , , and . Ramanujan wrote his first formal paper for the ''Journal'' on the properties of
Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s. One property he discovered was that the denominators of the fractions of Bernoulli numbers are always divisible by six. He also devised a method of calculating based on previous Bernoulli numbers. One of these methods follows: It will be observed that if ''n'' is even but not equal to zero, # is a fraction and the numerator of in its lowest terms is a prime number, # the denominator of contains each of the factors 2 and 3 once and only once, # is an integer and consequently is an ''odd'' integer. In his 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries and three conjectures. His writing initially had many flaws. As ''Journal'' editor M. T. Narayana Iyengar noted:
Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary athematical reader unaccustomed to such intellectual gymnastics, could hardly follow him.
Ramanujan later wrote another paper and also continued to provide problems in the ''Journal''. In early 1912, he got a temporary job in the Madras Accountant General's office, with a monthly salary of 20 rupees. He lasted only a few weeks. Toward the end of that assignment, he applied for a position under the Chief Accountant of the Madras Port Trust. In a letter dated 9 February 1912, Ramanujan wrote:
Sir,
I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.
Attached to his application was a recommendation from E. W. Middlemast, a mathematics professor at the Presidency College, who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics". Three weeks after he applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making 30 rupees per month. At his office, Ramanujan easily and quickly completed the work he was given and spent his spare time doing mathematical research. Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits.


Contacting British mathematicians

In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. M. J. M. Hill of
University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public research university , endowment = £143 million (2020) , budget = ...
commented that Ramanujan's papers were riddled with holes. He said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the necessary educational background and foundation to be accepted by mathematicians. Although Hill did not offer to take Ramanujan on as a student, he gave thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University. The first two professors,
H. F. Baker Henry Frederick Baker FRS FRSE (3 July 1866 – 17 March 1956) was a British mathematician, working mainly in algebraic geometry, but also remembered for contributions to partial differential equations (related to what would become known as ...
and E. W. Hobson, returned Ramanujan's papers without comment. On 16 January 1913, Ramanujan wrote to
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
. Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible fraud. Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe". One of the theorems Hardy found amazing was on the bottom of page three (valid for ): : \int\limits_0^\infty \frac \times\frac\times\cdots\,dx = \frac \times\frac. Hardy was also impressed by some of Ramanujan's other work relating to infinite series: : 1 - 5\left(\frac12\right)^3 + 9\left(\frac\right)^3 - 13\left(\frac\right)^3 + \cdots = \frac : 1 + 9\left(\frac14\right)^4 + 17\left(\frac\right)^4 + 25\left(\frac\right)^4 + \cdots = \frac. The first result had already been determined by G. Bauer in 1859. The second was new to Hardy, and was derived from a class of functions called hypergeometric series, which had first been researched by Euler and Gauss. Hardy found these results "much more intriguing" than Gauss's work on integrals. After seeing Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy said the theorems "defeated me completely; I had never seen anything in the least like them before", and that they "must be true, because, if they were not true, no one would have the imagination to invent them". Hardy asked a colleague,
J. E. Littlewood John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to mathematical analysis, analysis, number theory, and differential equations, and had lengthy collaborations with G. H. H ...
, to take a look at the papers. Littlewood was amazed by Ramanujan's genius. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power". One colleague,
E. H. Neville Eric Harold Neville, known as E. H. Neville (1 January 1889 London, England – 22 August 1961 Reading, Berkshire, England) was an English mathematician. A heavily fictionalised portrayal of his life is rendered in the 2007 novel ''The Indian ...
, later remarked that "not one heoremcould have been set in the most advanced mathematical examination in the world". On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest in his work, adding that it was "essential that I should see proofs of some of your assertions". Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to plan for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip. In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to " go to a foreign land". Meanwhile, he sent Hardy a letter packed with theorems, writing, "I have found a friend in you who views my labour sympathetically." To supplement Hardy's endorsement,
Gilbert Walker Gilbert Walker may refer to: * Gilbert Walker (cricketer) (1888–1952), English cricketer *Gilbert Walker (physicist) (1868–1958), English physicist *Gilbert Carlton Walker Gilbert Carlton Walker (August 1, 1833 – May 11, 1885) was a Unit ...
, a former mathematical lecturer at
Trinity College, Cambridge Trinity College is a constituent college of the University of Cambridge. Founded in 1546 by Henry VIII, King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any college at either Cambridge ...
, looked at Ramanujan's work and expressed amazement, urging the young man to spend time at Cambridge. As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan". The board agreed to grant Ramanujan a monthly research scholarship of 75 rupees for the next two years at the
University of Madras The University of Madras (informally known as Madras University) is a public university, public State university (India), state university in Chennai, Tamil Nadu, India. Established in 1857, it is one of the oldest and among the most prestigiou ...
. While he was engaged as a research student, Ramanujan continued to submit papers to the ''Journal of the Indian Mathematical Society.'' In one instance, Iyer submitted some of Ramanujan's theorems on summation of series to the journal, adding, "The following theorem is due to S. Ramanujan, the mathematics student of Madras University." Later in November, British Professor Edward B. Ross of
Madras Christian College Madras Christian College (MCC) is a liberal arts and sciences college in Chennai, India. Founded in 1837, MCC is one of Asia's oldest extant colleges. The college is affiliated to the University of Madras but functions as an autonomous institu ...
, whom Ramanujan had met a few years before, stormed into his class one day with his eyes glowing, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had just arrived in the day's mail. In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to evaluate formerly unyielding integrals. Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England. Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparently had now accepted the proposal; Neville said, "Ramanujan needed no converting" and "his parents' opposition had been withdrawn". Apparently, Ramanujan's mother had a vivid dream in which the family goddess, the deity of Namagiri, commanded her "to stand no longer between her son and the fulfilment of his life's purpose". On 17 March 1914, Ramanujan traveled to England by ship, leaving his wife to stay with his parents in India.


Life in England

Ramanujan departed from Madras aboard the S.S. ''Nevasa'' on 17 March 1914. 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. 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. Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a
Jacobi Jacobi may refer to: * People with the surname Jacobi (surname), Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenva ...
", while Hardy said he "can compare him only with
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
or Jacobi." Ramanujan spent nearly five years in
Cambridge Cambridge ( ) is a university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge bec ...
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 Foundations of mathematics is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the natu ...
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 (the predecessor of the PhD degree) in March 1916 for his work on
highly composite number __FORCETOC__ A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller ...
s, sections of the first part of which had been published the preceding year in the ''
Proceedings of the London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...
.'' 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'. 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 Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural science, natural knowledge, incl ...
, the second Indian admitted, after
Ardaseer Cursetjee Ardaseer Cursetjee Wadia FRS (6 October 1808 – 16 November 1877) was an Indian Parsi shipbuilder and engineer belonging to the Wadia ship building family. He is noted for having been the first Indian to be elected a Fellow of the Royal Socie ...
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 function In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...
s and the Theory of Numbers." On 13 October 1918, he was the first Indian to be elected a Fellow of Trinity College, Cambridge.


Illness and death

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 Tuberculosis (TB) is an infectious disease usually caused by '' Mycobacterium tuberculosis'' (MTB) bacteria. Tuberculosis generally affects the lungs, but it can also affect other parts of the body. Most infections show no symptoms, in ...
and a severe
vitamin A vitamin is an organic molecule (or a set of molecules closely related chemically, i.e. vitamers) that is an Nutrient#Essential nutrients, essential micronutrient that an organism needs in small quantities for the proper functioning of its ...
deficiency, and confined to a
sanatorium A sanatorium (from Latin '' sÄnÄre'' 'to heal, make healthy'), also sanitarium or sanitorium, are antiquated names for specialised hospitals, for the treatment of specific diseases, related ailments and convalescence. Sanatoriums are often ...
. In 1919, he returned to
Kumbakonam Kumbakonam (formerly spelt as Coombaconum or Combaconum) or Kudanthai is a city municipal corporation in the Thanjavur district in the States of India, Indian state of Tamil Nadu. It is located from Thanjavur and from Chennai and is the headq ...
,
Madras Presidency The Madras Presidency, or the Presidency of Fort St. George, also known as Madras Province, was an administrative subdivision (presidency) of British India. At its greatest extent, the presidency included most of southern India, including the ...
, 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. Ramanujan's widow,
Smt. Shrimati or Shreemati ( sa, शà¥à¤°à¥€à¤®à¤¤à¥€, translit=ÅšrÄ«matÄ«), abbreviated Smt., is a widely accepted Indian honorific (akin to Ms. in English) used when referring to an adult woman in some Indian languages, including Bengali, Hindi ...
Janaki Ammal, moved to
Bombay Mumbai (, ; also known as Bombay — the official name until 1995) is the capital city of the Indian state of Maharashtra and the ''de facto'' financial centre of India. According to the United Nations, as of 2018, Mumbai is the second- ...
. In 1931, she returned to Madras and settled in
Triplicane Triplicane, known in the vernacular as Thiruvallikeni, is one of the oldest neighbourhoods of Chennai, India. It is situated on the Bay of Bengal coast and about from Fort St George. The average elevation of the neighbourhood is 14 metres ...
, 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 State Bank of India (SBI) is an Indian multinational public sector bank and financial services statutory body headquartered in Mumbai, Maharashtra. SBI is the 49th largest bank in the world by total assets and ranked 221st in the ''Fortune ...
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 The Indian National Science Academy (INSA) is a national academy in New Delhi for Indian scientists in all branches of science and technology. In August 2019, Dr. Chandrima Shaha was appointed as the president of Indian National Science Academ ...
and the state governments of
Tamil Nadu Tamil Nadu (; , TN) is a States and union territories of India, state in southern India. It is the List of states and union territories of India by area, tenth largest Indian state by area and the List of states and union territories of India ...
,
Andhra Pradesh Andhra Pradesh (, abbr. AP) is a state in the south-eastern coastal region of India. It is the seventh-largest state by area covering an area of and tenth-most populous state with 49,386,799 inhabitants. It is bordered by Telangana to the ...
and
West Bengal West Bengal (, Bengali: ''Poshchim Bongo'', , abbr. WB) is a state in the eastern portion of India. It is situated along the Bay of Bengal, along with a population of over 91 million inhabitants within an area of . West Bengal is the fourt ...
. 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 Bruce Carl Berndt (born March 13, 1939, in St. Joseph, Michigan) is an American mathematician. Berndt attended college at Albion College, graduating in 1961, where he also ran track. He received his master's and doctoral degrees from the Universi ...
and
Béla Bollobás Béla Bollobás FRS (born 3 August 1943) is a Hungarian-born British mathematician who has worked in various areas of mathematics, including functional analysis, combinatorics, graph theory, and percolation. He was strongly influenced by Paul E ...
made it a point to visit her while in India. She died at her Triplicane residence in 1994. A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D. A. B. Young concluded that his medical
symptom Signs and symptoms are the observed or detectable signs, and experienced symptoms of an illness, injury, or condition. A sign for example may be a higher or lower temperature than normal, raised or lowered blood pressure or an abnormality showin ...
s—including his past relapses, fevers, and hepatic conditions—were much closer to those resulting from hepatic
amoebiasis Amoebiasis, or amoebic dysentery, is an infection of the intestines caused by a parasitic amoeba ''Entamoeba histolytica''. Amoebiasis can be present with no, mild, or severe symptoms. Symptoms may include lethargy, loss of weight, colonic ulce ...
, an illness then widespread in Madras, than tuberculosis. He had two episodes of
dysentery Dysentery (UK pronunciation: , US: ), historically known as the bloody flux, is a type of gastroenteritis that results in bloody diarrhea. Other symptoms may include fever, abdominal pain, and a feeling of incomplete defecation. Complications ...
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. At the time, if properly diagnosed, amoebiasis was a treatable and often curable disease; British soldiers who contracted it during the First World War were being successfully cured of amoebiasis around the time Ramanujan left England.


Personality and spiritual life

Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners. He lived a simple life at Cambridge. 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 Namakkal () is a special grade municipality and the headquarters of Namakkal district in the Indian state of Tamil Nadu. It is the first ISO 14001-2004 certified municipality in Asia for environmental management, specifically the provision a ...
. He looked to her for inspiration in his work and said he dreamed of blood drops that symbolised her consort,
Narasimha Narasimha ( sa, नरसिंह, lit=man-lion, ), sometimes rendered Narasingha, is the fourth avatar of the Hindu god Vishnu. He is regarded to have incarnated in the form of a part-lion, part-man being to slay Hiranyakashipu, to end rel ...
. Later he had visions of scrolls of complex mathematical content unfolding before his eyes. He often said, "An equation for me has no meaning unless it expresses a thought of God." Hardy cites Ramanujan as remarking that all religions seemed equally true to him. 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
vegetarian Vegetarianism is the practice of abstaining from the consumption of meat (red meat, poultry, seafood, insects, and the flesh of any other animal). It may also include abstaining from eating all by-products of animal slaughter. Vegetarianism m ...
ism. 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.


Mathematical achievements

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 Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...
for , one of which is given below: : \frac = \frac \sum^\infty_ \frac. This result is based on the negative
fundamental discriminant In mathematics, a fundamental discriminant ''D'' is an integer invariant in the theory of integral binary quadratic forms. If is a quadratic form with integer coefficients, then is the discriminant of ''Q''(''x'', ''y''). Conversely, every integer ...
with class number . Further, and , which is related to the fact that : e^ = 396^4 - 104.000000177\dots. This might be compared to
Heegner number In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factoriza ...
s, which have class number 1 and yield similar formulae. Ramanujan's series for converges extraordinarily rapidly and forms the basis of some of the fastest algorithms currently used to calculate . Truncating the sum to the first term also gives the approximation for , 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 In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as, :\frac = \frac \sum_^\infty \frac \frac to the form :\frac = \sum_^\infty s(k) \frac by using other well-defined sequences of integers s(k) obeying a cer ...
. 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 Prasanta Chandra Mahalanobis OBE, FNA, FASc, FRS (29 June 1893– 28 June 1972) was an Indian scientist and statistician. He is best remembered for the Mahalanobis distance, a statistical measure, and for being one of the members of the first ...
posed a problem: His intuition also led him to derive some previously unknown identities, such as : \begin & \left ( 1+2\sum_^\infty \frac \right )^ + \left (1+2\sum_^\infty \frac \right )^ \\ pt= & \frac \pi = \frac \end for all such that , \Re (\theta), <\pi and , \Im (\theta), <\pi, where is the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
, and related to a special value of the
Dedekind eta function In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string t ...
. Expanding into series of powers and equating coefficients of , , and gives some deep identities for the
hyperbolic secant In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the un ...
. In 1918, Hardy and Ramanujan studied the partition function extensively. They gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. In 1937,
Hans Rademacher Hans Adolph Rademacher (; 3 April 1892, Wandsbeck, now Hamburg-Wandsbek – 7 February 1969, Haverford, Pennsylvania, USA) was a German-born American mathematician, known for work in mathematical analysis and number theory. Biography Rademacher r ...
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 A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
. In the last year of his life, Ramanujan discovered
mock theta function In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight . The first examples of mock theta functions were described by Srinivasa Ramanu ...
s. For many years, these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms.


The Ramanujan conjecture

Although there are numerous statements that could have borne the name ''Ramanujan conjecture,'' one was highly influential on later work. In particular, the connection of this conjecture with conjectures of
André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
in algebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the size of the tau-function, which has as generating function the discriminant modular form Δ(''q''), a typical
cusp form In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion. Introduction A cusp form is distinguished in the case of modular forms for the modular gro ...
in the theory of
modular forms In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
. It was finally proven in 1973, as a consequence of
Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
's proof of the
Weil conjectures In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. Th ...
. The reduction step involved is complicated. Deligne won a
Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
in 1978 for that work. In his paper "On certain arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called (the
Ramanujan tau function The Ramanujan tau function, studied by , is the function \tau : \mathbb \rarr\mathbb defined by the following identity: :\sum_\tau(n)q^n=q\prod_\left(1-q^n\right)^ = q\phi(q)^ = \eta(z)^=\Delta(z), where with , \phi is the Euler function, is the ...
). He proved many congruences for these numbers, such as for primes . This congruence (and others like it that Ramanujan proved) inspired
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
(1954 Fields Medalist) to conjecture that there is a theory of Galois representations that "explains" these congruences and more generally all modular forms. is the first example of a modular form to be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The proof of
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
proceeds by first reinterpreting
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s and modular forms in terms of these Galois representations. Without this theory, there would be no proof of Fermat's Last Theorem.


Ramanujan's notebooks

While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of
looseleaf A loose leaf is a piece of paper of any kind that is not bound in place, or available on a continuous roll, and may be punched so as to be organized in a ring binder. Loose leaf paper may be sold as free sheets, or made up into notepads, where ...
paper. They were mostly written up without any derivations. This is probably the origin of the misapprehension that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician
Bruce C. Berndt Bruce Carl Berndt (born March 13, 1939, in St. Joseph, Michigan) is an American mathematician. Berndt attended college at Albion College, graduating in 1961, where he also ran track. He received his master's and doctoral degrees from the Universi ...
, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to prove most of his results, but chose not to record the proofs in his notes. This may have been for any number of reasons. Since paper was very expensive, Ramanujan did most of his work and perhaps his proofs on
slate Slate is a fine-grained, foliated, homogeneous metamorphic rock derived from an original shale-type sedimentary rock composed of clay or volcanic ash through low-grade regional metamorphism. It is the finest grained foliated metamorphic rock. ...
, after which he transferred the final results to paper. At the time, slates were commonly used by mathematics students in the
Madras Presidency The Madras Presidency, or the Presidency of Fort St. George, also known as Madras Province, was an administrative subdivision (presidency) of British India. At its greatest extent, the presidency included most of southern India, including the ...
. He was also quite likely to have been influenced by the style of
G. S. Carr George Shoobridge Carr (1837–1914) was a British mathematician. He wrote ''Synopsis of Pure Mathematics'' (1886). This book, first published in England in 1880, was read and studied closely by mathematician Srinivasa Ramanujan when he was a t ...
's book, which stated results without proofs. It is also possible that Ramanujan considered his work to be for his personal interest alone and therefore recorded only the results. The first notebook has 351 pages with 16 somewhat organised chapters and some unorganised material. The second has 256 pages in 21 chapters and 100 unorganised pages, and the third 33 unorganised pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself wrote papers exploring material from Ramanujan's work, as did G. N. Watson, B. M. Wilson, and Bruce Berndt. In 1976, George Andrews rediscovered a fourth notebook with 87 unorganised pages, the so-called Ramanujan's lost notebook, "lost notebook".


Hardy–Ramanujan number 1729

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:
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 Interesting number paradox, 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." The two different ways are: :1729=1^3+12^3=9^3+10^3. Generalisations of this idea have created the notion of "taxicab numbers".


Mathematicians' views of Ramanujan

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: Hardy further said: 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."Srinivasa Ramanujan
. Retrieved 2 December 2010.
He also said that he had "never met his equal, and can compare him only with
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
or
Jacobi Jacobi may refer to: * People with the surname Jacobi (surname), Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenva ...
". K. Srinivasa Rao has said, "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 John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to mathematical analysis, analysis, number theory, and differential equations, and had lengthy collaborations with G. H. H ...
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.


Posthumous recognition

The year after his death, ''Nature (journal), Nature'' listed Ramanujan among other distinguished scientists and mathematicians on a "Calendar of Scientific Pioneers" who had achieved eminence. Ramanujan's home state of
Tamil Nadu Tamil Nadu (; , TN) is a States and union territories of India, state in southern India. It is the List of states and union territories of India by area, tenth largest Indian state by area and the List of states and union territories of India ...
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. Since Ramanujan's centennial year, his birthday, 22 December, has been annually celebrated as Ramanujan Day by the
Government Arts College, Kumbakonam The Government Arts College, previously known as the Government Arts College for Men, is an autonomous college based in the town of Kumbakonam in Tamil Nadu, India. This college is functioning under Bharathidasan University. The Vice-Chancellor ...
, where he studied, and at the Indian Institute of Technology Madras, IIT Madras in
Chennai Chennai (, ), formerly known as Madras ( the official name until 1996), is the capital city of Tamil Nadu, the southernmost Indian state. The largest city of the state in area and population, Chennai is located on the Coromandel Coast of th ...
. 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. Shanmugha Arts, Science, Technology & Research Academy, SASTRA University, a private university based in
Tamil Nadu Tamil Nadu (; , TN) is a States and union territories of India, state in southern India. It is the List of states and union territories of India by area, tenth largest Indian state by area and the List of states and union territories of India ...
, 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. 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. 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''. Then Indian Prime Minister Manmohan Singh also declared that 2012 would be celebrated as National Mathematics Year and 22 December as National Mathematics Day (India), National Mathematics Day of India. Ramanujan IT City is an information technology (IT) special economic zone (SEZ) in
Chennai Chennai (, ), formerly known as Madras ( the official name until 1996), is the capital city of Tamil Nadu, the southernmost Indian state. The largest city of the state in area and population, Chennai is located on the Coromandel Coast of th ...
that was built in 2011. Situated next to the TIDEL Park, Tidel Park, it includes with two zones, with a total area of , including of office space.


Commemorative postal stamps

Commemorative stamps released by India Post (by year):


In popular culture

* ''The Man Who Loved Numbers'' is a 1988 PBS NOVA documentary about Ramanujan (S15, E9). * ''The Man Who Knew Infinity'' is a 2015 film based on Robert Kanigel, Kanigel's The Man Who Knew Infinity (book), book of the same name. British actor Dev Patel portrays Ramanujan. * ''Ramanujan (film), Ramanujan'', an Indo-British collaboration film chronicling Ramanujan's life, was released in 2014 by the independent film company Ramanujan (film)#Camphor Cinema, Camphor Cinema. The cast and crew include director Gnana Rajasekaran, cinematographer Sunny Joseph and editor B. Lenin. Indian and English stars Abhinay Vaddi, Suhasini Maniratnam, Bhama, Kevin McGowan and Michael Lieber star in pivotal roles. * 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. * ''Ramanujan (The Man Who Reshaped 20th Century Mathematics)'', an Indian docudrama film directed by Akashdeep released in 2018. * M. N. Krish's thriller novel ''The Steradian Trail'' weaves Ramanujan and his accidental discovery into its plot connecting religion, mathematics, finance and economics. * ''Partition'', a play by Ira Hauptman about Hardy and Ramanujan, was first performed in 2013. * The play ''First Class Man'' by Alter Ego Productions 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 (actor), Siddharth. * ''A Disappearing Number'' is a British stage production by the company Complicite that explores the relationship between Hardy and Ramanujan. * David Leavitt's novel ''The Indian Clerk'' explores the events following Ramanujan's letter to Hardy. * Google honoured Ramanujan on his 125th birth anniversary by replacing its logo with a Google Doodles, doodle on its home page. * 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.


Selected papers

* * * * * * * * * * * * * * * * Posthumously published extract of a longer, unpublished manuscript.


Further works of Ramanujan's mathematics

* George E. Andrews and
Bruce C. Berndt Bruce Carl Berndt (born March 13, 1939, in St. Joseph, Michigan) is an American mathematician. Berndt attended college at Albion College, graduating in 1961, where he also ran track. He received his master's and doctoral degrees from the Universi ...
, ''Ramanujan's Lost Notebook: Part I'' (Springer, 2005, ) * George E. Andrews and Bruce C. Berndt, ''Ramanujan's Lost Notebook: Part II'', (Springer, 2008, ) * George E. Andrews and Bruce C. Berndt, ''Ramanujan's Lost Notebook: Part III'', (Springer, 2012, ) * George E. Andrews and Bruce C. Berndt, ''Ramanujan's Lost Notebook: Part IV'', (Springer, 2013, ) * George E. Andrews and Bruce C. Berndt, ''Ramanujan's Lost Notebook: Part V'', (Springer, 2018, ) * M. P. Chaudhary, ''A simple solution of some integrals given by Srinivasa Ramanujan,'' (Resonance: J. Sci. Education – publication of Indian Academy of Science, 2008) * 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.


Selected publications on Ramanujan and his work

* * * * * * * * * * * * * * * * * *


Selected publications on works of Ramanujan

* :This book was originally published in 1927 after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime. The third reprint contains additional commentary by Bruce C. Berndt. * :These books contain photocopies of the original notebooks as written by Ramanujan. * :This book contains photo copies of the pages of the "Lost Notebook".
Problems posed by Ramanujan
Journal of the Indian Mathematical Society. * :This was produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai.


See also

* 1729 (number) * Brown numbers * List of amateur mathematicians * List of Indian mathematicians * Ramanujan graph * Ramanujan summation * Ramanujan's constant * Ramanujan's ternary quadratic form * Rank of a partition


References


External links


Media links

*
Feature Film on Mathematics Genius Ramanujan by Dev Benegal and Stephen Fry

BBC radio programme about Ramanujan – episode 5

A biographical song about Ramanujan's life
*


Biographical links

* * *


"Our Devoted Site for Great Mathematical Genius"


Other links

* * A Study Group For Mathematics
Srinivasa Ramanujan Iyengar


– An international journal devoted to Ramanujan
International Math Union Prizes
including a Ramanujan Prize * Hindu.com



* Hindu.com
The sponsor of Ramanujan
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Ramanujan
on ''Fried Eye'' * {{DEFAULTSORT:Ramanujan, Srinivasa Srinivasa Ramanujan, 1887 births 1920 deaths Scientists from Tamil Nadu 20th-century Indian mathematicians Indian Hindus Mental calculators Indian combinatorialists Indian number theorists Fellows of Trinity College, Cambridge Fellows of the Royal Society Pi-related people People from Erode district University of Madras alumni People from Thanjavur district 19th-century Indian mathematicians Number theorists 19th-century Hindus Infectious disease deaths in India