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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When he was 15, he obtained an out-of-date book called A Synopsis of Elementary Results in Pure and Applied Mathematics , Ramanujan set about feverishly and obsessively studying its thousands of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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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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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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 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.

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• 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
• Other: London Learned Societies
• Other: London individuals N-R
• Other: Most popular biographies – 2024
• Other: Oxford individuals
• Other: Popular biographies 2018

• IAS Preparation
• UPSC Preparation Strategy
• 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. 

Other Related Links:

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

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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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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.

About The Author

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I am Komal Gupta, the founder of Knowledge Glow, and my team and I aim to fuel dreams and help the readers achieve success. While you prepare for your competitive exams, we will be right here to assist you in improving your general knowledge and gaining maximum numbers from objective questions. We started this website in 2021 to help students prepare for upcoming competitive exams. Whether you are preparing for civil services or any other exam, our resources will be valuable in the process.

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• Srinivasa Ramanujan

Srinivasa Ramanujan was a famous Indian mathematician . In a lifespan of 32 years, Ramanujan contributed more to mathematics than many other accomplished mathematicians. English mathematician G. H. Hardy, who worked with him for a number of years, described him as a natural mathematical genius. Although he had no formal training in mathematics, he made significant contributions to mathematical analysis, infinite series, continued fractions and the number theory.

Ramanujan’s Early Life

Ramanujan was born on December 22, 1887, in the town of Erode in the South Indian state of Tamilnadu. He was born in an orthodox Hindu Brahmin family. His father’s name was K Srinivasa Iyengar and his mother was Komalatammal.

Even at a young age of 10, when mathematics was first introduced to him, Ramanujan had tremendous natural ability. He mastered trigonometry by the time he was 12 years old and developed theorems on his own. By the age of 17, he was conducting his own research in fields such as Bernoulli numbers and the Euler-Mascheroni constant.

Ramanujan’s Education

Ramanujan was a brilliant student, but his obsession with mathematics took a toll on the other subjects and he had to drop out of college as he was unable to get through his college examinations.

When he was 16 years old, he got a book entitled A Synopsis of Elementary Results in Pure and Applied Mathematics , which turned his life around. The book was just a compilation of thousands of mathematical facts, published mainly as a study aid for students. The book fascinated Ramanujan and he started working with the mathematical results given in it.

With no job and coming from a poor family, life was tough for him and he had to seek the help of friends to support himself while he worked on his mathematical discoveries and tried to get it noticed from accomplished mathematicians. Eventually an Indian mathematician, Ramachandra Rao, helped him get the post of a clerk at the Madras Port Trust.

Ramanujan Breaks into Mathematics

His life changed for the better in 1913 when he wrote to G. H. Hardy, an English mathematician. As a mathematician, Hardy was used to receiving prank letters from people claiming to have discovered something new in the field. Something about Ramanujan’s letter made him take a closer look and he and J. E. Littlewood, his collaborator, concluded that this one was different. The letter contained 120 statements on theorems related to the infinite series, improper integrals, continued fractions and the number theory.

Hardy wrote back to Ramanujan and his acknowledgement changed everything for the young mathematician. He became a research scholar at the University of Madras earning almost double what his job as a clerk was paying him. However, Hardy wanted him to come over to England.

Ramanujan’s Research

Ramanujan worked with Hardy for five years. Hardy was astonished by the genius of the young mathematician and said that he had never met anyone like him. His years at England were very decisive. He gained recognition and fame. Cambridge University gave him a Bachelor of Science degree just for his research in 1916 and he was elected a Fellow of the Royal Society in 1918.

Death and Legacy

Being a strict vegetarian and a religious person himself, the cultural differences and climatic conditions took a toll on his health. In 1917, he was hospitalized in a serious condition. His health improved in 1918 and he returned to India in 1919. However, his health problems got worse again and he died on April 26, 1920, in Chennai.

Ramanujan did not offer any proof for most of his mathematical results, but other mathematicians have validated and proved many of them. Some were known earlier and a few were found to be wrong, but the vast majority have been tested and shown to be correct.

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On January 16, 1913, a letter revealed a genius of mathematics. The missive came from Madras, a city – now known as Chennai – located in the south of India. The sender was a young 26-year-old clerk at the customs port, with a salary of £20 a year, enclosing nine sheets of formulas, incomprehensible at first sight. “Dear Sir, I have no University education but I have undergone the ordinary school course. I have made special investigation of divergent series in general and the results I get are termed by the local mathematicians as startling,” began the writing signed by S. Ramanujan. A century later, the legacy of this Indian genius continues to influence mathematics, physics or computation.

The renowned British mathematician G. H. Hardy was the stunned recipient of the document. It contained 120 formulas among which he identified one for knowing how many prime numbers there are between 1 and a certain number, and others that allowed one to calculate quickly the infinite decimals of the number pi. In some cases, Ramanujan had unwittingly arrived at conclusions already reached by western mathematicians , such as one of Bauer’s formulas for the decimals of pi, but many other formulas were entirely new. The formulas came alone, isolated, without formal demonstrations or statements. This lack of methodology almost led Hardy to throw the letter into the rubbish. However, in the end he concluded that: “They must be true because, if not, no one would have had the imagination to invent them.”

This statement resulted in the journey of Srinivasa Ramanujan (1887-1920) to Cambridge, where Hardy invited him to move in order to try to unravel the secret of this self-taught genius . Ramanujan arrived at Trinity College that same spring of 1913 at a time when colonialism was still justified on the basis of inferior races, a conviction that the extraordinary capacity of the Indian showed to be nonsense. However, during his nearly six years in Britain, Ramanujan had to endure the racism and contempt of English society.

Captivated by the number pi

Ramanujan is the icon of mathematical intuition. His case is a spectacular example of how mathematical language is inscribed in the brains of all human beings. In the same way that Mozart visualized music, this young Indian had the ability to sprout mathematical formulas with which he tried to explain the world. Coming from a poor family, Ramanujan formulated his first theorems at age 13, and by the age of 23 he was already a recognized local figure in the Indian mathematical community, even though he had no college education. He had been rejected twice in the entrance exam for leaving unanswered all those questions that were not related to mathematics.

However, this event did not stop him from continuing his training, which from 1906 became strictly self-taught. In this period, Ramanujan had a great obsession that would follow him until the end of his days: the number pi. From his hand came hundreds of different ways of calculating approximate values ​​of pi. In just the two notebooks he wrote before arriving at Cambridge are found 400 pages of formulas and theorems. Thanks to the theoretical foundations that Ramanujan laid a century ago, powerful computers have calculated the first 10 trillion decimals of the number pi. Going further is considered a test of fire in the world of computing.

Early death

Ramanujan’s method: intuitive and without formal demonstrations, clashed with the form of scientific work that demanded that the result be replicable, that is, that another mathematician could follow the approach. The mathematician used to claim that it was the protective goddess of his family, Namagiri, who showed him in dreams the equations of his formulas.

In spite of the peculiarities in his way of working, his results and the support that Hardy always gave him took him to the Royal Society and he became a member of the faculty of Trinity College . However, he was not able to enjoy much of these honours. Ramanujan, who had very fragile health throughout his life, contracted tuberculosis and was confined to a sanatorium in 1918. A year later he returned to his homeland, where he died in the following months aged only 32 years. This early death prevented him from completing the full proofs of his notes. His legacy, which has recently been portrayed by Hollywood in the film The Man Who Knew Infinity , goes beyond its exoticism and is a pillar of modern number theory.

By Beatriz Guillén for Ventana al Conocimiento

@BeaGTorres

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Stephen Wolfram

Who Was Ramanujan?

This week’s release of the movie The Man Who Knew Infinity (which I saw in rough form last fall through its mathematican-producers Manjul Bhargava and Ken Ono ) leads me to write about its subject, Srinivasa Ramanujan …

They used to come by physical mail. Now it’s usually email. From around the world, I have for many years received a steady trickle of messages that make bold claims — about prime numbers, relativity theory, AI, consciousness or a host of other things — but give little or no backup for what they say. I’m always so busy with my own ideas and projects that I invariably put off looking at these messages. But in the end I try to at least skim them — in large part because I remember the story of Ramanujan.

On about January 31, 1913 a mathematician named G. H. Hardy in Cambridge, England received a package of papers with a cover letter that began: “Dear Sir, I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £20 per annum. I am now about 23 years of age….” and went on to say that its author had made “startling” progress on a theory of divergent series in mathematics, and had all but solved the longstanding problem of the distribution of prime numbers . The cover letter ended: “Being poor, if you are convinced that there is anything of value I would like to have my theorems published…. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you. I remain, Dear Sir, Yours truly, S. Ramanujan”.

What followed were at least 11 pages of technical results from a range of areas of mathematics (at least 2 of the pages have now been lost). There are a few things that on first sight might seem absurd, like that the sum of all positive integers can be thought of as being equal to –1/12:

The concepts are familiar from college-level calculus. But these are not just complicated college-level calculus exercises. Instead, when one looks closely, each one has something more exotic and surprising going on — and seems to involve a quite different level of mathematics.

Will Knight

Today we can use Mathematica or Wolfram|Alpha to check the results — at least numerically. And sometimes we can even just type in the question and immediately get out the answer:

Needless to say, there’s a human story behind this: the remarkable story of Srinivasa Ramanujan .

He was born in a smallish town in India on December 22, 1887 (which made him not “about 23”, but actually 25 , when he wrote his letter to Hardy). His family was of the Brahmin (priests, teachers, …) caste but of modest means. The British colonial rulers of India had put in place a very structured system of schools, and by age 10 Ramanujan stood out by scoring top in his district in the standard exams. He also was known as having an exceptional memory, and being able to recite digits of numbers like pi as well as things like roots of Sanskrit words. When he graduated from high school at age 17 he was recognized for his mathematical prowess, and given a scholarship for college.

While in high school Ramanujan had started studying mathematics on his own — and doing his own research (notably on the numerical evaluation of Euler’s constant , and on properties of the Bernoulli numbers ). He was fortunate at age 16 (in those days long before the web!) to get a copy of a remarkably good and comprehensive (at least as of 1886) 1055-page summary of high-end undergraduate mathematics , organized in the form of results numbered up to 6165. The book was written by a tutor for the ultra-competitive Mathematical Tripos exams in Cambridge — and its terse “just the facts” format was very similar to the one Ramanujan used in his letter to Hardy.

Ramanujan seems to have supported himself by doing math tutoring — but soon became known around Madras as a math whiz, and began publishing in the recently launched Journal of the Indian Mathematical Society . His first paper — published in 1911 — was on computational properties of Bernoulli numbers (the same Bernoulli numbers that Ada Lovelace had used in her 1843 paper on the Analytical Engine). Though his results weren’t spectacular, Ramanujan’s approach was an interesting and original one that combined continuous (“what’s the numerical value?”) and discrete (“what’s the prime factorization?”) mathematics.

Meanwhile, Ramanujan’s expat friends were continuing to look for support for him — and he decided to start writing to British mathematicians himself, though with some significant help at composing the English in his letters. We don’t know exactly who all he wrote to first — although Hardy’s long-time collaborator John Littlewood mentioned two names shortly before he died 64 years later: H. F. Baker and E. W. Hobson . Neither were particularly good choices: Baker worked on algebraic geometry and Hobson on mathematical analysis, both subjects fairly far from what Ramanujan was doing. But in any event, neither of them responded.

And so it was that on Thursday, January 16, 1913 , Ramanujan sent his letter to G. H. Hardy.

The way the British academic system worked at that time — and basically until the 1960s — was that as soon as they graduated, top students could be elected to “college fellowships” that could last the rest of their lives. Hardy was at Trinity College — the largest and most scientifically distinguished college at Cambridge University — and when he graduated in 1900, he was duly elected to a college fellowship.

Hardy’s first research paper was about doing integrals like these:

His papers weren’t grand or visionary, but they were good examples of state-of-the-art mathematical craftsmanship. (As a colleague of Bertrand Russell ’s, he dipped into the new area of transfinite numbers, but didn’t do much with them.) Then in 1908, he wrote a textbook entitled A Course of Pure Mathematics — which was a good book, and was very successful in its time, even if its preface began by explaining that it was for students “whose abilities reach or approach something like what is usually described as ‘scholarship standard’”.

By 1910 or so, Hardy had pretty much settled into a routine of life as a Cambridge professor, pursuing a steady program of academic work. But then he met John Littlewood. Littlewood had grown up in South Africa and was eight years younger than Hardy, a recent Senior Wrangler, and in many ways much more adventurous. And in 1911 Hardy — who had previously always worked on his own — began a collaboration with Littlewood that ultimately lasted the rest of his life.

As a person, Hardy gives me the impression of a good schoolboy who never fully grew up. He seemed to like living in a structured environment, concentrating on his math exercises, and displaying cleverness whenever he could. He could be very nerdy — whether about cricket scores, proving the non-existence of God, or writing down rules for his collaboration with Littlewood. And in a quintessentially British way, he could express himself with wit and charm, but was personally stiff and distant — for example always theming himself as “G. H. Hardy”, with “Harold” basically used only by his mother and sister.

So in early 1913 there was Hardy: a respectable and successful, if personally reserved, British mathematician, who had recently been energized by starting to collaborate with Littlewood — and was being pulled in the direction of number theory by Littlewood’s interests there. But then he received the letter from Ramanujan.

Ramanujan’s letter began in a somewhat unpromising way, giving the impression that he thought he was describing for the first time the already fairly well-known technique of analytic continuation for generalizing things like the factorial function to non-integers . He made the statement that “My whole investigations are based upon this and I have been developing this to a remarkable extent so much so that the local mathematicians are not able to understand me in my higher flights.” But after the cover letter, there followed more than nine pages that listed over 120 different mathematical results.

Again, they began unpromisingly, with rather vague statements about having a method to count the number of primes up to a given size. But by page 3, there were definite formulas for sums and integrals and things. Some of them looked at least from a distance like the kinds of things that were, for example, in Hardy’s papers. But some were definitely more exotic. Their general texture, though, was typical of these types of math formulas. But many of the actual formulas were quite surprising — often claiming that things one wouldn’t expect to be related at all were actually mathematically equal.

At least two pages of the original letter have gone missing. But the last page we have again seems to end inauspiciously — with Ramanujan describing achievements of his theory of divergent series, including the seemingly absurd result about adding up all the positive integers, 1+2+3+4+…, and getting –1/12.

So what was Hardy’s reaction? First he consulted Littlewood. Was it perhaps a practical joke? Were these formulas all already known, or perhaps completely wrong? Some they recognized, and knew were correct. But many they did not. But as Hardy later said with characteristic clever gloss, they concluded that these too “must be true because, if they were not true, no one would have the imagination to invent them.”

Bertrand Russell wrote that by the next day he “found Hardy and Littlewood in a state of wild excitement because they believe they have found a second Newton , a Hindu clerk in Madras making 20 pounds a year.” Hardy showed Ramanujan’s letter to lots of people, and started making enquiries with the government department that handled India. It took him a week to actually reply to Ramanujan, opening with a certain measured and precisely expressed excitement: “I was exceedingly interested by your letter and by the theorems which you state.”

Then he went on: “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.” It was an interesting thing to say. To Hardy, it wasn’t enough to know what was true; he wanted to know the proof — the story — of why it was true. Of course, Hardy could have taken it upon himself to find his own proofs. But I think part of it was that he wanted to get an idea of how Ramanujan thought — and what level of mathematician he really was.

His letter went on — with characteristic precision — to group Ramanujan’s results into three classes: already known, new and interesting but probably not important, and new and potentially important. But the only things he immediately put in the third category were Ramanujan’s statements about counting primes, adding that “almost everything depends on the precise rigour of the methods of proof which you have used.”

Hardy had obviously done some background research on Ramanujan by this point, since in his letter he makes reference to Ramanujan’s paper on Bernoulli numbers . But in his letter he just says, “I hope very much that you will send me as quickly as possible… a few of your proofs,” then closes with, “Hoping to hear from you again as soon as possible.”

Ramanujan did indeed respond quickly to Hardy’s letter, and his response is fascinating. First, he says he was expecting the same kind of reply from Hardy as he had from the “Mathematics Professor at London”, who just told him “not [to] fall into the pitfalls of divergent series.” Then he reacts to Hardy’s desire for rigorous proofs by saying, “If I had given you my methods of proof I am sure you will follow the London Professor.” He mentions his result 1+2+3+4+…=–1/12 and says that “If I tell you this you will at once point out to me the lunatic asylum as my goal.” He goes on to say, “I dilate on this simply to convince you that you will not be able to follow my methods of proof… [based on] a single letter.” He says that his first goal is just to get someone like Hardy to verify his results — so he’ll be able to get a scholarship, since “I am already a half starving man. To preserve my brains I want food…”

So how was he getting his results? I’ll say more about this later. But he was certainly doing all sorts of calculations with numbers and formulas — in effect doing experiments. And presumably he was looking at the results of these calculations to get an idea of what might be true. It’s not clear how he figured out what was actually true — and indeed some of the results he quoted weren’t in the end true. But presumably he used some mixture of traditional mathematical proof, calculational evidence, and lots of intuition. But he didn’t explain any of this to Hardy.

Instead, he just started conducting a correspondence about the details of the results, and the fragments of proofs he was able to give. Hardy and Littlewood seemed intent on grading his efforts — with Littlewood writing about some result, for example, “(d) is still wrong, of course, rather a howler.” Still, they wondered if Ramanujan was “an Euler ”, or merely “a Jacobi ”. But Littlewood had to say, “The stuff about primes is wrong” — explaining that Ramanujan incorrectly assumed the Riemann zeta function didn’t have complex zeros, even though it actually has an infinite number of them, which are the subject of the whole Riemann hypothesis . (The Riemann hypothesis is still a famous unsolved math problem , even though an optimistic teacher suggested it to Littlewood as a project when he was an undergraduate…)

What about Ramanujan’s strange 1+2+3+4+… = –1/12? Well, that has to do with the Riemann zeta function as well. For positive integers, ζ(s) is defined as the sum

But back to the story. Hardy and Littlewood didn’t really have a good mental model for Ramanujan. Littlewood speculated that Ramanujan might not be giving the proofs they assumed he had because he was afraid they’d steal his work. (Stealing was a major issue in academia then as it is now.) Ramanujan said he was “pained” by this speculation, and assured them that he was not “in the least apprehensive of my method being utilized by others.” He said that actually he’d invented the method eight years earlier, but hadn’t found anyone who could appreciate it, and now he was “willing to place unreservedly in your possession what little I have.”

Meanwhile, even before Hardy had responded to Ramanujan’s first letter, he’d been investigating with the government department responsible for Indian students how he could bring Ramanujan to Cambridge. It’s not quite clear quite what got communicated, but Ramanujan responded that he couldn’t go — perhaps because of his Brahmin beliefs, or his mother, or perhaps because he just didn’t think he’d fit in. But in any case, Ramanujan’s supporters started pushing instead for him to get a graduate scholarship at the University of Madras . More experts were consulted, who opined that “His results appear to be wonderful; but he is not, now, able to present any intelligible proof of some of them,” but “He has sufficient knowledge of English and is not too old to learn modern methods from books.”

The university administration said their regulations didn’t allow a graduate scholarship to be given to someone like Ramanujan who hadn’t finished an undergraduate degree. But they helpfully suggested that “Section XV of the Act of Incorporation and Section 3 of the Indian Universities Act, 1904, allow of the grant of such a scholarship [by the Government Educational Department], subject to the express consent of the Governor of Fort St George in Council.” And despite the seemingly arcane bureaucracy, things moved quickly, and within a few weeks Ramanujan was duly awarded a scholarship for two years, with the sole requirement that he provide quarterly reports.

By the time he got his scholarship, Ramanujan had started writing more papers, and publishing them in the Journal of the Indian Mathematical Society . Compared to his big claims about primes and divergent series, the topics of these papers were quite tame. But the papers were remarkable nevertheless.

What’s immediately striking about them is how calculational they are — full of actual, complicated formulas. Most math papers aren’t that way. They may have complicated notation, but they don’t have big expressions containing complicated combinations of roots, or seemingly random long integers.

(As an aside, back in the late 1970s I started writing papers that involved formulas generated by computer. And in one particular paper , the formulas happened to have lots of occurrences of the number 9. But the experienced typist who typed the paper — yes, from a manuscript — replaced every “9” with a “g”. When I asked her why, she said, “Well, there are never explicit 9’s in papers!”)

Looking at Ramanujan’s papers, another striking feature is the frequent use of numerical approximations in arguments leading to exact results. People tend to think of working with algebraic formulas as an exact process — generating, for example, coefficients that are exactly 16, not just roughly 15.99999. But for Ramanujan, approximations were routinely part of the story, even when the final results were exact.

In some sense it’s not surprising that approximations to numbers are useful. Let’s say we want to know which is larger:

And of course if the numbers are very close one has to be careful about numerical round-off and so on. But for example in Mathematica and the Wolfram Language today — particularly with their built-in precision tracking for numbers — we often use numerical approximations internally as part of deriving exact results, much like Ramanujan did.

When Hardy asked Ramanujan for proofs, part of what he wanted was to get a kind of story for each result that explained why it was true. But in a sense Ramanujan’s methods didn’t lend themselves to that. Because part of the “story” would have to be that there’s this complicated expression, and it happens to be numerically greater than this other expression. It’s easy to see it’s true — but there’s no real story of why it’s true.

And the same happens whenever a key part of a result comes from pure computation of complicated formulas, or in modern times, from automated theorem proving. Yes, one can trace the steps and see that they’re correct. But there’s no bigger story that gives one any particular understanding of the results.

For most people it’d be bad news to end up with some complicated expression or long seemingly random number — because it wouldn’t tell them anything. But Ramanujan was different. Littlewood once said of Ramanujan that “every positive integer was one of his personal friends.” And between a good memory and good ability to notice patterns, I suspect Ramanujan could conclude a lot from a complicated expression or a long number. For him, just the object itself would tell a story.

Ramanujan was of course generating all these things by his own calculational efforts. But back in the late 1970s and early 1980s I had the experience of starting to generate lots of complicated results automatically by computer. And after I’d been doing it awhile, something interesting happened: I started being able to quickly recognize the “texture” of results — and often immediately see what might be likely to be true. If I was dealing, say, with some complicated integral, it wasn’t that I knew any theorems about it. I just had an intuition about, for example, what functions might appear in the result. And given this, I could then get the computer to go in and fill in the details — and check that the result was correct. But I couldn’t derive why the result was true, or tell a story about it; it was just something that intuition and calculation gave me.

Now of course there’s a fair amount of pure mathematics where one can’t (yet) just routinely go in and do an explicit computation to check whether or not some result is correct. And this often happens for example when there are infinite or infinitesimal quantities or limits involved. And one of the things Hardy had specialized in was giving proofs that were careful in handling such things. In 1910 he’d even written a book called Orders of Infinity that was about subtle issues that come up in taking infinite limits. (In particular, in a kind of algebraic analog of the theory of transfinite numbers, he talked about comparing growth rates of things like nested exponential functions — and we even make some use of what are now called Hardy fields in dealing with generalizations of power series in the Wolfram Language.)

So when Hardy saw Ramanujan’s “fast and loose” handling of infinite limits and the like, it wasn’t surprising that he reacted negatively — and thought he would need to “tame” Ramanujan, and educate him in the finer European ways of doing such things, if Ramanujan was actually going to reliably get correct answers.

Ramanujan was surely a great human calculator, and impressive at knowing whether a particular mathematical fact or relation was actually true. But his greatest skill was, I think, something in a sense more mysterious: an uncanny ability to tell what was significant, and what might be deduced from it.

Take for example his paper “ Modular Equations and Approximations to π ”, published in 1914, in which he calculates (without a computer of course):

It’s interesting to see in Ramanujan’s paper that even he occasionally didn’t know what was and wasn’t significant. For example, he noted:

To Hardy, Ramanujan’s way of working must have seemed quite alien. For Ramanujan was in some fundamental sense an experimental mathematician : going out into the universe of mathematical possibilities and doing calculations to find interesting and significant facts — and only then building theories based on them.

Hardy on the other hand worked like a traditional mathematician, progressively extending the narrative of existing mathematics. Most of his papers begin — explicitly or implicitly — by quoting some result from the mathematical literature, and then proceed by telling the story of how this result can be extended by a series of rigorous steps. There are no sudden empirical discoveries — and no seemingly inexplicable jumps based on intuition from them. It’s mathematics carefully argued, and built, in a sense, brick by brick.

A century later this is still the way almost all pure mathematics is done. And even if it’s discussing the same subject matter, perhaps anything else shouldn’t be called “mathematics”, because its methods are too different. In my own efforts to explore the computational universe of simple programs , I’ve certainly done a fair amount that could be called “mathematical” in the sense that it, for example, explores systems based on numbers .

Over the years, I’ve found all sorts of results that seem interesting. Strange structures that arise when one successively adds numbers to their digit reversals. Bizarre nested recurrence relations that generate primes. Peculiar representations of integers using trees of bitwise xors. But they’re empirical facts — demonstrably true, yet not part of the tradition and narrative of existing mathematics.

For many mathematicians — like Hardy — the process of proof is the core of mathematical activity. It’s not particularly significant to come up with a conjecture about what’s true; what’s significant is to create a proof that explains why something is true, constructing a narrative that other mathematicians can understand.

Particularly today, as we start to be able to automate more and more proofs, they can seem a bit like mundane manual labor, where the outcome may be interesting but the process of getting there is not. But proofs can also be illuminating. They can in effect be stories that introduce new abstract concepts that transcend the particulars of a given proof, and provide raw material to understand many other mathematical results.

For Ramanujan, though, I suspect it was facts and results that were the center of his mathematical thinking, and proofs felt a bit like some strange European custom necessary to take his results out of his particular context, and convince European mathematicians that they were correct.

But let’s return to the story of Ramanujan and Hardy. In the early part of 1913, Hardy and Ramanujan continued to exchange letters. Ramanujan described results; Hardy critiqued what Ramanujan said, and pushed for proofs and traditional mathematical presentation. Then there was a long gap, but finally in December 1913, Hardy wrote again, explaining that Ramanujan’s most ambitious results — about the distribution of primes — were definitely incorrect, commenting that “…the theory of primes is full of pitfalls, to surmount which requires the fullest of trainings in modern rigorous methods.” He also said that if Ramanujan had been able to prove his results it would have been “about the most remarkable mathematical feat in the whole history of mathematics.”

In January 1914 a young Cambridge mathematician named E. H. Neville came to give lectures in Madras, and relayed the message that Hardy was (in Ramanujan’s words) “anxious to get [Ramanujan] to Cambridge”. Ramanujan responded that back in February 1913 he’d had a meeting, along with his “superior officer”, with the Secretary to the Students Advisory Committee of Madras, who had asked whether he was prepared to go to England. Ramanujan wrote that he assumed he’d have to take exams like the other Indian students he’d seen go to England, which he didn’t think he’d do well enough in — and also that his superior officer, a “very orthodox Brahman having scruples to go to foreign land replied at once that I could not go”.

But then he said that Neville had “cleared [his] doubts”, explaining that there wouldn’t be an issue with his expenses, that his English would do, that he wouldn’t have to take exams, and that he could remain a vegetarian in England. He ended by saying that he hoped Hardy and Littlewood would “be good enough to take the trouble of getting me [to England] within a very few months.”

Hardy had assumed it would be bureaucratically trivial to get Ramanujan to England, but actually it wasn’t. Hardy’s own Trinity College wasn’t prepared to contribute any real funding. Hardy and Littlewood offered to put up some of the money themselves. But Neville wrote to the registrar of the University of Madras saying that “the discovery of the genius of S. Ramanujan of Madras promises to be the most interesting event of our time in the mathematical world” — and suggested the university come up with the money. Ramanujan’s expat supporters swung into action, with the matter eventually reaching the Governor of Madras — and a solution was found that involved taking money from a grant that had been given by the government five years earlier for “establishing University vacation lectures”, but that was actually, in the bureaucratic language of “Document No. 182 of the Educational Department”, “not being utilized for any immediate purpose”.

There are strange little notes in the bureaucratic record, like on February 12: “What caste is he? Treat as urgent.” But eventually everything was sorted out, and on March 17, 1914 , after a send-off featuring local dignitaries, Ramanujan boarded a ship for England, sailing up through the Suez Canal , and arriving in London on April 14 . Before leaving India, Ramanujan had prepared for European life by getting Western clothes, and learning things like how to eat with a knife and fork, and how to tie a tie. Many Indian students had come to England before, and there was a whole procedure for them. But after a few days in London, Ramanujan arrived in Cambridge — with the Indian newspapers proudly reporting that “Mr. S. Ramanujan, of Madras, whose work in the higher mathematics has excited the wonder of Cambridge, is now in residence at Trinity.”

(In addition to Hardy and Littlewood, two other names that appear in connection with Ramanujan’s early days in Cambridge are Neville and Barnes . They’re not especially famous in the overall history of mathematics, but it so happens that in the Wolfram Language they’re both commemorated by built-in functions: NevilleThetaS and BarnesG .)

Ramanujan described the war in a letter to his mother, saying for example, “They fly in aeroplanes at great heights, bomb the cities and ruin them. As soon as enemy planes are sighted in the sky, the planes resting on the ground take off and fly at great speeds and dash against them resulting in destruction and death.”

Ramanujan nevertheless continued to pursue mathematics, explaining to his mother that “war is waged in a country that is as far as Rangoon is away from [Madras] ”. There were practical difficulties, like a lack of vegetables, which caused Ramanujan to ask a friend in India to send him “some tamarind (seeds being removed) and good cocoanut oil by parcel post”. But of more importance, as Ramanujan reported it, was that the “professors here… have lost their interest in mathematics owing to the present war”.

Ramanujan told a friend that he had “changed [his] plan of publishing [his] results”. He said that he would wait to publish any of the old results in his notebooks until the war was over. But he said that since coming to England he had learned “their methods”, and was “trying to get new results by their methods so that I can easily publish these results without delay”.

In 1915 Ramanujan published a long paper entitled “ Highly Composite Numbers ” about maxima of the function ( DivisorSigma in the Wolfram Language) that counts the number of divisors of a given number. Hardy seems to have been quite involved in the preparation of this paper — and it served as the centerpiece of Ramanujan’s analog of a PhD thesis.

For the next couple of years, Ramanujan prolifically wrote papers — and despite the war, they were published. A notable paper he wrote with Hardy concerns the partition function ( PartitionsP in the Wolfram Language) that counts the number of ways an integer can be written as a sum of positive integers. The paper is a classic example of mixing the approximate with the exact. The paper begins with the result for large n :

But then in May 1917, there was another problem: Ramanujan got sick. From what we know now, it’s likely that what he had was a parasitic liver infection picked up in India. But back then nobody could diagnose it. Ramanujan went from doctor to doctor, and nursing home to nursing home. He didn’t believe much of what he was told, and nothing that was done seemed to help much. Some months he would be well enough to do a significant amount of mathematics; others not. He became depressed, and at one point apparently suicidal. It didn’t help that his mother had prevented his wife back in India from communicating with him, presumably fearing it would distract him.

Hardy tried to help — sometimes by interacting with doctors, sometimes by providing mathematical input. One doctor told Hardy he suspected “some obscure Oriental germ trouble imperfectly studied at present”. Hardy wrote, “Like all Indians, [Ramanujan] is fatalistic, and it is terribly hard to get him to take care of himself.” Hardy later told the now-famous story that he once visited Ramanujan at a nursing home, telling him that he came in a taxicab with number 1729, and saying that it seemed to him a rather dull number — to which Ramanujan replied: “No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways”:

But through all of this, Ramanujan’s mathematical reputation continued to grow. He was elected a Fellow of the Royal Society (with his supporters including Hobson and Baker, both of whom had failed to respond to his original letter) — and in October 1918 he was elected a fellow of Trinity College, assuring him financial support. A month later World War I was over — and the threat of U-boat attacks, which had made travel to India dangerous, was gone.

And so on March 13, 1919 , Ramanujan returned to India — now very famous and respected, but also very ill. Through it all, he continued to do mathematics, writing a notable letter to Hardy about “mock” theta functions on January 12, 1920 . He chose to live humbly, and largely ignored what little medicine could do for him. And on April 26, 1920, at the age of 32, and three days after the last entry in his notebook, he died.

From when he first started doing mathematics research, Ramanujan had recorded his results in a series of hardcover notebooks — publishing only a very small fraction of them. When Ramanujan died, Hardy began to organize an effort to study and publish all 3000 or so results in Ramanujan’s notebooks. Several people were involved in the 1920s and 1930s, and quite a few publications were generated. But through various misadventures the project was not completed — to be taken up again only in the 1970s.

When Ramanujan died, it took only days for his various relatives to start asking for financial support. There were large medical bills from England, and there was talk of selling Ramanujan’s papers to raise money.

Ramanujan’s wife was 21 when he died, but as was the custom, she never remarried. She lived very modestly, making her living mostly from tailoring. In 1950 she adopted the son of a friend of hers who had died. By the 1960s, Ramanujan was becoming something of a general Indian hero, and she started receiving various honors and pensions. Over the years, quite a few mathematicians had come to visit her — and she had supplied them for example with the passport photo that has become the most famous picture of Ramanujan.

She lived a long life, dying in 1994 at the age of 95, having outlived Ramanujan by 73 years.

Hardy was 35 when Ramanujan’s letter arrived, and was 43 when Ramanujan died. Hardy viewed his “discovery” of Ramanujan as his greatest achievement, and described his association with Ramanujan as the “one romantic incident of [his] life”. After Ramanujan died, Hardy put some of his efforts into continuing to decode and develop Ramanujan’s results, but for the most part he returned to his previous mathematical trajectory. His collected works fill seven large volumes (while Ramanujan’s publications make up just one fairly slim volume). The word clouds of the titles of his papers show only a few changes from before he met Ramanujan to after:

Hardy’s mathematics was always of the finest quality. He dreamed of doing something like solving the Riemann hypothesis — but in reality never did anything truly spectacular. He wrote two books, though, that continue to be read today: An Introduction to the Theory of Numbers , with E. M. Wright ; and Inequalities , with Littlewood and G. Pólya .

Hardy lived his life in the stratum of the intellectual elite. In the 1920s he displayed a picture of Lenin in his apartment, and was briefly president of the “scientific workers” trade union. He always wrote elegantly, mostly about mathematics, and sometimes about Ramanujan. He eschewed gadgets and always lived along with students and other professors in his college. He never married, though near the end of his life his younger sister joined him in Cambridge (she also had never married, and had spent most of her life teaching at the girls’ school where she went as a child).

In 1940 Hardy wrote a small book called A Mathematician’s Apology . I remember when I was about 12 being given a copy of this book. I think many people viewed it as a kind of manifesto or advertisement for pure mathematics. But I must say it didn’t resonate with me at all. It felt to me at once sanctimonious and austere, and I wasn’t impressed by its attempt to describe the aesthetics and pleasures of mathematics, or by the pride with which its author said that “nothing I have ever done is of the slightest practical use” (actually, he co-invented the Hardy-Weinberg law used in genetics). I doubt I would have chosen the path of a pure mathematician anyway, but Hardy’s book helped make certain of it.

A quite different point is that while making specific contributions to an existing area (as Hardy did) is something that can potentially be done by the young, creating a whole new structure tends to require the broader knowledge and experience that comes with age.

But back to Hardy. I suspect it was a lack of motivation rather than ability, but in his last years, he became quite dispirited and all but dropped mathematics. He died in 1947 at the age of 70.

Littlewood, who was a decade younger than Hardy, lived on until 1977. Littlewood was always a little more adventurous than Hardy, a little less austere, and a little less august. Like Hardy, he never married — though he did have a daughter (with the wife of the couple who shared his vacation home) whom he described as his “niece” until she was in her forties. And — giving a lie to Hardy’s claim about math being a young man’s game — Littlewood (helped by getting early antidepressant drugs at the age of 72) had remarkably productive years of mathematics in his 80s.

What became of Ramanujan’s mathematics? For many years, not too much. Hardy pursued it some, but the whole field of number theory — which was where the majority of Ramanujan’s work was concentrated — was out of fashion. Here’s a plot of the fraction of all math papers tagged as “number theory” as a function of time in the Zentralblatt database :

In the 1970s, though, number theory suddenly became more popular again, driven by advances in algebraic number theory. (Other subcategories showing substantial increases at that time include automorphic forms, elementary number theory and sequences.)

Back in the late 1970s, I had certainly heard of Ramanujan — though more in the context of his story than his mathematics. And I was pleased in 1982, when I was writing about the vacuum in quantum field theory , that I could use results of Ramanujan’s to give closed forms for particular cases (of infinite sums in various dimensions of modes of a quantum field — corresponding to Epstein zeta functions):

A significant part of what Ramanujan did was to study so-called special functions — and to invent some new ones. Special functions — like the zeta function, elliptic functions, theta functions, and so on — can be thought of as defining convenient “packets” of mathematics. There are an infinite number of possible functions one can define, but what get called “special functions” are ones whose definitions survive because they turn out to be repeatedly useful.

And today, for example, in Mathematica and the Wolfram Language we have RamanujanTau , RamanujanTauL , RamanujanTauTheta and RamanujanTauZ as special functions. I don’t doubt that in the future we’ll have more Ramanujan-inspired functions. In the last year of his life, Ramanujan defined some particularly ambitious special functions that he called “mock theta functions” — and that are still in the process of being made concrete enough to routinely compute.

If one looks at the definition of Ramanujan’s tau function it seems quite bizarre (notice the “24”):

In antiquity, the Pythagoreans made much of the fact that 1+2+3+4=10. But to us today, this just seems like a random fact of mathematics, not of any particular significance. When I look at Ramanujan’s results, many of them also seem like random facts of mathematics. But the amazing thing that’s emerged over the past century, and particularly over the past few decades, is that they’re not. Instead, more and more of them are being found to be connected to deep, elegant mathematical principles.

To enunciate these principles in a direct and formal way requires layers of abstract mathematical concepts and language which have taken decades to develop. But somehow, through his experiments and intuition, Ramanujan managed to find concrete examples of these principles. Often his examples look quite arbitrary — full of seemingly random definitions and numbers. But perhaps it’s not surprising that that’s what it takes to express modern abstract principles in terms of the concrete mathematical constructs of the early twentieth century. It’s a bit like a poet trying to express deep general ideas — but being forced to use only the imperfect medium of human natural language.

It’s turned out to be very challenging to prove many of Ramanujan’s results. And part of the reason seems to be that to do so — and to create the kind of narrative needed for a good proof — one actually has no choice but to build up much more abstract and conceptually complex structures, often in many steps.

So how is it that Ramanujan managed in effect to predict all these deep principles of later mathematics? I think there are two basic logical possibilities. The first is that if one drills down from any sufficiently surprising result, say in number theory, one will eventually reach a deep principle in the effort to explain it. And the second possibility is that while Ramanujan did not have the wherewithal to express it directly, he had what amounts to an aesthetic sense of which seemingly random facts would turn out to fit together and have deeper significance.

I’m not sure which of these possibilities is correct, and perhaps it’s a combination. But to understand this a little more, we should talk about the overall structure of mathematics. In a sense mathematics as it’s practiced is strangely perched between the trivial and the impossible. At an underlying level, mathematics is based on simple axioms. And it could be — as it is, say, for the specific case of Boolean algebra — that given the axioms there’s a straightforward procedure to figure out whether any particular result is true. But ever since Gödel’s theorem in 1931 (which Hardy must have been aware of, but apparently never commented on) it’s been known that for an area like number theory the situation is quite different: there are statements one can give within the context of the theory whose truth or falsity is undecidable from the axioms.

It was proved in the early 1960s that there are polynomial equations involving integers where it’s undecidable from the axioms of arithmetic — or in effect from the formal methods of number theory — whether or not the equations have solutions. The particular examples of classes of equations where it’s known that this happens are extremely complex. But from my investigations in the computational universe , I’ve long suspected that there are vastly simpler equations where it happens too. Over the past several decades, I’ve had the opportunity to poll some of the world’s leading number theorists on where they think the boundary of undecidability lies. Opinions differ, but it’s certainly within the realm of possibility that for example cubic equations with three variables could exhibit undecidability.

So the question then is, why should the truth of what seem like random facts of number theory even be decidable? In other words, it’s perfectly possible that Ramanujan could have stated a result that simply can’t be proved true or false from the axioms of arithmetic. Conceivably the Goldbach conjecture will turn out to be an example. And so could many of Ramanujan’s results.

Some of Ramanujan’s results have taken decades to prove — but the fact that they’re provable at all is already important information. For it suggests that in a sense they’re not just random facts; they’re actually facts that can somehow be connected by proofs back to the underlying axioms.

And I must say that to me this tends to support the idea that Ramanujan had intuition and aesthetic criteria that in some sense captured some of the deeper principles we now know, even if he couldn’t express them directly.

It’s pretty easy to start picking mathematical statements, say at random, and then getting empirical evidence for whether they’re true or not. Gödel’s theorem effectively implies that you’ll never know how far you’ll have to go to be certain of any particular result. Sometimes it won’t be far, but sometimes it may in a sense be arbitrarily far.

Ramanujan no doubt convinced himself of many of his results by what amount to empirical methods — and often it worked well. In the case of the counting of primes, however, as Hardy pointed out, things turn out to be more subtle, and results that might work up to very large numbers can eventually fail.

So let’s say one looks at the space of possible mathematical statements, and picks statements that appear empirically at least to some level to be true. Now the next question: are these statements connected in any way?

Imagine one could find proofs of the statements that are true. These proofs effectively correspond to paths through a directed graph that starts with the axioms, and leads to the true results. One possibility is then that the graph is like a star — with every result being independently proved from the axioms. But another possibility is that there are many common “waypoints” in getting from the axioms to the results. And it’s these waypoints that in effect represent general principles.

If there’s a certain sparsity to true results, then it may be inevitable that many of them are connected through a small number of general principles. It might also be that there are results that aren’t connected in this way, but these results, perhaps just because of their lack of connections, aren’t considered “interesting” — and so are effectively dropped when one thinks about a particular subject.

I have to say that these considerations lead to an important question for me. I have spent many years studying what amounts to a generalization of mathematics: the behavior of arbitrary simple programs in the computational universe. And I’ve found that there’s a huge richness of complex behavior to be seen in such programs. But I have also found evidence — not least through my Principle of Computational Equivalence — that undecidability is rife there.

But now the question is, when one looks at all that rich and complex behavior, are there in effect Ramanujan-like facts to be found there? Ultimately there will be much that can’t readily be reasoned about in axiom systems like the ones in mathematics. But perhaps there are networks of facts that can be reasoned about — and that all connect to deeper principles of some kind.

We know from the idea around the Principle of Computational Equivalence that there will always be pockets of “computational reducibility”: places where one will be able to identify abstract patterns and make abstract conclusions without running into undecidability. Repetitive behavior and nested behavior are two almost trivial examples. But now the question is whether among all the specific details of particular programs there are other general forms of organization to be found.

Of course, whereas repetition and nesting are seen in a great many systems, it could be that another form of organization would be seen only much more narrowly. But we don’t know. And as of now, we don’t really have much of a handle on finding out — at least until or unless there’s a Ramanujan-like figure not for traditional mathematics but for the computational universe.

Will there ever be another Ramanujan? I don’t know if it’s the legend of Ramanujan or just a natural feature of the way the world is set up, but for at least 30 years I’ve received a steady stream of letters that read a bit like the one Hardy got from Ramanujan back in 1913. Just a few months ago, for example, I received an email (from India, as it happens) with an image of a notebook listing various mathematical expressions that are numerically almost integers — very much like Ramanujan’s

Needless to say, the definition of “interesting” isn’t an easy or objective one. And in fact the issues are very much the same as Hardy faced with Ramanujan’s letter. If one can see how what’s being presented fits into some bigger picture — some narrative — that one understands, then one can tell whether, at least within that framework, something is “interesting”. But if one doesn’t have the bigger picture — or if what’s being presented is just “too far out” — then one really has no way to tell if it should be considered interesting or not.

When I first started studying the behavior of simple programs, there really wasn’t a context for understanding what was going on in them. The pictures I got certainly seemed visually interesting. But it wasn’t clear what the bigger intellectual story was. And it took quite a few years before I’d accumulated enough empirical data to formulate hypotheses and develop principles that let one go back and see what was and wasn’t interesting about the behavior I’d observed.

I’ve put a few decades into developing a science of the computational universe. But it’s still young, and there is much left to discover — and it’s a highly accessible area, with no threshold of elaborate technical knowledge. And one consequence of this is that I frequently get letters that show remarkable behavior in some particular cellular automaton or other simple program. Often I recognize the general form of the behavior, because it relates to things I’ve seen before, but sometimes I don’t — and so I can’t be sure what will or won’t end up being interesting.

Back in Ramanujan’s day, mathematics was a younger field — not quite as easy to enter as the study of the computational universe, but much closer than modern academic mathematics. And there were plenty of “random facts” being published: a particular type of integral done for the first time, or a new class of equations that could be solved. Many years later we would collect as many of these as we could to build them into the algorithms and knowledge base of Mathematica and the Wolfram Language . But at the time probably the most significant aspect of their publication was the proofs that were given: the stories that explained why the results were true. Because in these proofs, there was at least the potential that concepts were introduced that could be reused elsewhere, and build up part of the fabric of mathematics.

It would take us too far afield to discuss this at length here, but there is a kind of analog in the study of the computational universe: the methodology for computer experiments. Just as a proof can contain elements that define a general methodology for getting a mathematical result, so the particular methods of search, visualization or analysis can define something in computer experiments that is general and reusable, and can potentially give an indication of some underlying idea or principle.

And so, a bit like many of the mathematics journals of Ramanujan’s day, I’ve tried to provide a journal and a forum where specific results about the computational universe can be reported — though there is much more that could be done along these lines.

When a letter one receives contains definite mathematics, in mathematical notation, there is at least something concrete one can understand in it. But plenty of things can’t usefully be formulated in mathematical notation. And too often, unfortunately, letters are in plain English (or worse, for me, other languages) and it’s almost impossible for me to tell what they’re trying to say. But now there’s something much better that people increasingly do: formulate things in the Wolfram Language . And in that form, I’m always able to tell what someone is trying to say — although I still may not know if it’s significant or not.

Over the years, I’ve been introduced to many interesting people through letters they’ve sent. Often they’ll come to our Summer School , or publish something in one of our various channels . I have no story (yet) as dramatic as Hardy and Ramanujan. But it’s wonderful that it’s possible to connect with people in this way, particularly in their formative years. And I can’t forget that a long time ago, I was a 14-year-old who mailed papers about the research I’d done to physicists around the world…

Ramanujan did his calculations by hand — with chalk on slate, or later pencil on paper. Today with Mathematica and the Wolfram Language we have immensely more powerful tools with which to do experiments and make discoveries in mathematics (not to mention the computational universe in general).

It’s fun to imagine what Ramanujan would have done with these modern tools. I rather think he would have been quite an adventurer — going out into the mathematical universe and finding all sorts of strange and wonderful things, then using his intuition and aesthetic sense to see what fits together and what to study further.

Ramanujan unquestionably had remarkable skills. But I think the first step to following in his footsteps is just to be adventurous: not to stay in the comfort of well-established mathematical theories, but instead to go out into the wider mathematical universe and start finding — experimentally — what’s true.

It’s taken the better part of a century for many of Ramanujan’s discoveries to be fitted into a broader and more abstract context. But one of the great inspirations that Ramanujan gives us is that it’s possible with the right sense to make great progress even before the broader context has been understood. And I for one hope that many more people will take advantage of the tools we have today to follow Ramanujan’s lead and make great discoveries in experimental mathematics — whether they announce them in unexpected letters or not.

Peter Guest

Hemal Jhaveri

Steven Levy

Virginia Heffernan

Robert Peck

The man who taught infinity: how GH Hardy tamed Srinivasa Ramanujan’s genius

Professor of Pure Mathematics, University of Cambridge

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Throughout the history of mathematics, there has been no one remotely like Srinivasa Ramanujan. There is no doubt that he was a great mathematician, but had he had simply a good university education and been taught by a good professor in his field, we wouldn’t have a film about him .

As the years pass, I admire more and more the astonishing body of work Ramanujan produced in India before he made contact with any top mathematicians. Not because the results he got at the time changed the face of mathematics, far from it, but because, working by himself, he fearlessly attacked many important and some not so important problems in analysis and, especially, number theory – simply for the love of mathematics.

It cannot be understated, however, the role played by Ramanujan’s tutor Godfrey Harold Hardy in his life story. The Cambridge mathematician worked tirelessly with the Indian genius, to tame his creativity within the then current understanding of the field. It was only with Hardy’s care and mentoring that Ramanujan became the scholar we know him as today.

Determined and obsessed

In December 1903, at the age of 16, Ramanujan passed the matriculation exam for the University of Madras. But as he concentrated on mathematics to the exclusion of all other subjects, he did not progress beyond the second year. In 1909 he married a nine-year-old girl, but failed to secure any steady income until the beginning of 1912, when he became a clerk in the Madras Port Trust office on a meagre salary.

All this time, Ramanujan remained obsessed with mathematics and kept working on continued fractions, divergent series, elliptic integrals, hypergeometric series and the distribution of primes. By 1911, Ramanujan was desperate to gain recognition from leading mathematicians, especially those in England. So, at the beginning of 1913, when he was just past 25, he dispatched a letter to Hardy in Cambridge with a long list of his discoveries –- a letter which changed both their lives.

Although only 36 when he received Ramanujan’s letter, Hardy was already the leading mathematician in England . The mathematical scene in England in the first half of the 20th century was dominated by Hardy and another titan of Trinity College, J.E. Littlewood . The two formed a legendary partnership, unique to this day, writing an astounding 100 joint papers. They were instrumental in turning England into a superpower in mathematics, especially in number theory and analysis.

Hardy was not the first mathematician to whom Ramanujan had sent his results, however the first two to whom he had written judged him to be a crank. But Hardy was not only an outstanding mathematician, he was also a wonderful teacher, eager to nurture talent.

Genius unknown

After dinner in Trinity one evening, some of the fellows adjourned to the combination room. Over their claret and port Hardy mentioned to Littlewood some of the claims he had received in the mail from an unknown Indian. Some assertions they knew well, others they could prove, others they could disprove, but many they found not only fascinating and unusual but also impossible to resolve.

This toing and froing between Hardy and Littlewood continued the next day and beyond, and soon they were convinced that their correspondent was a genius. So Hardy sent an encouraging reply to Ramanujan, which led to a frequent exchange of letters.

It was clear to Hardy that Ramanujan was totally exceptional: however, in spite of his amazing feats in mathematics, he lacked the basic tools of the trade of a professional mathematician. Hardy knew that if Ramanujan was to fulfil his potential, he had to have a solid foundation in mathematics, at least as much as the best Cambridge graduates.

It was for Ramanujan’s good that Hardy invited him to Cambridge, then, and he was taken aback when, due to caste prejudices, Ramanujan did not jump at the chance. As a Brahmin , Ramanujan was not allowed to cross the ocean and his mother was totally opposed to the idea of the voyage. When, in early 1914, Ramanujan gained his mother’s consent, Hardy swang into action. He asked E.H Neville, another fellow of Trinity College , who was on a serendipitous trip to Madras, to secure Ramanujan a scholarship from the University of Madras. Neville’s wrote in a letter to the university that “the discovery of the genius of S. Ramanujan of Madras promises to be the most interesting event of our time in the mathematical world …”

Ramanujan sailed for England in the company of Neville, and arrived in Cambridge in April 1914.

Fearless mentoring

I cannot but admire Hardy for his care in mentoring Ramanujan. His main worry was how to teach this astounding talent much mathematics without destroying his confidence. The last thing Hardy wanted was to dent Ramanujan’s fearless approach to the most difficult problems. To quote Hardy:

The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations, and theorems of complex multiplication, to orders unheard of, whose mastery of continued fractions was, on the formal side at any rate, beyond that of any mathematician in the world … It was impossible to ask such a man to submit to systematic instruction, to try to learn mathematics from the beginning once more. On the other hand there were things of which it was impossible that he would remain in ignorance … so I had to try to teach him, and in a measure I succeeded, though I obviously learnt from him much more than he learnt from me.

For almost three years, things went extremely well. In 1916 Ramanujan got his BA from Cambridge and his research went from strength to strength. He published one excellent paper after another, with a great deal of Hardy’s help in the proofs and presentation. They also collaborated on several great projects, and published wonderful joint papers. Sadly, in the spring of 1917 Ramanujan fell ill, and was in and out of sanatoria for the rest of his stay in Cambridge.

By early 1919 Ramanujan seemed to have recovered sufficiently, and decided to travel back to India. Hardy was alarmed not to have heard from him for a considerable time, but a letter in February 1920 made it clear that Ramanujan was very active in research.

Ramanujan’s letter contained some examples of his latest discovery, mock theta functions , which have turned out to be very important. A main conjecture about them was solved 80 years later , and these functions are now seen as interesting examples of a much larger class of mock modular forms in mathematics, which have applications to elliptic curves, Borcherds products , Eichler cohomology and Galois representations – and the nature of black holes.

Sadly, Ramanujan’s recovery was short-lived. His illness returned and killed him, aged just 32, on April 26 1920, leaving him only a short time to benefit from his fellowship of the Royal Society and fellowship of Trinity.

Ramanujan’s death at the height of his powers was a tremendous blow to mathematics. His like may never be seen again, and certainly such a partnership as that which Hardy and Ramanujan built will not either.

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Essay on Srinivasa Ramanujan

500 words essay on srinivasa ramanujan.

Srinivasa Ramanujan is one of the world’s greatest mathematicians of all time. Furthermore, this man, from a poor Indian family, rose to prominence in the field of mathematics. This essay on Srinivasa Ramanujan will throw more light on the life of this great personality.

                                                                                             Essay On Srinivasa Ramanujan

Early Life of Srinivasa Ramanujan

Ramanujan was born in Erode on December 22, 1887, in his grandmother’s house.  Furthermore, he went to primary school in Kumbakonamwas when he was five years old.  Moreover, he would attend several different primary schools before his entry took place to the Town High School in Kumbakonam in January 1898.

At the Town High School, Ramanujan proved himself as a talented student and did well in all of his school subjects. In 1900, he became involved with mathematics and began summing geometric and arithmetic series on his own.

In the Town High School, Ramanujan began reading a mathematics book called ‘Synopsis of Elementary Results in Pure Mathematics’. Furthermore, this book was by G. S. Carr.

With the help of this book, Ramanujan began to teach himself mathematics . Furthermore, the book contained theorems, formulas and short proofs. It also contained an index to papers on pure mathematics.

His Contribution to Mathematics

By 1904, Ramanujan’s focus was on deep research. Moreover, an investigation took place by him of the series (1/n). Moreover, calculation took place by him of Euler’s constant to 15 decimal places. This was entirely his own independent discovery.

Ramanujan gained a scholarship because of his outstanding performance in his studies. Consequently, he was a brilliant student at Kumbakonam’s Government College. Moreover, his fascination and passion for mathematics kept on growing.

In the spring of 1913, there was the presentation of Ramanujan’s work to British mathematicians by Narayana Iyer, Ramachandra Rao and E. W. Middlemast. Afterwards, M.J.M Hill did not made an offer to take Ramanujan on as a student, rather, he provided professional advice to him. With the help of friends, Ramanujan sent letters to leading mathematicians at Cambridge University and was ultimately selected.

Ramanujan spent a significant time period of five years at Cambridge. At Cambridge, collaboration took place of Ramanujan with Hardy and Littlewood. Most noteworthy, the publishing of his findings took place there.

Ramanujan received the honour of a Bachelor of Arts by Research degree in March 1916. This honour was due to his work on highly composite numbers, sections of the first part whose publishing had taken place the preceding year. Moreover, the paper’s size was more than fifty pages long.

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Conclusion of the Essay on Srinivasa Ramanujan

Srinivasa Ramanujan is a man whose contributions to the field of mathematics are unmatchable. Furthermore, experts in mathematics worldwide all recognize his tremendous worth. Most noteworthy, Srinivasa Ramanujan made his country proud at a time when India was still under British occupation.

FAQs For Essay on Srinivasa Ramanujan

Question 1: What is Srinivasa Ramanujan famous for?

Answer 1: Srinivas Ramanujan is famous for his discoveries that have influenced several areas of mathematics. Furthermore, he is famous for his contributions to number theory and infinite series. Moreover, he came up with fascinating formulas that facilitate in the calculation of the digits of pi in unusual ways.

Question 2: What is the special quality of number 1729 discovered by Srinivasa Ramanujan?

Answer 2:  Srinivas Ramanujan discovered that the number 1729 had a special characteristic.  Furthermore, this quality is that the number 1729 is the only number whose expression can take place as the sum of the cubes of two different sets of numbers. Consequently, people call 1729 the magic number.

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Books on Ramanujan and Busts of Ramanujan

• First Online: 31 May 2021

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• K. Srinivasa Rao 2  

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The most authentic sources about the life and work of the mathematical genius Srinivasa Ramanujan are the articles of Ramachandra Rao and Seshu Aiyer at the beginning of the Collected Papers by Srinivasa Ramanujan , first published in 1928, and G.H. Hardy’s Ramanujan: Twelve Lectures on subjects suggested by his life and work .

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Collected Papers by Srinivasa Ramanujan , edited by G.H. Hardy, P.V. Seshu Aiyar and B.M. Wilson, Chelsea, New York (1962), first published by Cambridge University Press, Cambridge (1928), and more recently reprinted on the occasion of the 125th Birth Anniversary of Ramanujan, in 2012, by the Am. Math. Soc. and London Math. Soc. (2012).

G. H. Hardy, Ramanujan: Twelve Lectures on subjects suggested by his life and work , Chelsea, New York (1940). This book has also been reprinted by the AMS-LMS in (2012).

P.K. Srinivasan, Editor, Ramanujan: Letters and Reminiscences , Memorial Number, Vol. 1, Muthialpet High School, Madras (1968).

P.K. Srinivasan, Editor, Ramanujan: An Inspiration , Memorial Number, Vol. 2, Muthialpet High School, Madras (1968).

S.R. Ranganathan, Ramanujan: The Man and the Mathematician , Asia Publishing House (1967).

Kanigel should be complemented for finding this most apt word in Sanskrit to describe the genius of Ramanujan, which he uses as a heading to a section in his book The Man Who Knew Infinity .

The Copley Medal was awarded to Dr. S. Chandrasekhar, the Indian-American Astrophysicist.

Robert Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan , Charles Scribner’s Sons, New York (1991), was priced at $ 27.95. In 1994, Rupa & Co. brought out an Indian paper back edition at a cost of only Rs. 195/-.

Bruce C. Berndt and Robert A. Rankin, Ramanujan: Letters and Commentary , AM–LMS publication (1995).

Bruce C. Berndt and Robert A. Rankin, Ramanujan: Letters and Commentary , Indian edition was brought out by K. Srinivasa Rao, through Affiliated East West Press Pvt. Ltd., in 1997.

On my first visit to Paris, France, in 1972, my father told me that I should not miss seeing the tomb of Napoleon at the Invalides, in France, and the Shakespeare Museum at Stratford-upon-Avon in England—and I did visit these two places and brought souvenirs for my father, a small statue of Napoleon on a horse after one of his battles on the Alps, and an illustrated complete works of William Shakespeare, as a gift from the Curator of the Museum!

The British government, which did not insist on a passport when he left for England, insisted on his getting one on his return to India, with a photograph, in March 1919, when he was emaciated after being in different sanatoria for nearly 2 and 1/2 years!

The author visited Trinity College, Cambridge University, as a guest of Prof. Macfarlane and wishes to express his thanks for this opportunity which enabled him to ask for and see the cardboard boxes in their archives—more about this later.

This was easy for me, since my wife, Dr. Geetha Srinivasa Rao, was a professor at the Ramanujan Institute for Advanced Study of the University of Madras and so was well-known to all those in the Registrar’s office.

In the Wren Library, the works of Newton and the originals of Winnie the Pooh, have been preserved as antiques and exhibited prominently by the Curators of the Library.

Subbiah Muthiah, MBE (1930–2019), is an Indian writer, journalist, cartographer, amateur historian and heritage activist known for his writings on the political and cultural history of Chennai. He is known for his books on the History of Chennai— Madras Discovered , East West Books (Madras) Pvt. Ltd. (1981) and a collection of articles under the column “Madras Miscellany”, in the daily newspaper The Hindu, from November 1999. On 7 March 2002, Muthiah was made an “Honorary Member of the Civil Division of the Most Excellent Order of the British Empire”, the citation for which read, for: “service by those who are not British citizens but who have pursued ideals which Britain values and shares”. He is the founder of the fortnightly newspaper Madras Musings and the principal organizer of the annual Madras Day celebrations. Muthiah is the founder-President of the Madras Book Club.- Wikipedia.

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Rao, K.S. (2021). Books on Ramanujan and Busts of Ramanujan. In: Srinivasa Ramanujan. Springer, Singapore. https://doi.org/10.1007/978-981-16-0447-8_7

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Published : 31 May 2021

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