srinivasa ramanujan biography in english wikipedia

Srinivasa Ramanujan

(1887-1920)

Who Was Srinivasa Ramanujan?

After demonstrating an intuitive grasp of mathematics at a young age, Srinivasa Ramanujan began to develop his own theories and in 1911, he published his first paper in India. Two years later Ramanujan began a correspondence with British mathematician G. H. Hardy that resulted in a five-year-long mentorship for Ramanujan at Cambridge, where he published numerous papers on his work and received a B.S. for research. His early work focused on infinite series and integrals, which extended into the remainder of his career. After contracting tuberculosis, Ramanujan returned to India, where he died in 1920 at 32 years of age.

Srinivasa Ramanujan was born on December 22, 1887, in Erode, India, a small village in the southern part of the country. Shortly after this birth, his family moved to Kumbakonam, where his father worked as a clerk in a cloth shop. Ramanujan attended the local grammar school and high school and early on demonstrated an affinity for mathematics.

When he was 15, he obtained an out-of-date book called A Synopsis of Elementary Results in Pure and Applied Mathematics , Ramanujan set about feverishly and obsessively studying its thousands of theorems before moving on to formulate many of his own. At the end of high school, the strength of his schoolwork was such that he obtained a scholarship to the Government College in Kumbakonam.

A Blessing and a Curse

However, Ramanujan’s greatest asset proved also to be his Achilles heel. He lost his scholarship to both the Government College and later at the University of Madras because his devotion to math caused him to let his other courses fall by the wayside. With little in the way of prospects, in 1909 he sought government unemployment benefits.

Yet despite these setbacks, Ramanujan continued to make strides in his mathematical work, and in 1911, published a 17-page paper on Bernoulli numbers in the Journal of the Indian Mathematical Society . Seeking the help of members of the society, in 1912 Ramanujan was able to secure a low-level post as a shipping clerk with the Madras Port Trust, where he was able to make a living while building a reputation for himself as a gifted mathematician.

Around this time, Ramanujan had become aware of the work of British mathematician G. H. Hardy — who himself had been something of a young genius — with whom he began a correspondence in 1913 and shared some of his work. After initially thinking his letters a hoax, Hardy became convinced of Ramanujan’s brilliance and was able to secure him both a research scholarship at the University of Madras as well as a grant from Cambridge.

The following year, Hardy convinced Ramanujan to come study with him at Cambridge. During their subsequent five-year mentorship, Hardy provided the formal framework in which Ramanujan’s innate grasp of numbers could thrive, with Ramanujan publishing upwards of 20 papers on his own and more in collaboration with Hardy. Ramanujan was awarded a bachelor of science degree for research from Cambridge in 1916 and became a member of the Royal Society of London in 1918.

Doing the Math

"[Ramanujan] made many momentous contributions to mathematics especially number theory," states George E. Andrews, an Evan Pugh Professor of Mathematics at Pennsylvania State University. "Much of his work was done jointly with his benefactor and mentor, G. H. Hardy. Together they began the powerful "circle method" to provide an exact formula for p(n), the number of integer partitions of n. (e.g. p(5)=7 where the seven partitions are 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1). The circle method has played a major role in subsequent developments in analytic number theory. Ramanujan also discovered and proved that 5 always divides p(5n+4), 7 always divides p(7n+5) and 11 always divides p(11n+6). This discovery led to extensive advances in the theory of modular forms."

But years of hard work, a growing sense of isolation and exposure to the cold, wet English climate soon took their toll on Ramanujan and in 1917 he contracted tuberculosis. After a brief period of recovery, his health worsened and in 1919 he returned to India.

The Man Who Knew Infinity

Ramanujan died of his illness on April 26, 1920, at the age of 32. Even on his deathbed, he had been consumed by math, writing down a group of theorems that he said had come to him in a dream. These and many of his earlier theorems are so complex that the full scope of Ramanujan’s legacy has yet to be completely revealed and his work remains the focus of much mathematical research. His collected papers were published by Cambridge University Press in 1927.

Of Ramanujan's published papers — 37 in total — Berndt reveals that "a huge portion of his work was left behind in three notebooks and a 'lost' notebook. These notebooks contain approximately 4,000 claims, all without proofs. Most of these claims have now been proved, and like his published work, continue to inspire modern-day mathematics."

A biography of Ramanujan titled The Man Who Knew Infinity was published in 1991, and a movie of the same name starring Dev Patel as Ramanujan and Jeremy Irons as Hardy, premiered in September 2015 at the Toronto Film Festival.

QUICK FACTS

Name: Srinivasa Ramanujan
Birth Year: 1887
Birth date: December 22, 1887
Birth City: Erode
Birth Country: India
Gender: Male
Best Known For: Srinivasa Ramanujan was a mathematical genius who made numerous contributions in the field, namely in number theory. The importance of his research continues to be studied and inspires mathematicians today.
Education and Academia
Astrological Sign: Sagittarius
University of Madras
Cambridge University
Nacionalities
Death Year: 1920
Death date: April 26, 1920
Death City: Kumbakonam
Death Country: India

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CITATION INFORMATION

Article Title: Srinivasa Ramanujan Biography
Author: Biography.com Editors
Website Name: The Biography.com website
Url: https://www.biography.com/scientists/srinivasa-ramanujan
Access Date:
Publisher: A&E; Television Networks
Last Updated: September 10, 2019
Original Published Date: September 10, 2015

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MacTutor

Srinivasa aiyangar ramanujan.

A short uncouth figure, stout, unshaven, not over clean, with one conspicuous feature-shining eyes- walked in with a frayed notebook under his arm. He was miserably poor. ... He opened his book and began to explain some of his discoveries. I saw quite at once that there was something out of the way; but my knowledge did not permit me to judge whether he talked sense or nonsense. ... I asked him what he wanted. He said he wanted a pittance to live on so that he might pursue his researches.

I have passed the Matriculation Examination and studied up to the First Arts 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 strongly recommend the applicant. He is a young man of quite exceptional capacity in mathematics and especially in work relating to numbers. He has a natural aptitude for computation and is very quick at figure work.

I have had no university education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as 'startling'.

I was exceedingly interested by your letter and by the theorems which you state. You will however understand that, before I can judge properly of the value of what you have done, it is essential that I should see proofs of some of your assertions. Your results seem to me to fall into roughly three classes: (1) there are a number of results that are already known, or easily deducible from known theorems; (2) there are results which, so far as I know, are new and interesting, but interesting rather from their curiosity and apparent difficulty than their importance; (3) there are results which appear to be new and important...

I have found a friend in you who views my labours sympathetically. ... I am already a half starving man. To preserve my brains I want food and this is my first consideration. Any sympathetic letter from you will be helpful to me here to get a scholarship either from the university of from the government.

What was to be done in the way of teaching him modern mathematics? The limitations of his knowledge were as startling as its profundity.

... that it was extremely difficult because every time some matter, which it was thought that Ramanujan needed to know, was mentioned, Ramanujan's response was an avalanche of original ideas which made it almost impossible for Littlewood to persist in his original intention.

Batty Shaw found out, what other doctors did not know, that he had undergone an operation about four years ago. His worst theory was that this had really been for the removal of a malignant growth, wrongly diagnosed. In view of the fact that Ramanujan is no worse than six months ago, he has now abandoned this theory - the other doctors never gave it any support. Tubercle has been the provisionally accepted theory, apart from this, since the original idea of gastric ulcer was given up. ... Like all Indians he is fatalistic, and it is terribly hard to get him to take care of himself.

I think we may now hope that he has turned to corner, and is on the road to a real recovery. His temperature has ceased to be irregular, and he has gained nearly a stone in weight. ... There has never been any sign of any diminuation in his extraordinary mathematical talents. He has produced less, naturally, during his illness but the quality has been the same. .... He will return to India with a scientific standing and reputation such as no Indian has enjoyed before, and I am confident that India will regard him as the treasure he is. His natural simplicity and modesty has never been affected in the least by success - indeed all that is wanted is to get him to realise that he really is a success.

References ( show )

O Ore, Biography in Dictionary of Scientific Biography ( New York 1970 - 1990) . See THIS LINK .
Biography in Encyclopaedia Britannica. http://www.britannica.com/biography/Srinivasa-Ramanujan
B C Berndt and R A Rankin, Ramanujan : Letters and commentary ( Providence, Rhode Island, 1995) .
G H Hardy, Ramanujan ( Cambridge, 1940) .
R Kanigel, The man who knew infinity : A life of the genius Ramanujan ( New York, 1991) .
J N Kapur ( ed. ) , Some eminent Indian mathematicians of the twentieth century ( Kapur, 1989) .
S Ram, Srinivasa Ramanujan ( New Delhi, 1979) .
S Ramanujan, Collected Papers ( Cambridge, 1927) .
S R Ranganathan, Ramanujan : the man and the mathematician ( London, 1967) .
P K Srinivasan, Ramanujan : Am inspiration 2 Vols. ( Madras, 1968) .
P V Seshu Aiyar, The late Mr S Ramanujan, B.A., F.R.S., J. Indian Math. Soc. 12 (1920) , 81 - 86 .
G E Andrews, An introduction to Ramanujan's 'lost' notebook, Amer. Math. Monthly 86 (1979) , 89 - 108 .
B Berndt, Srinivasa Ramanujan, The American Scholar 58 (1989) , 234 - 244 .
B Berndt and S Bhargava, Ramanujan - For lowbrows, Amer. Math. Monthly 100 (1993) , 644 - 656 .
B Bollobas, Ramanujan - a glimpse of his life and his mathematics, The Cambridge Review (1988) , 76 - 80 .
B Bollobas, Ramanujan - a glimpse of his life and his mathematics, Eureka 48 (1988) , 81 - 98 .
J M Borwein and P B Borwein, Ramanujan and pi, Scientific American 258 (2) (1988) , 66 - 73 .
S Chandrasekhar, On Ramanujan, in Ramanujan Revisited ( Boston, 1988) , 1 - 6 .
L Debnath, Srinivasa Ramanujan (1887 - 1920) : a centennial tribute, International journal of mathematical education in science and technology 18 (1987) , 821 - 861 .
G H Hardy, The Indian mathematician Ramanujan, Amer. Math. Monthly 44 (3) (1937) , 137 - 155 .
G H Hardy, Srinivasa Ramanujan, Proc. London Math, Soc. 19 (1921) , xl-lviii.
E H Neville, Srinivasa Ramanujan, Nature 149 (1942) , 292 - 294 .
C T Rajagopal, Stray thoughts on Srinivasa Ramanujan, Math. Teacher ( India ) 11 A (1975) , 119 - 122 , and 12 (1976) , 138 - 139 .
K Ramachandra, Srinivasa Ramanujan ( the inventor of the circle method ) , J. Math. Phys. Sci. 21 (1987) , 545 - 564 .
K Ramachandra, Srinivasa Ramanujan ( the inventor of the circle method ) , Hardy-Ramanujan J. 10 (1987) , 9 - 24 .
R A Rankin, Ramanujan's manuscripts and notebooks, Bull. London Math. Soc. 14 (1982) , 81 - 97 .
R A Rankin, Ramanujan's manuscripts and notebooks II, Bull. London Math. Soc. 21 (1989) , 351 - 365 .
R A Rankin, Srinivasa Ramanujan (1887 - 1920) , International journal of mathematical education in science and technology 18 (1987) , 861 -.
R A Rankin, Ramanujan as a patient, Proc. Indian Ac. Sci. 93 (1984) , 79 - 100 .
R Ramachandra Rao, In memoriam S Ramanujan, B.A., F.R.S., J. Indian Math. Soc. 12 (1920) , 87 - 90 .
E Shils, Reflections on tradition, centre and periphery and the universal validity of science : the significance of the life of S Ramanujan, Minerva 29 (1991) , 393 - 419 .
D A B Young, Ramanujan's illness, Notes and Records of the Royal Society of London 48 (1994) , 107 - 119 .

Additional Resources ( show )

Other pages about Srinivasa Ramanujan:

Multiple entries in The Mathematical Gazetteer of the British Isles ,
Miller's postage stamps
Heinz Klaus Strick biography

Other websites about Srinivasa Ramanujan:

Dictionary of Scientific Biography
Dictionary of National Biography
Encyclopaedia Britannica
Ramanujan's last letter
Srinivasa Rao
Plus Magazine
A Sen ( An article about the influence of Carr's book on Ramanujan )
Kevin Brown ( Something else about 1729)
The mathematician and his legacy ( YouTube video )
Sci Hi blog
Google doodle
Mathematical Genealogy Project
MathSciNet Author profile
zbMATH entry

Honours ( show )

Honours awarded to Srinivasa Ramanujan

Fellow of the Royal Society 1918
Popular biographies list Number 1
Google doodle 2012

Cross-references ( show )

History Topics: Squaring the circle
Famous Curves: Ellipse
Societies: Indian Academy of Sciences
Societies: Indian Mathematical Society
Societies: Ramanujan Mathematical Society
Other: 16th March
Other: 1st April
Other: 2009 Most popular biographies
Other: 22nd December
Other: 27th February
Other: 8th February
Other: Cambridge Colleges
Other: Cambridge Individuals
Other: Earliest Known Uses of Some of the Words of Mathematics (D)
Other: Earliest Known Uses of Some of the Words of Mathematics (H)
Other: Jeff Miller's postage stamps
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Biography of Srinivasa Ramanujan, Mathematical Genius

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Srinivasa Ramanujan (born December 22, 1887 in Erode, India) was an Indian mathematician who made substantial contributions to mathematics—including results in number theory, analysis, and infinite series—despite having little formal training in math.

Fast Facts: Srinivasa Ramanujan

Full Name: Srinivasa Aiyangar Ramanujan
Known For: Prolific mathematician
Parents’ Names: K. Srinivasa Aiyangar, Komalatammal
Born: December 22, 1887 in Erode, India
Died: April 26, 1920 at age 32 in Kumbakonam, India
Spouse: Janakiammal
Interesting Fact: Ramanujan's life is depicted in a book published in 1991 and a 2015 biographical film, both titled "The Man Who Knew Infinity."

Early Life and Education

Ramanujan was born on December 22, 1887, in Erode, a city in southern India. His father, K. Srinivasa Aiyangar, was an accountant, and his mother Komalatammal was the daughter of a city official. Though Ramanujan’s family was of the Brahmin caste , the highest social class in India, they lived in poverty.

Ramanujan began attending school at the age of 5. In 1898, he transferred to Town High School in Kumbakonam. Even at a young age, Ramanujan demonstrated extraordinary proficiency in math, impressing his teachers and upperclassmen.

However, it was G.S. Carr’s book, "A Synopsis of Elementary Results in Pure Mathematics," which reportedly spurred Ramanujan to become obsessed with the subject. Having no access to other books, Ramanujan taught himself mathematics using Carr’s book, whose topics included integral calculus and power series calculations. This concise book would have an unfortunate impact on the way Ramanujan wrote down his mathematical results later, as his writings included too few details for many people to understand how he arrived at his results.

Ramanujan was so interested in studying mathematics that his formal education effectively came to a standstill. At the age of 16, Ramanujan matriculated at the Government College in Kumbakonam on a scholarship, but lost his scholarship the next year because he had neglected his other studies. He then failed the First Arts examination in 1906, which would have allowed him to matriculate at the University of Madras, passing math but failing his other subjects.

For the next few years, Ramanujan worked independently on mathematics, writing down results in two notebooks. In 1909, he began publishing work in the Journal of the Indian Mathematical Society, which gained him recognition for his work despite lacking a university education. Needing employment, Ramanujan became a clerk in 1912 but continued his mathematics research and gained even more recognition.

Receiving encouragement from a number of people, including the mathematician Seshu Iyer, Ramanujan sent over a letter along with about 120 mathematical theorems to G. H. Hardy, a lecturer in mathematics at Cambridge University in England. Hardy, thinking that the writer could either be a mathematician who was playing a prank or a previously undiscovered genius, asked another mathematician J.E. Littlewood, to help him look at Ramanujan’s work.

The two concluded that Ramanujan was indeed a genius. Hardy wrote back, noting that Ramanujan’s theorems fell into roughly three categories: results that were already known (or which could easily be deduced with known mathematical theorems); results that were new, and that were interesting but not necessarily important; and results that were both new and important.

Hardy immediately began to arrange for Ramanujan to come to England, but Ramanujan refused to go at first because of religious scruples about going overseas. However, his mother dreamed that the Goddess of Namakkal commanded her to not prevent Ramanujan from fulfilling his purpose. Ramanujan arrived in England in 1914 and began his collaboration with Hardy.

In 1916, Ramanujan obtained a Bachelor of Science by Research (later called a Ph.D.) from Cambridge University. His thesis was based on highly composite numbers, which are integers that have more divisors (or numbers that they can be divided by) than do integers of smaller value.

In 1917, however, Ramanujan became seriously ill, possibly from tuberculosis, and was admitted to a nursing home at Cambridge, moving to different nursing homes as he tried to regain his health.

In 1919, he showed some recovery and decided to move back to India. There, his health deteriorated again and he died there the following year.

Personal Life

On July 14, 1909, Ramanujan married Janakiammal, a girl whom his mother had selected for him. Because she was 10 at the time of marriage, Ramanujan did not live together with her until she reached puberty at the age of 12, as was common at the time.

Honors and Awards

1918, Fellow of the Royal Society
1918, Fellow of Trinity College, Cambridge University

In recognition of Ramanujan’s achievements, India also celebrates Mathematics Day on December 22, Ramanjan’s birthday.

Ramanujan died on April 26, 1920 in Kumbakonam, India, at the age of 32. His death was likely caused by an intestinal disease called hepatic amoebiasis.

Legacy and Impact

Ramanujan proposed many formulas and theorems during his lifetime. These results, which include solutions of problems that were previously considered to be unsolvable, would be investigated in more detail by other mathematicians, as Ramanujan relied more on his intuition rather than writing out mathematical proofs.

His results include:

An infinite series for π, which calculates the number based on the summation of other numbers. Ramanujan’s infinite series serves as the basis for many algorithms used to calculate π.
The Hardy-Ramanujan asymptotic formula, which provided a formula for calculating the partition of numbers—numbers that can be written as the sum of other numbers. For example, 5 can be written as 1 + 4, 2 + 3, or other combinations.
The Hardy-Ramanujan number, which Ramanujan stated was the smallest number that can be expressed as the sum of cubed numbers in two different ways. Mathematically, 1729 = 1 3 + 12 3 = 9 3 + 10 3 . Ramanujan did not actually discover this result, which was actually published by the French mathematician Frénicle de Bessy in 1657. However, Ramanujan made the number 1729 well known. 1729 is an example of a “taxicab number,” which is the smallest number that can be expressed as the sum of cubed numbers in n different ways. The name derives from a conversation between Hardy and Ramanujan, in which Ramanujan asked Hardy the number of the taxi he had arrived in. Hardy replied that it was a boring number, 1729, to which Ramanujan replied that it was actually a very interesting number for the reasons above.
Kanigel, Robert. The Man Who Knew Infinity: A Life of the Genius Ramanujan . Scribner, 1991.
Krishnamurthy, Mangala. “The Life and Lasting Influence of Srinivasa Ramanujan.” Science & Technology Libraries , vol. 31, 2012, pp. 230–241.
Miller, Julius. “Srinivasa Ramanujan: A Biographical Sketch.” School Science and Mathematics , vol. 51, no. 8, Nov. 1951, pp. 637–645.
Newman, James. “Srinivasa Ramanujan.” Scientific American , vol. 178, no. 6, June 1948, pp. 54–57.
O'Connor, John, and Edmund Robertson. “Srinivasa Aiyangar Ramanujan.” MacTutor History of Mathematics Archive , University of St. Andrews, Scotland, June 1998, www-groups.dcs.st-and.ac.uk/history/Biographies/Ramanujan.html.
Singh, Dharminder, et al. “Srinvasa Ramanujan's Contributions in Mathematics.” IOSR Journal of Mathematics , vol. 12, no. 3, 2016, pp. 137–139.
“Srinivasa Aiyangar Ramanujan.” Ramanujan Museum & Math Education Centre , M.A.T Educational Trust, www.ramanujanmuseum.org/aboutramamujan.htm.
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Srinivasa Ramanujan

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Srinivasa Ramanujan [lower-alpha 1] (22 December 1887 – 26 April 1920) was an Indian mathematician . Though he had almost no formal training in pure mathematics , he made substantial contributions to mathematical analysis , number theory , infinite series , and continued fractions , including solutions to mathematical problems then considered unsolvable.

Ramanujan initially developed his own mathematical research in isolation. According to Hans Eysenck , "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". [4] Seeking mathematicians who could better understand his work, in 1913 he began a postal correspondence with the English mathematician G. H. Hardy at the University of Cambridge , 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 theorems , including some that "defeated me completely; I had never seen anything in the least like them before", [5] and some recently proven but highly advanced results.

During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations ). [6] Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime , the Ramanujan theta function , partition formulae and mock theta functions , have opened entire new areas of work and inspired further research. [7] Of his thousands of results, most have been proven correct. [8] The Ramanujan Journal , a scientific journal , was established to publish work in all areas of mathematics influenced by Ramanujan, [9] 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. [10] [11] He became one of the youngest Fellows of the Royal Society and only the second Indian member, and the first Indian to be elected a Fellow of Trinity College, Cambridge .

In 1919, ill health—now believed to have been hepatic amoebiasis (a complication from episodes of dysentery 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 ", containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976.

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Srinivasa Ramanujan (1887-1920) was an Indian mathematician who made great and original contributions to many mathematical fields, including complex analysis , number theory , infinite series , and continued fractions . He was "discovered" by G. H. Hardy and J. E. Littlewood, two world-class mathematicians at Cambridge, and enjoyed an extremely fruitful period of collaboration with them from 1914 to 1919. Unfortunately, his mathematical career was curtailed by health problems; he returned to India and died when he was only 32 years old.

Hardy, who was a great mathematician in his own right, recognized Ramanujan's genius from a series of letters that Ramanujan sent to mathematicians at Cambridge in 1913. Like much of his writing, the letters contained a dizzying array of unique and difficult results, stated without much explanation or proof. The contrast between Hardy, who was above all concerned with mathematical rigor and purity, and Ramanujan, whose writing was difficult to read and peppered with mistakes but bespoke an almost supernatural insight, produced a rich partnership.

Since his death, Ramanujan's writings (many contained in his famous notebooks) have been studied extensively. Some of his conjectures and assertions have led to the creation of new fields of study. Some of his formulas are believed to be true but as yet unproven.

There are many existing biographies of Ramanujan. The Man Who Knew Infinity , by Robert Kanigel, is an accessible and well-researched historical account of his life. The rest of this wiki will give a brief and light summary of the mathematical life of Ramanujan. As an appetizer, here is an anecdote from Kanigel's book.

In 1914, Ramanujan's friend P. C. Mahalanobis gave him a problem he had read in the English magazine Strand . The problem was to determine the number $ x $ of a particular house on a street where the houses were numbered $ 1,2,3,\ldots,n $. The house with number $ x $ had the property that the sum of the house numbers to the left of it equaled the sum of the house numbers to the right of it. The problem specified that $ 50 < n < 500 $.

Ramanujan quickly dictated a continued fraction for Mahalanobis to write down. The numerators and denominators of the convergents to that continued fraction gave all solutions $ (n,x) $ to the problem $($not just the particular one where $ 50 < n < 500). $ Mahalanobis was astonished, and asked Ramanujan how he had found the solution.

Ramanujan responded, "...It was clear that the solution should obviously be a continued fraction; I then thought, which continued fraction? And the answer came to my mind."

This is not the most illuminating answer! If we cannot duplicate the genius of Ramanujan, let us at least find the solution to the original problem. What is $ x $?

 Bonus: Which continued fraction did Ramanujan give Mahalanobis?

This anecdote and problem is taken from The Man Who Knew Infinity , a biography of Ramanujan by Robert Kanigel.

Taxicab numbers, nested radicals and continued fractions, ramanujan primes, ramanujan sums, the ramanujan $ \tau $ function and ramanujan's conjecture.

Many of Ramanujan's mathematical formulas are difficult to understand, let alone prove. For instance, an identity such as

\[\frac1{\pi} = \frac{2\sqrt{2}}{9801}\sum_{k=0}^{\infty} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\]

is not particularly easy to get a handle on. Perhaps this is why the most famous mathematical fact about Ramanujan is trivial and uninteresting, compared to the many brilliant theorems he proved.

The story goes that Hardy was visiting Ramanujan in the hospital, and remarked offhandedly that the taxi he had taken had a "dull number," 1729. Instantly Ramanujan replied, "No, it is a very interesting number! It is the smallest positive integer expressible as the sum of two positive cubes in two different ways."

That is, $ 1729 = 1^3+12^3 = 9^3+10^3 $.

Hardy and Wright proved in 1938 that for every $ n $, there is a positive integer $ \text{Ta}(n) $ that is expressible as the sum of two positive cubes in $ n $ different ways. So $ \text{Ta}(2) = 1729 $. $($The value of $ \text{Ta}(2) $ had been known since the $17^\text{th}$ century, which is in some sense characteristic of Ramanujan as well: as he was largely self-taught, he was often rediscovering theorems that were already well-known at the same time as he was constructing entirely new ones.$)$ The numbers $ \text{Ta}(n) $ are called taxicab numbers in honor of Hardy and Ramanujan.

Ramanujan developed several formulas that allowed him to evaluate nested radicals such as \[ 3 = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}}. \] This is a special case of a result from his notebooks, which is proved in the wiki on nested functions .

He also contributed greatly to the theory of continued fractions . One of the identities in his letter to Hardy was \[ 1+\frac{e^{-2\pi}}{1+\frac{e^{-4\pi}}{1+\frac{e^{-6\pi}}{\cdots}}} = \left( \sqrt{\frac{5+\sqrt{5}}2} - \frac{1+\sqrt{5}}2 \right)e^{2\pi/5}. \] This and several others along these lines were among the results that convinced Hardy that Ramanujan was a brilliant mathematician. This result is in fact a special case of the Rogers-Ramanujan continued fraction , which is of the form \[ R(q) = \frac{q^{1/5}}{1+\frac{q}{1+\frac{q^2}{1+\frac{q^3}{\cdots}}}} \] and is related to the theory of modular forms, a deep branch of modern number theory.

Ramanujan's work with modular forms produced the following celebrated divisibility results involving the partition function $ p(n) $: \[ \begin{align} p(5k+4) &\equiv 0 \pmod 5 \\ p(7k+5) &\equiv 0 \pmod 7 \\ p(11k+6) &\equiv 0 \pmod{11}. \end{align} \] Ramanujan commented in the paper in which he proved these results that there did not appear to be any other simple results of the same type. But in fact there are similar congruences of the form $ p(ak+b) \equiv 0 \pmod n $ for any $ n $ relatively prime to $ 6$; this is due to Ken Ono (2000). (Even for small $ n$, the values of $ a $ and $ b $ in the congruences are quite large.) The topic remains the subject of much contemporary research.

Ramanujan proved a generalization of Bertrand's postulate , as follows: Let $ \pi(x) $ be the number of positive prime numbers $ \le x $; then for every positive integer $ n $, there exists a prime number $ R_n $ such that \[ \pi(x)-\pi(x/2) \ge n \text{ for all } x \ge R_n. \] $($The case $ n = 1 $, $ R_n = 2 $ is Bertrand's postulate.$)$

The $ R_n $ are called Ramanujan primes .

The sum $ c_q(n) $ of the $n^\text{th}$ powers of the primitive $ q^\text{th}$ roots of unity is called a Ramanujan sum . It can be shown that these are multiplicative arithmetic functions , and in fact that \[c_q(n) = \frac{\mu\left(\frac qd\right)\phi(q)}{\phi\left(\frac qd\right)},\] where $ d = \text{gcd}(q,n)$, and $ \mu $ and $ \phi $ are the Mobius function and Euler's totient function , respectively.

Let $c_{2015}(n)$ be the sum of the $n^\text{th}$ powers of all the primitive $2015^\text{th}$ roots of unity, $\omega.$ Find the minimal value of $c_{2015}(n)$ for all positive integers $n$.

This year's problem

Ramanujan found nice infinite sums of the form $ \sum a_n c_q(n) $ or $ \sum a_q c_q(n) $ representing the standard arithmetic functions that are important in number theory. For instance, \[ d(n) = -\frac1{2\gamma+\ln(n)} \sum_{q=1}^{\infty} \frac{\ln(q)^2}{q} c_q(n), \] where $ \gamma $ is the Euler-Mascheroni constant .

Another example: the identity \[ \sum_{q=1}^{\infty} \frac{c_q(n)}{q} = 0 \] turns out to be equivalent to the prime number theorem .

Sums involving $ c_q(n) $ are known as Ramanujan sums ; these were also used in applications including the proof of Vinogradov's theorem that every sufficiently large odd positive integer is the sum of three primes.

Ramanujan's $ \tau $ function is defined by the formula \[ \sum_{n=1}^{\infty} \tau(n) q^n = q\prod_{n=1}^{\infty} (1-q^n)^{24} \] and is related to the theory of modular forms.

Ramanujan conjectured several properties of the $ \tau $ function, including \[ |\tau(p)| \le 2p^{11/2} \text{ for all primes } p. \] This turned out to be an extremely important and deep result, which was proved in 1974 by Pierre Deligne in his Fields-medal-winning proofs of the Weil conjectures on points on algebraic varieties over finite fields.

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May 1, 2014

Ramanujan’s Long Legacy

Although he died young, the mathematical prodigy’s genius lives on

By The Editors

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Self-taught mathematical prodigy Srinivasa Ramanujan had a brilliant but brief life. In 1920, at the age of 32, he died from a combination of illness and malnutrition. Before he passed, he filled various notebooks and manuscripts with nearly 4,000 results and conjectures. These documents have inspired mathematicians ever since, helping solve various conundrums and inspiring new fields of math (See “The Oracle” by Ariel Bleicher in the May issue of Scientific American ). Here is a timeline tracing Ramanujan’s intellectual legacy. 12/22/1887 Ramanujan (R) born in what is now Tamil Nadu, India. He shows an immense talent for math from a very young age 1/16/1913 Isolated from the greater mathematical community, R sends letters to several prominent English mathematicians. On this date he mails his first fateful letter to G.H. Hardy, who invites him to the University of Cambridge 4/14/1914 R arrives at the University of Cambridge. He has a fruitful five years of collaboration with Hardy 1/12/1920 R's last letter to Hardy 4/26/1920 R dies in Madras, India, weak and malnourished 11/14/1935 English mathematician G. N. Watson, at his retirement, describes his findings on R's "mock theta functions” 1943 German mathematician Hans Rademacher perfects R's asymptotic formula and arrives at a formula that's accurate enough to compute individual values of the partition function p(n) 7/1/1952 M. Rushforth publishes some of Ramanujan's previously unpublished manuscripts 1957 M. Newman proves some of R's claims for the function lambda(n) 1959 O. Kolberg proves that p(n) takes infinitely many even and odd values 1976 American mathematician G.E. Andrews rediscovers R's "lost notebook" in a box of his belongings at the University of Cambridge 1979 J. H. Conway and S. P. Norton use Ramanujan’s work to formulate the so-called Monstrous Moonshine Conjectures 1988 Frank Garvan of the University of Florida proves one of R's formulas and uses it to give a new proof of R's congruence for the partition function p(n) 2002 Dutch mathematician Sanders Zwegers formally defines the mock theta functions originally described by Ramanujan 2004 Inspired by Ramanujan, Jan Hendrik Bruinier and Jens Funke introduce harmonic Maass forms 2005 Andrews and Pedro Freitas establish an infinite family of extensions of Abel's Lemma and apply their results to obtain q-series identities 2007 W.Y.C. Chen and K Q. Ji provide combinatorial proofs of sums of tails of Euler's partition products 2007 Ken Ono and his colleagues use their development of the mock theta functions as the holomorphic parts of Maass wave forms to obtain a general theorem that has corollaries to the mock theta conjectures 1/2011 Ono and collaborators find solution linking the partition function p(n)to higher prime numbers 1/2011 Ono and collaborator describe first formula that directly calculates p(n) for any n 12/2011 Indian government declares R’s birthday National Mathematics Day

Srinivasa Ramanujan, 1887-1920

Some links to websites that discuss the life and work of ramanujan., wikipedia article , a good place to start., review of the book ramanujans lost notebook, part i by george e. andrews and bruce c. berndt, published in 2005., srinivasa ramanujan complete collection of his published papers and unpublished notebooks., ramanujan's notebooks photographic copy of ramanujan's first two notebooks., relevance of srinivasa ramanujan at the dawn of the new millennium by . srinivasa rao , chapter from a confercne proceeding, number theory and discrete mathematics (google books) the chapter begins on page 261. the story about 1729 appears on page 264., srinivasa ramanujan by james r. newman , chapter in the 4-volume anthology the world of mathematics (google books) includes ramnanujan's 1913 letter to hardy and some of the formuals in contained., st. andrews biography of ramanujan, srinivasa ramanujan , biography by mike hoffman, a friend of mine at the u. s. naval academy., the ramanujan journal the ramanujan journal will publish original research papers of the highest quality in all areas of mathematics influenced by srinivasa ramanujan. his remarkable discoveries have made a great impact on several branches of mathematics, revealing deep and fundamental connections., ramanujan: essay and surveys edited by bruce berndt and robert rankin, 2001 (google books), the ramanujan pages a collection of very accessible papers about ramanujan's work by titus piezas iii (who is this guy), sarah zubairy, '04, a ur math graduate who wrote three papers on ramanujan's work while she was an undergraduate here., short video about ramanujan, trailer for "the man who knew infinity.", ramanujan pi formula ., return to doug's teaching page.

The City of Madras, Now Called Chennai

Demographics of madras (chennai), poverty and the mathematical education of ramanujan, the health of ramanujan, ramanujan's family and friends, ramanujan's journey over the deep, dark sea to england, the mathematical style of ramanujan.

srinivasa ramanujan biography in english wikipedia

Ramanujan and Asperger's Syndrome

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Srinivasa Ramanujan Biography: Education, Contribution, Interesting Facts

Srinivasa Ramanujan: Srinivasa Ramanujan (1887–1920) was an Indian mathematician known for his brilliant, self-taught contributions to number theory and mathematical analysis. His work, including discoveries in infinite series and modular forms, has had a lasting impact on mathematics.

In this article, We have covered the Complete Biography of Srinivasa Ramanujan including his early childhood and education, Srinivasa Ramanujan’s Contribution to Mathematics, Interesting Facts about him, and many more.

Let’s dive right in.

Srinivasa Ramanujan Biography

Table of Content

Srinivasa Ramanujan Biography Overview

Srinivasa ramanujan early life and education, srinivasa ramanujan in england, srinivasa ramanujan contribution to mathematics, srinivasa ramanujan discovery, interesting facts about srinivasa ramanujan, awards and achievements of srinivasa ramanujan.

Here are some major details about Srinivasa Ramanujan FRS as mentioned below:

Srinivasa Ramanujan FRS was an Indian mathematician who was the mathematics god in contemporary times. The genius proposed some theories and works in the 20th century that are still relevant in this 21st century.

Birth of Srinivasa Ramanujan

Srinivasa Ramanujan was born on December 22, 1887, in Erode, India. A self-taught mathematician, he made significant contributions to number theory and mathematical analysis, despite facing limited formal education.He was born in a poor family. His father was a clerk. His mother was a homemaker.

He was born on 22nd December 1887. His native place is a south Indian town of Tamil Nadu, named Erode. His father Mr. Kuppuswamy Srinivasa Iyengar worked as a clerk in a saree shop. His mother Mrs. Komalatamma was a housewife.

Education of Srinivasa Ramanujan

Srinivasa Ramanujan did his early schooling in Madras. He was a self taught mathematician. He won so many academic prizes in his high school. In his college life started to study mathematics only. He performed bad in all other subjects. He dropped out of college due to the academic reasons. His theories got a final breakdown at this stage.

His early education was started in Madras. He fall in love with Mathematics at a very young age. He got many academis prizes in his school life. He continued to study one subject in collge and kept failing in other subjects. For this he became a dropped out student.

Final Breakthrough in life of Srinivasa Ramanujan

At this time Ramanujan sent his works to the International mathematicians. In 1912, he was working as a clerk in the Madras Post Trust Office. At this time he reached out to the famous mathematician G.H. Hardy. In 1913, he sent his 120 theorems to the famous mathematician G.H. Hardy. G.H. Hardy analysed his work and from here Ramanujan became a genius for the world. He moved to abroad to work more on these theories.

After dropping out from college, he started to send his work to International mathematicians. In 1912, he was appointed as a clerk of Madras Post Trust Office. The manager of Madras Post Trust Office, SN Aiyar helped him to communicate with G.H. Hardy.

Srinivasa Ramanujan’s time in England, particularly at Cambridge University, was a crucial period in his life marked by significant mathematical contributions, collaboration. Here is his time in England chronologically.

1914: Ramanujan arrived in England in April 1914, initially facing challenges in adapting to the climate and culture.
Collaboration with G. H. Hardy: Upon his arrival, he started collaborating with G. H. Hardy at Cambridge University. Hardy recognized Ramanujan’s exceptional talent and the two worked closely on various mathematical problems.
1916: Despite lacking formal academic credentials, Ramanujan was admitted to Cambridge University based on the strength of his mathematical work. He became a research student.
Contributions to Mathematics: Between 1914 and 1919, Ramanujan produced over 30 research papers, making profound contributions to number theory, modular forms, and elliptic functions, among other areas.
Recognition and Fellowships: In 1918, Ramanujan was elected a Fellow of the Royal Society, a prestigious recognition of his outstanding contributions to mathematics.
Health Challenges: Ramanujan faced health challenges during his time in England, exacerbated by malnutrition. His dedication to mathematics often led him to neglect his well-being.
Return to India: Due to deteriorating health, Ramanujan returned to India in 1919. His contributions to mathematics during his time in England left an indelible mark on the field.

Here are some major contributions of Srinivasa Ramanujan as mentioned below:

Developed advanced formulas for hypergeometric series and discovered relationships between different series.
Contributed to the theory of q-series and modular forms.
Identified the famous number 1729 as the smallest positive integer expressible as the sum of two cubes in two distinct ways.
Introduced and studied mock theta functions, extending the theory of theta functions in modular forms.
Investigated the partition function, yielding groundbreaking results and congruences that significantly advanced number theory.
Proposed the concept of the Ramanujan prime, contributing to the understanding of prime numbers.
Worked on the tau function, providing insights into modular forms and elliptic functions.
Made profound contributions to the theory of theta functions and elliptic functions, impacting the field of complex analysis.
Strived to unify different areas of mathematics, demonstrating a deep understanding of mathematical structures.
Collaborated with G. H. Hardy at Cambridge University, resulting in joint publications that enriched the field of mathematics.
Developed theorems in calculus, showcasing his ability to provide rigorous mathematical proofs for his intuitive results.

The following are some of the some of the notable discoveries of Srinivasa Ramanujan:

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Ramanujan had no formal training in mathematics and was largely self-taught. His early exposure to advanced mathematical concepts was through books he obtained and studied on his own.
Ramanujan was known for his intuitive approach to mathematics. He often presented results without formal proofs, and many of his theorems were later proven by other mathematicians.
By the age of 13, Ramanujan had independently developed theorems in advanced trigonometry and infinite series. His mathematical talent was evident from a young age.
As a child, Ramanujan discovered the formula for the sum of an infinite geometric series at the age of 14, which was published in the Journal of the Indian Mathematical Society.
During a visit to Ramanujan in the hospital, G. H. Hardy mentioned taking a rather dull taxi with the number 1729. Ramanujan immediately replied that 1729 is an interesting number as it is the smallest number that can be expressed as the sum of two cubes in two different ways: 1729=13+123=93+1031729=13+123=93+103. This incident led to the term “taxicab number.”
Ramanujan made substantial contributions to number theory, particularly in the areas of prime numbers, modular forms, and elliptic functions.
In 1918, Ramanujan was elected a Fellow of the Royal Society, a prestigious recognition of his outstanding contributions to mathematics.
Ramanujan faced health issues during his time in England, partly due to nutritional deficiencies. His dedication to mathematics sometimes led him to neglect his well-being.

Srinivasa Ramanujan FRS was a briliant personality from his childhood. He achieved so many things in his 35 years of life. Here is his Awards and Achievements given below.

He had completely read Loney’s book on Plane trigimetry at the age of 12.

He became the first Indian to be honored as a Fellow of the Royal Society.
In 1997, The Ramanujan Journal was launched to publish about his work.
2012 was declared as the National Mathematical Year in India.
Since 2021 in India, his birth anniversary has been observed as the National Mathematicians Day every year.
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FAQs on Srinivasa Ramanujan Biography

What is the meaning of frs in srinivasa ramanujan.

The meaning of FRS is Fellow of Royal Society.

When did Ramanujan got FRS?

On 2nd May 1918 Ramanujan got FRS .

Why is 1729 called Ramanujan number?

1729 as the sum of two positive cubes. It is known as the Hardy–Ramanujan number.

What is Ramanujan famous for?

Ramanujan’s contribution extends to mathematical fields such as complex analysis, number theory, infinite series, and continued fractions.

Why did Ramanujan died at 32?

At the age of 32 Ramanujan died due to tuberculosis.

What was the invention of Srinivasa Ramanujan?

Srinivasa Ramanujan made groundbreaking contributions to mathematics, discovering formulas for infinite series, introducing concepts like modular forms and mock theta functions, and making significant advancements in number theory. His work has had a lasting impact on diverse mathematical fields.

Who was the wife of Srinivasa Ramanujan?

Srinivasa Ramanujan’s wife was Janaki Ammal. They got married in July 1909 when Ramanujan was 21 years old, and Janaki was 10 years old. Their marriage was arranged, following the customs of the time in India.

Did Srinivasa Ramanujan have Child?

Yes, Srinivasa Ramanujan and his wife Janaki Ammal had a son named Namagiri Thayar. The couple named their son after the goddess Namagiri Thayar, to whom Ramanujan attributed the inspiration for some of his mathematical insights.

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Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence

Krishnaswami Alladi 0 ,
George E. Andrews 1 ,
Bruce C. Berndt 2 ,
Frank Garvan 3 ,
Ken Ono 4 ,
Peter Paule 5 ,
S. Ole Warnaar 6 ,
Ae Ja Yee 7

Department of Mathematics, University of Florida, Gainesville, USA

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Department of Mathematics, The Pennsylvania State University, University Park, USA

Department of mathematics, university of illinois, urbana, usa, department of mathematics, university of virginia, charlottesville, usa, risc, johannes kepler university linz, linz, austria, school of mathematics and physics, the university of queensland, brisbane, australia.

Authoritative book on an important mathematical legacy
Curated by foremost experts on Ramanujan
For students and researchers on a broad spectrum of mathematics and mathematical physics

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This authoritative volume covers aspects of the life and enduring mathematical research of Srinivasa Ramanujan. Born in the late 19th century, Ramanujan had little formal training in pure mathematics. This iconic figure made extraordinary contributions to many facets of mathematical analysis and number theory. During his short life, Ramanujan published 37 papers and curated in notebooks more than 3900 identities which he recorded without proof. Nearly all of his claims that were new have now been proven correct. He stated numerous results that were both original and highly unconventional. Many of these identities have led to major achievements in a wide range of areas of mathematics and theoretical physics. The eight editors of this Handbook have assembled articles on many aspects of Ramanujan’s life and mathematical legacy with a focus on the evolution of his discoveries into many important sub-disciplines of current mathematical research. Included are 234 articles supplied by 88 authors. The book will be of interest to students, teachers, researchers and anyone who is intrigued by the legacy of one of the most striking figures in the history of mathematics.

Srinivasa Ramanujan
Handbook Ramanujan
mathematical legacy
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sub-disciplines mathematical research

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Krishnaswami Alladi, Frank Garvan

George E. Andrews, Ae Ja Yee

Bruce C. Berndt

Peter Paule

S. Ole Warnaar

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Book Title : Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence

Editors : Krishnaswami Alladi, George E. Andrews, Bruce C. Berndt, Frank Garvan, Ken Ono, Peter Paule, S. Ole Warnaar, … Ae Ja Yee

Publisher : Springer Cham

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Hardcover ISBN : 978-3-031-50146-3 Due: 31 May 2024

Softcover ISBN : 978-3-031-50149-4 Due: 31 May 2024

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Srinivasa Ramanujan

Srinivasa Ramanujan (1887-1920)

One of the greatest mathematicians of all time, Srinivasa Ramanujan was born in 1887 in the Southern part of India. He is still remembered for his contributions to the field of mathematics. Theorems formulated by him are to date studied by students across the world and within very few years of his lifespan, he made some exceptional discoveries in mathematics.

His biography and achievements prove a lot about him and his struggles to contribute to the field of this subject. All this is also an essential part of the syllabus for aspirants preparing for the upcoming IAS Exam .

The facts, achievements and contributions presented by Srinivasa Ramanujan have not just been acknowledged within India, but also globally by leading mathematicians. Aspirants can also learn about other Indian mathematicians and their contributions , by visiting the linked article.

Srinivasa Ramanujan Biography [UPSC Notes]:- Download PDF Here

Indian Mathematician S. Ramanujan – Biography

Born in 1887, Ramanujan’s life, as said by Sri Aurobindo, was a “rags to mathematical riches” life story. His geniuses of the 20th century are still giving shape to 21st-century mathematics.

Discussed below is the history, achievements, contributions, etc. of Ramanujan’s life journey.

Birth –

Srinivasa Ramanujan was born on 22nd December 1887 in the south Indian town of Tamil Nad, named Erode.
His father, Kuppuswamy Srinivasa Iyengar worked as a clerk in a saree shop and his mother, Komalatamma was a housewife.
Since a very early age, he had a keen interest in mathematics and had already become a child prodigy

Srinivasa Ramanujan Education –

He attained his early education and schooling from Madras , where he was enrolled in a local school
His love for mathematics had grown at a very young age and was mostly self-taught
He was a promising student and had won many academic prizes in high school
But his love for mathematics proved to be a disadvantage when he reached college. As he continued to excel in only one subject and kept failing in all others . This resulted in him dropping out of college
However, he continued to work on his collection of mathematical theorems, ideologies and concepts until he got his final breakthrough

Final Break Through –

S. Ramanujam did not keep all his discoveries to himself but continued to send his works to International mathematicians
In 1912, he was appointed at the position of clerk in the Madras Post Trust Office, where the manager, S.N. Aiyar encouraged him to reach out to G.H. Hardy, a famous mathematician at the Cambridge University
In 1913, he had sent the famous letter to Hardy, in which he had attached 120 theorems as a sample of his work
Hardy along with another mathematician at Cambridge, J.E.Littlewood analysed his work and concluded it to be a work of true genius
It was after this that his journey and recognition as one of the greatest mathematicians had started

Death –

In 1919, Ramanujan’s health had started to deteriorate, after which he decided to move back to India
After his return in 1920, his health further worsened and he died at the age of just 32 years

The life of such great Indians and their contribution in various fields is an important part of the UPSC Syllabus . Candidates preparing for the upcoming civil services exam must analyse this information carefully.

Srinivasa Ramanujan Contributions

Between 1914 and 1914, while Ramanujan was in England, he along with Hardy published over a dozen research papers
During the time period of three years, he had published around 30 research papers
Hardy and Ramanujan had developed a new method, now called the circle method , to derive an asymptomatic formula for this function
His first paper published, a 17-page work on Bernoulli numbers that appeared in 1911 in the Journal of the Indian Mathematical Society
One remarkable result of the Hardy-Ramanujan collaboration was a formula for the number p(n) of partitions of a number ‘n’

Achievements of Srinivasa Ramanujan

At the age of 12, he had completely read Loney’s book on Plane Trignimetry and A Synopsis of Elementary Results in Pure and Applied Mathematics , which were way beyond the standard of a high school student
In 1916 , he was granted a Bachelor of Science degree “by research” at the Cambridge University
In 1918 , he became the first Indian to be honoured as a Fellow of the Royal Society
In 1997, The Ramanujan Journal was launched to publish work “in areas of mathematics influenced by Ramanujan”
The year 2012 was declared as the National Mathematical Year as it marked the 125th birth year of one of the greatest Indian mathematicians
Since 2021, his birth anniversary, December 22, is observed as the National Mathematicians Day every year in India

The intention behind encouraging the significance of mathematics was mainly to boost youngsters who are the future of the country and influence them to have a keen interest in analysing the scope of this subject.

Also, aspirants appearing in the civil services exam can choose mathematics as an optional and the success stories of IAS Toppers from the past have shown the scope of this subject.

To get details of UPSC 2024 , candidates can visit the linked article.

For any further information about the upcoming civil services examination , study material, preparation tips and strategy, candidates can visit the linked article.

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Srinivasa Ramanujan | Biography, Contributions & Speech in English

Srinivasa Ramanujan Speech in English

The story of Srinivasa Ramanujan is one that can inspire anyone. His work in mathematics was remarkable and his life was full of challenges, but he persevered through them all. In this post, we’ll explore some of the key factors that make Srinivasa Ramanujan’s story so inspirational.

Who Was Srinivasa Ramanujan?

Srinivasa Ramanujan was an Indian mathematician who made significant contributions to a number of fields, including number theory, analysis, and combinatorics. He was born in 1887 in Erode, Tamil Nadu, and began showing signs of his mathematical genius at a young age. When he was just 12 years old, he taught himself advanced trigonometry from a book borrowed from a friend. Ramanujan’s breakthrough came when he met English mathematician G. H. Hardy at the University of Cambridge in 1913. Hardy recognized Ramanujan’s potential and helped him publish his work in prestigious mathematical journals. Ramanujan made major contributions to the field of number theory and developed novel techniques for solving mathematical problems. He also worked on approximating pi and discovered an infinite series that can be used to do so. Ramanujan returned to India in 1919 and continued working on mathematics until his untimely death in 1920 at the age of 32. Despite his short career, Ramanujan left a lasting legacy and is considered one of the greatest mathematicians of all time.

Ramanujan number speciality

Ramanujan numbers are a special class of integers that are named after the Indian mathematician Srinivasa Ramanujan. They are characterized by the fact that they are the smallest numbers that can be expressed as the sum of two cubes in more than one way. The first Ramanujan number is 1, which can be expressed as 1 = 1^3 + 0^3. The second Ramanujan number is 33, which can be expressed as 33 = 3^3 + 3^3. Ramanujan numbers have been studied extensively by mathematicians and have been found to have a variety of interesting properties. For example, it is known that there are infinitely many Ramanujan numbers, and that they become increasingly rare as they get larger. The study of Ramanujan numbers has led to the development of some deep mathematical results, including a connection with modular forms and theta functions.

The Early Life of Srinivasa Ramanujan

Srinivasa Ramanujan was born on December 22, 1887, in the small village of Erode, Tamil Nadu, India. His father, Kuppuswamy Srinivasa Iyengar, worked as a clerk in a sari shop and his mother, Nagammal, was a housewife. He was the couple’s second child; they had another son named Lakshmi Narasimhan and a daughter named Thanuja. Ramanujan showed an early interest in mathematics. At the age of five he gave his first public lecture on the topic. When he was eleven years old he obtained a copy of George Shoobridge Carr’s Synopsis of Elementary Results in Pure Mathematics. He mastered this book and went on to teach himself advanced mathematics from books borrowed from local libraries. In 1903 Ramanujan entered Pachaiyappa’s College in Madras where he studied subjects including English, Telugu, Tamil, Arithmetic and Geometry. He excelled in mathematics but struggled with other subjects due to his poor English skills. In 1904 Ramanujan failed his first-year examinations but passed them after taking them again the following year.

Also Read: Important Maths Formulas for Class 8

Ramanujan’s Contribution to Mathematics

Ramanujan was an Indian mathematician who made significant contributions to the field of mathematics. He is best known for his work on integer partitions and his discovery of the Ramanujan prime. Ramanujan’s work on integer partitions was a major contribution to the field of number theory. He developed a method to calculate the number of ways a positive integer can be expressed as a sum of other positive integers. This work has been credited with helping to pave the way for the development of combinatorial Theory. Ramanujan also made significant contributions to the field of analysis. He developed a new method for calculating pi that was more accurate than any previous method. He also discovered several new Infinite Series, including the Ramanujan Prime Series. Ramanujan’s work has had a lasting impact on mathematics and has inspired many other mathematicians to make their own contributions to the field.

The Ramanujan Prime and the Ramanujan theta function

Ramanujan was an Indian mathematician who made significant contributions to the field of number theory. He is perhaps best known for his discovery of the Ramanujan prime and the Ramanujan theta function. The Ramanujan prime is a prime number that can be expressed as a sum of two cubes in more than one way. The first few Ramanujan primes are 7, 17, 37, 59, 67, 97, 101, 103, 137, 149, 163, 173, 179, 191, 193, 223, 227, 229… As you can see, the list goes on indefinitely. In fact, it is believed that there are infinitely many Ramanujan primes! The Ramanujan theta function is a special function that allows for the representation of certain modular forms. It has many applications in number theory and combinatorics.

The Legacy of Srinivasa Ramanujan

In his short life, Srinivasa Ramanujan made incredible strides in the field of mathematics. His work has inspired other mathematicians and thinkers for generations. Ramanujan was born in India in 1887. At a young age, he showed a remarkable aptitude for mathematics. He did not receive formal training in mathematics, but he taught himself advanced topics such as calculus and number theory. Ramanujan’s work on infinite series and continued fractions led to new insights in these fields. He also developed novel methods for solving mathematical problems. Ramanujan’s work has had a lasting impact on mathematics and has inspired many subsequent mathematicians.

Why is Ramanujan’s story so inspiring?

Ramanujan’s story is so inspiring because he was born in a poor family in India and worked hard to achieve greatness. He did not have any special ability, but he worked on the problem for years and years until he finally solved it. In his later years, he was able to travel across Europe and speak at conferences about his work with infinite precision.

Ramanujan’s genius was not just limited to mathematics; it also extended into other fields such as physics and music theory.

Also Check Out : Geometry Formulas For Class 8

How can we learn from Ramanujan’s example?

To be a mathematician, you have to be a genius. And to be a genius, you have to work hard. You must study mathematics for years and years before becoming good enough at it that people will call your name out when they hear about new discoveries in mathematics (or any subject). Then once again, there are some very specific requirements for being called “a great mathematician” or “a great genius”:

To write down your own theory so it is not just an idea but something that exists in reality somehow;
To show how this new theory works on its own without needing anyone else’s help; and (this one applies more often than not)

Frequently Asked Questions of Srinivasa Ramanujan

Where and when was srinivasa ramanujan born.

Srinivasa Ramanujan was born on December 22nd 1887 in Erode, India. His father was a clerk at the government railway office, and his mother was a housewife.

What are some of Ramanujan’s contributions to mathematics?

Ramanujan has made many contributions to mathematics, including:

The Ramanujan theta functions, which are used in number theory and analysis.
Some of the earliest work on modular forms and harmonic numbers.
A formula for a partition function that is important in statistical mechanics.

What is Srinivasa Ramanujan famous for?

Srinivasa Ramanujan is famous for his contributions to mathematical analysis, number theory and infinite series. He was also known for his ability to make accurate predictions about the behavior of numbers without having any formal training in mathematics.

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Srinivasa Ramanujan

Adulthood in India

Pursuit of career in mathematics, contacting british mathematicians, life in england, illness and death, personality and spiritual life, mathematical achievements, the ramanujan conjecture, ramanujan's notebooks, hardy–ramanujan number 1729, mathematicians' views of ramanujan, posthumous recognition, commemorative postal stamps, in popular culture, selected papers, further works of ramanujan's mathematics, selected publications on ramanujan and his work, selected publications on works of ramanujan, external links, media links, biographical links, other links.

Ramanujan (literally, "younger brother of Rama ", a Hindu deity) [12] was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode , in present-day Tamil Nadu . [13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district , worked as a clerk in a sari shop. [14] [2] His mother, Komalatammal, was a housewife and sang at a local temple. [15] They lived in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam . [16] 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. [12]

On 1 October 1892, Ramanujan was enrolled at the local school. [17] After his maternal grandfather lost his job as a court official in Kanchipuram, [18] Ramanujan and his mother moved back to Kumbakonam , and he was enrolled in Kangayan Primary School. [19] 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. [19]

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. [20] 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. [21] That year, Ramanujan entered Town Higher Secondary School , where he encountered formal mathematics for the first time. [21]

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. [22] [23] 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. [24] 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. [25]

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. [26] [27] Ramanujan reportedly studied the contents of the book in detail. [28] The next year, Ramanujan independently developed and investigated the Bernoulli numbers and calculated the Euler–Mascheroni constant up to 15 decimal places. [29] His peers at the time said they "rarely understood him" and "stood in respectful awe" of him. [24]

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. [30] He received a scholarship to study at Government Arts College, Kumbakonam , [31] [32] 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. [33] In August 1905, Ramanujan ran away from home, heading towards Visakhapatnam , and stayed in Rajahmundry [34] for about a month. [33] 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. [35] 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. [36]

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 . [37]

On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 April 1994), [38] a girl his mother had selected for him a year earlier and who was ten years old when they married. [39] [40] [41] 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. [42] 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. [43]

After the marriage, Ramanujan developed a hydrocele testis . [44] 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. [45]

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. [46]

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 ." [47] After Ramanujan recovered and retrieved his notebooks from Iyer, he took a train from Kumbakonam to Villupuram , a city under French control. [48] [49] 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. [50] In May 1913, upon securing a research position at Madras University, Ramanujan moved with his family to Triplicane . [51]

In 1910, Ramanujan met deputy collector V. Ramaswamy Aiyer , who founded the Indian Mathematical Society. [52] 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 [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department. [53]

Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras. [52] Some of them looked at his work and gave him letters of introduction to R. Ramachandra Rao , the district collector for Nellore and the secretary of the Indian Mathematical Society. [54] [55] [56] 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 mathematician, in which Saldhana expressed a lack of understanding of his work but concluded that he was not a fraud. [57] 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 integrals , hypergeometric series , and his theory of divergent series , which Rao said ultimately convinced him of Ramanujan's brilliance. [57] 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. [58]

One of the first problems he posed in the journal [30] was to find the value of:

$Srinivasa Ramanujan$

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 [59] 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 radicals problem.

$Srinivasa Ramanujan$

Using this equation, the answer to the question posed in the Journal was simply 3, obtained by setting x = 2 , n = 1 , and a = 0 . [60] Ramanujan wrote his first formal paper for the Journal on the properties of Bernoulli numbers . One property he discovered was that the denominators of the fractions of Bernoulli numbers (sequence A027642 in the OEIS ) are always divisible by six. He also devised a method of calculating B n based on previous Bernoulli numbers. One of these methods follows:

It will be observed that if n is even but not equal to zero,

the denominator of B n contains each of the factors 2 and 3 once and only once,
2 n (2 n − 1) B n / n is an integer and 2(2 n − 1) B n 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. [61] 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 [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him. [62]

Ramanujan later wrote another paper and also continued to provide problems in the Journal . [63] 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. [64] 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. [65]

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". [66] 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. [67] 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. [68]

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 commented that Ramanujan's papers were riddled with holes. [69] 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. [70] 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. [71]

The first two professors, H. F. Baker and E. W. Hobson , returned Ramanujan's papers without comment. [72] On 16 January 1913, Ramanujan wrote to G. H. Hardy . [73] Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible fraud. [74] Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe". [75] : 494 One of the theorems Hardy found amazing was on the bottom of page three (valid for 0 < a < b + 1 / 2 ):

$Srinivasa Ramanujan$

Hardy was also impressed by some of Ramanujan's other work relating to infinite series:

$Srinivasa Ramanujan$

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. [76] 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", [77] and that they "must be true, because, if they were not true, no one would have the imagination to invent them". [77] Hardy asked a colleague, J. E. Littlewood , 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". [75] : 494–495 One colleague, E. H. Neville , later remarked that "not one [theorem] could have been set in the most advanced mathematical examination in the world". [63]

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". [78] 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. [79] In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to " go to a foreign land ". [80] Meanwhile, he sent Hardy a letter packed with theorems, writing, "I have found a friend in you who views my labour sympathetically." [81]

To supplement Hardy's endorsement, Gilbert Walker , a former mathematical lecturer at Trinity College, Cambridge , looked at Ramanujan's work and expressed amazement, urging the young man to spend time at Cambridge. [82] 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". [83] The board agreed to grant Ramanujan a monthly research scholarship of 75 rupees for the next two years at the University of Madras . [84]

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 , 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. [85] 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. [86]

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. [87] 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". [63] 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". [63] On 17 March 1914, Ramanujan traveled to England by ship, [88] leaving his wife to stay with his parents in India. [89]

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

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. [43]

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. [42] [43]

A 1994 analysis of Ramanujan's medical records and symptoms by 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]

While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.

—Srinivasa Ramanujan [104]

Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners. [105] He lived a simple life at Cambridge. [106] 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 [107] 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. [108] He often said, "An equation for me has no meaning unless it expresses a thought of God." [109]

Hardy cites Ramanujan as remarking that all religions seemed equally true to him. [110] 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 . [111]

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:

$Srinivasa Ramanujan$

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 = 396 2 , which is related to the fact that

$Srinivasa Ramanujan$

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

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." [112] [113]

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

$Srinivasa Ramanujan$

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 . [114]

In the last year of his life, Ramanujan discovered mock theta functions . [115] For many years, these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms .

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 in algebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the size of the tau-function , which has a generating function as the discriminant modular form Δ( q ), a typical cusp form in the theory of modular forms . It was finally proven in 1973, as a consequence of Pierre Deligne 's proof of the Weil conjectures . The reduction step involved is complicated. Deligne won a Fields Medal in 1978 for that work. [7] [116]

In his paper "On certain arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called τ ( n ) (the Ramanujan tau function ). [117] He proved many congruences for these numbers, such as τ ( p ) ≡ 1 + p 11 mod 691 for primes p . This congruence (and others like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to conjecture that there is a theory of Galois representations that "explains" these congruences and more generally all modular forms. Δ( z ) 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 proceeds by first reinterpreting elliptic curves and modular forms in terms of these Galois representations. Without this theory, there would be no proof of Fermat's Last Theorem. [118]

While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of looseleaf 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 , 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 , after which he transferred the final results to paper. At the time, slates were commonly used by mathematics students in the Madras Presidency . He was also quite likely to have been influenced by the style of G. S. Carr '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. [119]

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. [119]

In 1976, George Andrews rediscovered a fourth notebook with 87 unorganised pages, the so-called "lost notebook" . [101]

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: [120]

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." [121]

The two different ways are:

$Srinivasa Ramanujan$

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

"Of course, we're always hoping. That's one reason I always read letters that come in from obscure places and are written in an illegible scrawl. I always hope it might be from another Ramanujan."

—Freeman Dyson on how another such genius might appear anywhere [122]

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. [75]

Hardy further said: [123]

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." [124] He also said that he had "never met his equal, and can compare him only with Euler or Jacobi". [124] Littlewood reportedly said that helping Ramanujan catch up with European mathematics beyond what was available in India was very difficult, because each new point mentioned to Ramanujan caused him to produce original ideas that prevented Littlewood from continuing the lesson. [125]

K. Srinivasa Rao has said, [126] "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. [115] [127]

The year after his death, Nature listed Ramanujan among other distinguished scientists and mathematicians on a "Calendar of Scientific Pioneers" who had achieved eminence. [128] 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. [129]

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. [130]

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. [130]

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 . [131] Then Indian Prime Minister Manmohan Singh also declared that 2012 would be celebrated as National Mathematics Year and 22 December as National Mathematics Day of India. [132]

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 m 2 ) , including 4.5 million square feet (420,000 m 2 ) of office space. [133]

Commemorative stamps released by India Post (by year):

The Man Who Loved Numbers is a 1988 PBS NOVA documentary about Ramanujan (S15, E9). [134]
The Man Who Knew Infinity is a 2015 film based on Kanigel's book of the same name . British actor Dev Patel portrays Ramanujan. [135] [136] [137]
Ramanujan , an Indo-British collaboration film chronicling Ramanujan's life, was released in 2014 by the independent film company Camphor Cinema . [138] The cast and crew include director Gnana Rajasekaran , cinematographer Sunny Joseph and editor B. Lenin . [139] [140] Indian and English stars Abhinay Vaddi , Suhasini Maniratnam , Bhama , Kevin McGowan and Michael Lieber star in pivotal roles. [141]
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. [142]
Ramanujan (The Man Who Reshaped 20th Century Mathematics) , an Indian docudrama film directed by Akashdeep released in 2018. [143]
M. N. Krish's thriller novel The Steradian Trail weaves Ramanujan and his accidental discovery into its plot connecting religion, mathematics, finance and economics. [144] [145]
Partition , a play by Ira Hauptman about Hardy and Ramanujan, was first performed in 2013. [146] [147] [148] [149]
The play First Class Man by Alter Ego Productions [150] 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 . [151]
A Disappearing Number is a British stage production by the company Complicite that explores the relationship between Hardy and Ramanujan. [152]
David Leavitt 's novel The Indian Clerk explores the events following Ramanujan's letter to Hardy. [153] [154]
Google honoured Ramanujan on his 125th birth anniversary by replacing its logo with a doodle on its home page. [155] [156]
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. [157]
Ramanujan, S. (1914). "Some definite integrals connected with Gauss's sums" . Messenger Math . 44 : 75–85.
Ramanujan, S. (1915). "On certain infinite series" . Messenger Math . 45 : 11–15.
Ramanujan, S. (1915). "Highly Composite Numbers" . Proceedings of the London Mathematical Society . 14 (1): 347–409. doi : 10.1112/plms/s2_14.1.347 .
Ramanujan, S. (1915). "On the number of divisors of a number" . The Journal of the Indian Mathematical Society . 7 (4): 131–133.
Ramanujan, S. (1915). "Short Note: On the sum of the square roots of the first n natural numbers" . The Journal of the Indian Mathematical Society . 7 (5): 173–175.
Ramanujan, S. (1916). "Some formulae in the analytical theory of numbers" . Messenger Math . 45 : 81–84.
Ramanujan, S. (1916). "A Series for Euler's Constant γ" . Messenger Math . 46 : 73–80.
Ramanujan, S. (1917). "On the expression of numbers in the form ax 2 + by 2 + cz 2 + du 2 " . Mathematical Proceedings of the Cambridge Philosophical Society . 19 : 11–21.
Hardy, G. H.; Ramanujan, S. (1917). "Asymptotic Formulae for the Distribution of Integers of Various Types" . Proceedings of the London Mathematical Society . 16 (1): 112–132. doi : 10.1112/plms/s2-16.1.112 .
Hardy, G. H. ; Ramanujan, Srinivasa (1918). "Asymptotic Formulae in Combinatory Analysis" . Proceedings of the London Mathematical Society . 17 (1): 75–115. doi : 10.1112/plms/s2-17.1.75 .
Hardy, G. H.; Ramanujan, Srinivasa (1918). "On the coefficients in the expansions of certain modular functions" . Proc. R. Soc. A . 95 (667): 144–155. Bibcode : 1918RSPSA..95..144H . doi : 10.1098/rspa.1918.0056 .
Ramanujan, Srinivasa (1919). "Some definite integrals" . The Journal of the Indian Mathematical Society . 11 (2): 81–88.
Ramanujan, S. (1919). "A proof of Bertrand's postulate" . The Journal of the Indian Mathematical Society . 11 (5): 181–183.
Ramanujan, S. (1920). "A class of definite integrals" . Quart. J. Pure. Appl. Math . 48 : 294–309. hdl : 2027/uc1.$b417568 .
Ramanujan, S. (1921). "Congruence properties of partitions" . Math. Z . 9 (1–2): 147–153. doi : 10.1007/BF01378341 . S2CID 121753215 . Posthumously published extract of a longer, unpublished manuscript.
George E. Andrews and Bruce C. Berndt , Ramanujan's Lost Notebook: Part I (Springer, 2005, ISBN 0-387-25529-X ) [158]
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) [159]
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.
Berndt, Bruce C. (1998). Butzer, P. L.; Oberschelp, W.; Jongen, H. Th. (eds.). Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe (PDF) . Turnhout, Belgium: Brepols Verlag. pp. 119–146. ISBN 978-2-503-50673-9 . Archived (PDF) from the original on 9 September 2004.
Berndt, Bruce C.; Rankin, Robert A. (1995). Ramanujan: Letters and Commentary . Vol. 9. Providence, Rhode Island: American Mathematical Society . ISBN 978-0-8218-0287-8 .
Berndt, Bruce C. ; Rankin, Robert A. (2001). Ramanujan: Essays and Surveys . Vol. 22. Providence, Rhode Island: American Mathematical Society . ISBN 978-0-8218-2624-9 .
Berndt, Bruce C. (2006). Number Theory in the Spirit of Ramanujan . Vol. 9. Providence, Rhode Island: American Mathematical Society . ISBN 978-0-8218-4178-5 .
Berndt, Bruce C. (1985). Ramanujan's Notebooks: Part I . New York: Springer. ISBN 978-0-387-96110-1 .
Berndt, Bruce C. (1999). Ramanujan's Notebooks: Part II . New York: Springer. ISBN 978-0-387-96794-3 .
Berndt, Bruce C. (2004). Ramanujan's Notebooks: Part III . New York: Springer. ISBN 978-0-387-97503-0 .
Berndt, Bruce C. (1993). Ramanujan's Notebooks: Part IV . New York: Springer. ISBN 978-0-387-94109-7 .
Berndt, Bruce C. (2005). Ramanujan's Notebooks: Part V . New York: Springer. ISBN 978-0-387-94941-3 .
Hardy, G. H. (March 1937). "The Indian Mathematician Ramanujan". The American Mathematical Monthly . 44 (3): 137–155. doi : 10.2307/2301659 . JSTOR 2301659 .
Hardy, G. H. (1978). Ramanujan . New York: Chelsea Pub. Co. ISBN 978-0-8284-0136-4 .
Hardy, G. H. (1999). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work . Providence, Rhode Island: American Mathematical Society. ISBN 978-0-8218-2023-0 .
Henderson, Harry (1995). Modern Mathematicians . New York: Facts on File Inc. ISBN 978-0-8160-3235-8 .
Kanigel, Robert (1991). The Man Who Knew Infinity: a Life of the Genius Ramanujan . New York: Charles Scribner's Sons. ISBN 978-0-684-19259-8 .
Leavitt, David (2007). The Indian Clerk (paperback ed.). London: Bloomsbury. ISBN 978-0-7475-9370-6 .
Narlikar, Jayant V. (2003). Scientific Edge: the Indian Scientist From Vedic to Modern Times . New Delhi, India: Penguin Books. ISBN 978-0-14-303028-7 .
Ono, Ken ; Aczel, Amir D. (13 April 2016). My Search for Ramanujan: How I Learned to Count . Springer . ISBN 978-3319255668 .
Sankaran, T. M. (2005). Srinivasa Ramanujan- Ganitha lokathile Mahaprathibha (Report) (in Malayalam). Kochi, India: Kerala Sasthra Sahithya Parishad .
Ramanujan, Srinivasa; Hardy, G. H.; Seshu Aiyar, P. V.; Wilson, B. M. ; Berndt, Bruce C. (2000). Collected Papers of Srinivasa Ramanujan . AMS. ISBN 978-0-8218-2076-6 .
S. Ramanujan (1957). Notebooks (2 Volumes) . Bombay: Tata Institute of Fundamental Research.
S. Ramanujan (1988). The Lost Notebook and Other Unpublished Papers . New Delhi: Narosa. ISBN 978-3-540-18726-4 .
Problems posed by Ramanujan , Journal of the Indian Mathematical Society.
S. Ramanujan (2012). Notebooks (2 Volumes) . Bombay: Tata Institute of Fundamental Research.
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

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↑ > Kanigel 1991 , p. 72
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↑ Kanigel 1991 , pp. 74–75
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↑ Srinivasan (1968), p. 49.
↑ Kanigel 1991 , p. 96
↑ Berndt & Rankin (2001) , p. 97.
↑ Kanigel 1991 , p. 105
↑ Letter from M. J. M. Hill to a C. L. T. Griffith (a former student who sent the request to Hill on Ramanujan's behalf), 28 November 1912.
↑ Kanigel 1991 , p. 106
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1 2 Kanigel 1991 , p. 168
↑ Letter, Hardy to Ramanujan, 8 February 1913.
↑ Letter, Ramanujan to Hardy, 22 January 1914.
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Biswas, Soutik (16 March 2006). "Film to celebrate mathematics genius" . BBC . Retrieved 24 August 2006 .
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
"Why Did This Mathematician's Equations Make Everyone So Angry?" . Youtube.com . Thoughty2. 11 April 2022 . Retrieved 29 June 2022 .
Srinivasa Ramanujan at the Mathematics Genealogy Project
O'Connor, John J.; Robertson, Edmund F. , "Srinivasa Ramanujan" , MacTutor History of Mathematics Archive , University of St Andrews
Weisstein, Eric Wolfgang (ed.). "Ramanujan, Srinivasa (1887–1920)" . ScienceWorld .
A short biography of Ramanujan
"Our Devoted Site for Great Mathematical Genius"
Wolfram, Stephen (27 April 2016). "Who Was Ramanujan?" .
A Study Group For Mathematics: Srinivasa Ramanujan Iyengar
The Ramanujan Journal – An international journal devoted to Ramanujan
International Math Union Prizes , including a Ramanujan Prize
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Hindu.com: The sponsor of Ramanujan
Bruce C. Berndt; Robert A. Rankin (2000). "The Books Studied by Ramanujan in India". American Mathematical Monthly . 107 (7): 595–601. doi : 10.2307/2589114 . JSTOR 2589114 . MR 1786233 .
"Ramanujan's mock theta function puzzle solved"
Ramanujan's papers and notebooks
Sample page from the second notebook
Ramanujan on Fried Eye
Clark, Alex. "163 and Ramanujan Constant" . Numberphile . Brady Haran . Archived from the original on 4 February 2018 . Retrieved 23 June 2018 .
Āryabhaṭīya
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Hindu–Arabic numeral system
Symbol for zero (0)
Infinite series expansions for the trigonometric functions
Kerala school of astronomy and mathematics
Jantar Mantar ( Jaipur , New Delhi , Ujjain , Varanasi )
Bapudeva Sastri (1821–1900)
Shankar Balakrishna Dikshit (1853–1898)
Sudhakara Dvivedi (1855–1910)
M. Rangacarya (1861–1916)
P. C. Sengupta (1876–1962)
B. B. Datta (1888–1958)
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A. N. Singh (1901–1954)
C. T. Rajagopal (1903–1978)
T. A. Saraswati Amma (1918–2000)
S. N. Sen (1918–1992)
K. S. Shukla (1918–2007)
K. V. Sarma (1919–2005)
Walter Eugene Clark
David Pingree
Islamic mathematics
Indian Statistical Institute
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Chennai Mathematical Institute
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Indian Institute of Science
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Srinivasa Ramanujan
Srinivasa Ramanujan (22 December 1887 - 26 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, ... Indian and English stars Abhinay Vaddi, Suhasini Maniratnam, Bhama, Kevin McGowan and Michael Lieber star in pivotal roles.
Srinivasa Ramanujan
Srinivasa Ramanujan (born December 22, 1887, Erode, India—died April 26, 1920, Kumbakonam) was an Indian mathematician whose contributions to the theory of numbers include pioneering discoveries of the properties of the partition function.. When he was 15 years old, he obtained a copy of George Shoobridge Carr's Synopsis of Elementary Results in Pure and Applied Mathematics, 2 vol. (1880 ...
Srinivasa Ramanujan
Srinivasa Ramanujan Aiyangar (December 22, 1887 - April 26, 1920) was an Indian mathematician. His father's name was K. Srinivasa Iyengar and his mother's name was Komalatammal. On 1st October 1892 Ramanujan was enrolled at local school. He had no formal training in mathematics. However, he has made a large contribution to number theory ...
Srinivasa Ramanujan
Srinivasa Ramanujan was a mathematical genius who made numerous contributions in the field, namely in number theory. ... wet English climate soon took their toll on Ramanujan and in 1917 he ...
Srinivasa Ramanujan (1887
Biography. Srinivasa Ramanujan was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series. Ramanujan was born in his grandmother's house in Erode, a small village about 400 km southwest of Madras (now Chennai).
Biography of Srinivasa Ramanujan, Mathematical Genius
Parents' Names: K. Srinivasa Aiyangar, Komalatammal. Born: December 22, 1887 in Erode, India. Died: April 26, 1920 at age 32 in Kumbakonam, India. Spouse: Janakiammal. Interesting Fact: Ramanujan's life is depicted in a book published in 1991 and a 2015 biographical film, both titled "The Man Who Knew Infinity."
Srinivasa Ramanujan
Srinivasa Ramanujan was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable.
Srinivasa Ramanujan
Srinivasa Ramanujan (1887-1920) was an Indian mathematician who made great and original contributions to many mathematical fields, including complex analysis, number theory, infinite series, and continued fractions. He was "discovered" by G. H. Hardy and J. E. Littlewood, two world-class mathematicians at Cambridge, and enjoyed an extremely fruitful period of collaboration with them ...
Ramanujan's Long Legacy
Here is a timeline tracing Ramanujan's intellectual legacy. 12/22/1887 Ramanujan (R) born in what is now Tamil Nadu, India. He shows an immense talent for math from a very young age. 1/16/1913 ...
Srinivasa Ramanujan Aiyangar
The Indian mathematician Srinivasa Ramanujan Aiyangar (1887-1920) is best known for his work on hypergeometric series and continued fractions. Srinivasa Ramanujan, born into a poor Brahmin family at Erode on Dec. 22, 1887, attended school in nearby Kumbakonam. By the time he was 13, he could solve unaided every problem in Loney's Trigonometry ...
Srinivasa Ramanujan
1887-1920. Indian mathematician notable for his brilliant mathematical intuition, which inspired him to create numerous new theorems and insights into analytical number theory. Born into a poor Brahmin family, Ramanujan was largely self-taught. His genius was recognized by the British mathematician G. H. Hardy, who arranged for Ramanujan to ...
Srinivasa Ramanujan, 1887-1920
Srinivasa Ramanujan, biography by Mike Hoffman, a friend of mine at the U. S. Naval Academy. "It is one of the most romantic stories in the history of mathematics: in 1913, the English mathematician G. H. Hardy received a strange letter from an unknown clerk in Madras, India. The ten-page letter contained about 120 statements of theorems on ...
The Man Who Knew Infinity (book)
The Man Who Knew Infinity. The Man Who Knew Infinity: A Life of the Genius Ramanujan is a biography of the Indian mathematician Srinivasa Ramanujan, written in 1991 by Robert Kanigel. The book gives a detailed account of his upbringing in India, his mathematical achievements and his mathematical collaboration with mathematician G. H. Hardy.
Srinivasa Ramanujan, a Mathematician Brilliant Beyond Comparison
The value of π to 14 decimal places is 3.141592653589793, so Ramanujan's formula provided a result accurate to 9 places on the second step. Altogether Ramanujan had 17 series formulas for the reciprocal of π. There is no way anyone could have created such a formula without a touch of genius.
Srinivasa Ramanujan Biography: Education, Contribution, Interesting Facts
Srinivasa Ramanujan was born on December 22, 1887, in Erode, India. A self-taught mathematician, he made significant contributions to number theory and mathematical analysis, despite facing limited formal education.He was born in a poor family. His father was a clerk. His mother was a homemaker. He was born on 22nd December 1887.
Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence
Born in the late 19th century, Ramanujan had little formal training in pure mathematics. This iconic figure made extraordinary contributions to many facets of mathematical analysis and number theory. During his short life, Ramanujan published 37 papers and curated in notebooks more than 3900 identities which he recorded without proof.
Srinivasa Ramanujan (1887-1920)
Birth -. Srinivasa Ramanujan was born on 22nd December 1887 in the south Indian town of Tamil Nad, named Erode. His father, Kuppuswamy Srinivasa Iyengar worked as a clerk in a saree shop and his mother, Komalatamma was a housewife. Since a very early age, he had a keen interest in mathematics and had already become a child prodigy.
Srinivasa Ramanujan Biography
Childhood & Early Life. Srinivasa Ramanujan was born on 22 December 1887, in Erode, Madras Presidency, British India, to K. Srinivasa Iyengar and his wife Komalatammal. His family was a humble one and his father worked as a clerk in a sari shop. His mother gave birth to several children after Ramanujan, but none survived infancy.
PDF Life and work of the Mathemagician Srinivasa Ramanujan
Introduction. Srinivasa Ramanujan, hailed as one of the greatest mathematicians of this cen-tury, left behind an incredibly vast and formidable amount of original work, which has greatly influenced the development and growth of some of the best research work in mathematics of this century. He was born at Erode, on Dec. 22, 1887.
Srinivasa Ramanujan
He was born in 1887 in Erode, Tamil Nadu, and began showing signs of his mathematical genius at a young age. When he was just 12 years old, he taught himself advanced trigonometry from a book borrowed from a friend. Ramanujan's breakthrough came when he met English mathematician G. H. Hardy at the University of Cambridge in 1913.
Srinivasa Ramanujan
Srinivasa Ramanujan FRS (/ ˈ s r iː n ɪ v ɑː s ə r ɑː ˈ m ɑː n ʊ dʒ ən / SREE-nih-vah-sə rah-MAH-nuuj-ən; [1] born Srinivasa Ramanujan Aiyangar, Tamil: [sriːniʋaːsa ɾaːmaːnud͡ʑan ajːaŋgar]; 22 December 1887 - 26 April 1920) [2] [3] was an Indian mathematician.Though he had almost no formal training in pure mathematics, he made substantial contributions to ...
1729 (number)
1729 as the sum of two positive cubes. 1729 is the smallest nontrivial taxicab number, [1] and is known as the Hardy-Ramanujan number, [2] after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation: [3] [4] [5] [6]

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Why is Ramanujan’s story so inspiring?

How can we learn from Ramanujan’s example?

Frequently Asked Questions of Srinivasa Ramanujan

What are some of Ramanujan’s contributions to mathematics?

What is Srinivasa Ramanujan famous for?

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