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Srinivasa Ramanujan

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Institute of Mathematics & its Applications - Biography of Srinivasa Ramanujan (1887–1920): The Centenary of a Remarkable Mathematician
IndiaNetzone - Indian Personalities - Biography of Srinavasa Iyengar Ramanujan
San José State University - Srinivasa Ramanujan, a mathematician brilliant beyond comparison
U.S.Naval Institute - Srinivasa Ramanujan
Linda Hall Library - Srinivasa Ramanujan
Academia - Great Mathematician Ramanujan
Srinivasa Ramanujan - Student Encyclopedia (Ages 11 and up)

At age 15 Srinivasa Ramanujan obtained a mathematics book containing thousands of theorems , which he verified and from which he developed his own ideas. In 1903 he briefly attended the University of Madras . In 1914 he went to England to study at Trinity College, Cambridge , with British mathematician G.H. Hardy .

What were Srinivasa Ramanujan’s contributions?

Indian mathematician Srinivasa Ramanujan made contributions to the theory of numbers , including pioneering discoveries of the properties of the partition function. His papers were published in English and European journals, and in 1918 he was elected to the Royal Society of London.

What is Srinivasa Ramanujan remembered for?

Srinivasa Ramanujan is remembered for his unique mathematical brilliance, which he had largely developed by himself. In 1920 he died at age 32, generally unknown to the world at large but recognized by mathematicians as a phenomenal genius, without peer since Leonhard Euler (1707–83) and Carl Jacobi (1804–51).

Srinivasa Ramanujan (born December 22, 1887, Erode , India—died April 26, 1920, Kumbakonam) was an Indian mathematician whose contributions to the theory of numbers include pioneering discoveries of the properties of the partition function.

When he was 15 years old, he obtained a copy of George Shoobridge Carr’s Synopsis of Elementary Results in Pure and Applied Mathematics, 2 vol. (1880–86). This collection of thousands of theorems , many presented with only the briefest of proofs and with no material newer than 1860, aroused his genius. Having verified the results in Carr’s book, Ramanujan went beyond it, developing his own theorems and ideas. In 1903 he secured a scholarship to the University of Madras but lost it the following year because he neglected all other studies in pursuit of mathematics .

Ramanujan continued his work, without employment and living in the poorest circumstances. After marrying in 1909 he began a search for permanent employment that culminated in an interview with a government official, Ramachandra Rao. Impressed by Ramanujan’s mathematical prowess, Rao supported his research for a time, but Ramanujan, unwilling to exist on charity, obtained a clerical post with the Madras Port Trust.

In 1911 Ramanujan published the first of his papers in the Journal of the Indian Mathematical Society . His genius slowly gained recognition, and in 1913 he began a correspondence with the British mathematician Godfrey H. Hardy that led to a special scholarship from the University of Madras and a grant from Trinity College , Cambridge . Overcoming his religious objections, Ramanujan traveled to England in 1914, where Hardy tutored him and collaborated with him in some research.

Ramanujan’s knowledge of mathematics (most of which he had worked out for himself) was startling. Although he was almost completely unaware of modern developments in mathematics, his mastery of continued fractions was unequaled by any living mathematician. He worked out the Riemann series, the elliptic integrals , hypergeometric series, the functional equations of the zeta function , and his own theory of divergent series, in which he found a value for the sum of such series using a technique he invented that came to be called Ramanujan summation. On the other hand, he knew nothing of doubly periodic functions, the classical theory of quadratic forms, or Cauchy’s theorem, and he had only the most nebulous idea of what constitutes a mathematical proof. Though brilliant, many of his theorems on the theory of prime numbers were wrong.

In England Ramanujan made further advances, especially in the partition of numbers (the number of ways that a positive integer can be expressed as the sum of positive integers; e.g., 4 can be expressed as 4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1). His papers were published in English and European journals, and in 1918 he was elected to the Royal Society of London . In 1917 Ramanujan had contracted tuberculosis , but his condition improved sufficiently for him to return to India in 1919. He died the following year, generally unknown to the world at large but recognized by mathematicians as a phenomenal genius, without peer since Leonhard Euler (1707–83) and Carl Jacobi (1804–51). Ramanujan left behind three notebooks and a sheaf of pages (also called the “lost notebook”) containing many unpublished results that mathematicians continued to verify long after his death.

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Srinivasa Ramanujan

Srinivasa Ramanujan (1887-1920)

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

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

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

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

Kickstart your preparation now! Complement it with the links given below:

Indian Mathematician S. Ramanujan – Biography

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

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

Birth –

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

Srinivasa Ramanujan Education –

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

Final Break Through –

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

Death –

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

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

Srinivasa Ramanujan Contributions

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

Achievements of Srinivasa Ramanujan

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

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

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

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

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

IAS General Studies Notes Links

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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
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Publisher: A&E; Television Networks
Last Updated: September 10, 2019
Original Published Date: September 10, 2015

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Srinivasa aiyangar ramanujan.

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

I have passed the Matriculation Examination and studied up to the First Arts but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject.

I can strongly recommend the applicant. He is a young man of quite exceptional capacity in mathematics and especially in work relating to numbers. He has a natural aptitude for computation and is very quick at figure work.

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

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

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

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

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

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

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

References ( show )

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

Additional Resources ( show )

Other pages about Srinivasa Ramanujan:

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

Other websites about Srinivasa Ramanujan:

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

Honours ( show )

Honours awarded to Srinivasa Ramanujan

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

Cross-references ( show )

History Topics: Squaring the circle
Famous Curves: Ellipse
Societies: Indian Academy of Sciences
Societies: Indian Mathematical Society
Societies: Ramanujan Mathematical Society
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Other: 2009 Most popular biographies
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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)
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Biography of Srinivasa Ramanujan, Mathematical Genius

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

Fast Facts: Srinivasa Ramanujan

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

Early Life and Education

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

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

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

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

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

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

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

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

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

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

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

Personal Life

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

Honors and Awards

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

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

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

Legacy and Impact

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

His results include:

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

Srinivasa Ramanujan was a largely self-taught pure mathematician. Hindered by poverty and ill-health, his highly original work has considerably enriched number theory. More recently his discoveries have been applied to physics, where his theta function lies at the heart of string theory.

Srinivasa Ramanujan was born on December 22, 1887 in the town of Erode, in Tamil Nadu, in the south east of India. His father was K. Srinivasa Iyengar, an accounting clerk for a clothing merchant. His mother was Komalatammal, who earned a small amount of money each month as a singer at the local temple.

His family were Brahmins, the Hindu caste of priests and scholars. His mother ensured the boy was in tune with Brahmin traditions and culture. Although his family were high caste, they were very poor.

Ramanujan’s parents moved around a lot, and he attended a variety of different elementary schools.

Early Mathematics At age 10, Ramanujan was the top student in his district and he started high school at the Kumbakonam Town High School. Looking at the mathematics books in his school’s library, he quickly found his vocation. By age 12, he had begun serious self-study of mathematics, working through cubic equations and arithmetic and geometric series. He invented his own method of solving quartic equations.

As Ramanujan’s mathematical knowledge developed, his main source of inspiration and expertise became Synopsis of elementary results in pure mathematics by George S. Carr. This book presented a very large number of mathematical results – over 4000 theorems – but generally showed little working, cramming into its pages as many results as possible.

Entry 2478 from Carr’s Synopsis of elementary results in pure mathematics

With little other guidance, Ramanujan came to believe this was how mathematics was done, so he himself learned to show little working. Also, he could afford only a small amount of paper, doing most of his work on slate with chalk, transferring a minimal amount of his working and his results to paper.

His memory for mathematical formulas and constants seems to have been boundless: he amazed classmates with his ability to recite the values of irrational numbers like π, e, and √ 2 to as many decimal places as they asked for.

An Apparently Bright Future Fizzles Out In 1904, Ramanujan left high school; his future looked promising: he had won the school’s mathematics prize and, more importantly, a scholarship allowing him to study at the Government Arts College in the town of Kumbakonam.

Obsessed with mathematics, Ramanujan failed his non-mathematical exams and lost his scholarship. In 1905, he traveled to Madras and enrolled at Pachaiyappa’s College, but again failed his non-mathematical exams.

The Discovery of Ramanujan as a Mathematician of Genius

The Hungry Years At the beginning of 1907, at age 19, with minimal funds and a stomach all too often groaning with hunger, Ramanujan continued on the path he had chosen: total devotion to mathematics. The mathematics he was doing was highly original and very advanced.

Even though (or some might say because) he had very little formal mathematical education he was able to discover new theorems. He also independently discovered results originally discovered by some of the greatest mathematicians in history, such as Carl Friedrich Gauss and Leonhard Euler .

Ill-health was Ramanujan’s constant companion – as it would be for much of his short life.

By 1910, he realized he must find work to stay alive. In the city of Madras he found some students who needed mathematics tutoring and he also walked around the city offering to do accounting work for businesses.

And then a piece of luck came his way. Ramanujan tried to find work at the government revenue department, and there he met an official whose name was Ramaswamy Aiyer. Ramanujan did not have a resume to show Ramaswamy Aiyer; all he had were his notebooks – the results of his mathematical work.

Ramanujan’s good fortune was that Ramaswamy Aiyer was a mathematician. He had only recently founded the Indian Mathematical Society, and his jaw dropped when he saw Ramanujan’s work.

Things Begin to Look Up Ramaswamy Aiyer contacted the secretary of the Indian Mathematical Society, R. Ramachandra Rao, suggesting he provide financial support for Ramanujan. At first Rao resisted the idea, believing Ramanujan was simply copying the work of earlier great mathematicians. A meeting with Ramanujan, however, convinced Rao that he was dealing with a genuine mathematical genius. He agreed to provide support for Ramanujan, and Ramaswamy Aiyer began publishing Ramanujan’s work in the Journal of the Indian Mathematical Society .

Ramanujan’s work, however, was hard to understand. The style he had adopted as a schoolboy, after digesting George S. Carr’s book, contributed to the problem. His mathematics often left too few clues to allow anyone who wasn’t also a mathematical genius to see how he obtained his results.

In March 1912, his financial position improved when he got a job as an accounting clerk with the Madras Port Trust.

There he was encouraged to do mathematics at work after finishing his daily tasks by the port’s Chief Accountant, S. Narayana Iyer, who was treasurer of the Indian Mathematical Society, and by Sir Francis Spring, an engineer, who was Chairman of the Madras Port Trust.

Francis Spring began pressing for Ramanujan’s mathematical work to be supported by the government and for him to be appointed to a research position at one of the great British universities.

A Crank or a Genius? Ramanujan and his supporters contacted a number of British professors, but only one was receptive – an eminent pure mathematician at the University of Cambridge – Godfrey Harold Hardy, known to everyone as G. H. Hardy, who received a letter from Ramanujan in January 1913. By this time, Ramanujan had reached the age of 25.

Professor Hardy puzzled over the nine pages of mathematical notes Ramanujan had sent. They seemed rather incredible. Could it be that one of his colleagues was playing a trick on him?

Hardy reviewed the papers with J. E. Littlewood, another eminent Cambridge mathematician, telling Littlewood they had been written by either a crank or a genius, but he wasn’t quite sure which. After spending two and a half hours poring over the outlandishly original work, the mathematicians came to a conclusion. They were looking at the papers of a mathematical genius:

Hardy was eager for Ramanujan to move to Cambridge, but in accordance with his Brahmin beliefs, Ramanujan refused to travel overseas. Instead, an arrangement was made to fund two years of work at the University of Madras. During this time, Ramanujan’s mother had a dream in which the goddess Namagiri told her she should give her son permission to go to Cambridge, and this she did. Her decision led to several very heated quarrels with other devout family members.

Ramanujan at Cambridge

Ramanujan arrived in Cambridge in April 1914, three months before the outbreak of World War 1. Within days he had begun work with Hardy and Littlewood. Two years later, he was awarded the equivalent of a Ph.D. for his work – a mere formality.

Srinivasa Ramanujan after his Cambridge degree was awarded in March 1916.

Srinivasa Ramanujan at Cambridge

Ramanujan’s prodigious mathematical output amazed Hardy and Littlewood.

The notebooks he brought from India were filled with thousands of identities, equations, and theorems he discovered for himself in the years 1903 – 1914.

Some had been discovered by earlier mathematicians; some, through inexperience, were mistaken; many were entirely new.

Explaining Ramanujan’s Extraordinary Mathematical Output

Ramanujan had very little formal training in mathematics, and indeed large areas of mathematics were unknown to him. Yet in the areas familiar to him and in which he enjoyed working, his output of new results was phenomenal.

Ramanujan said the Hindu goddess Namagiri – who had appeared in his mother’s dream telling her to allow him to go to Cambridge – had appeared in one of his own dreams .

According to Hardy, Ramanujan’s ideas were:

It is possible that Ramanujan’s brain was wired differently from most mathematicians.

He seems to have had a personal window through which some problems in number theory appeared with a clarity denied to most people in the field. Results they fought for through days of arduous thought seemed obvious to Ramanujan.

Professor Bruce Berndt is an analytic number theorist who, since 1977, has spent decades researching Ramanujan’s theorems. He has published several books about them, establishing that the great majority are correct. He was told an interesting story by the great Hungarian mathematician Paul Erdős about something G. H. Hardy had once said to him:

Given that David Hilbert is regarded by many as the greatest mathematician of the early twentieth century, and Hardy and Littlewood were immensely influential mathematicians, it is fascinating to see how exceptional Hardy thought Ramanujan’s raw mathematical ability was.

Number Theory and String Theory In 1918 Ramanujan became the first Indian mathematician to be elected a Fellow of the British Royal Society:

“Distinguished as a pure mathematician particularly for his investigation in elliptic functions and the theory of numbers.”

In his short lifetime he produced almost 4000 proofs, identities, conjectures, and equations in pure mathematics.

His theta function lies at the heart of string theory in physics.

The Ramanujan theta function.

Some Personal Details and the End

In July 1909, Ramanujan married S. Janaki Ammal, who was then just 10 years old. The marriage had been arranged by Ramanujan’s mother. The couple began sharing a home in 1912.

When Ramanujan left to study at the University of Cambridge, his wife moved in with Ramanujan’s parents. Ramanujan’s scholarship was sufficient for his needs in Cambridge and the family’s needs in Kumbakonam.

For his first three years in Cambridge, Ramanujan was very happy. His health, however, had always been rather poor. The winter weather in England, much colder than anything he had ever imagined, made him ill for a time.

In 1917, he was diagnosed with tuberculosis and worryingly low vitamin levels. He spent months being cared for in sanitariums and nursing homes.

In February 1919, his health seemed to have recovered sufficiently for him to return to India, but sadly he lived for only one more year.

Srinivasa Ramanujan died aged 32 in Madras on April 26, 1920. His death was most likely caused by hepatic amoebiasis caused by liver parasites common in Madras. His body was cremated.

Sadly, some of Ramanujan’s Brahmin relatives refused to attend his funeral because he had traveled overseas.

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Creative Commons Images Image of Paul Erdős by Topsy Kretts, Creative Commons Attribution 3.0 Unported License.

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Further Reading Srinivasa Ramanujan Aiyangar, Godfrey Harold Hardy, P. Venkatesvara Seshu Aiyar, Bertram Martin Wilson Collected Papers of Srinivasa Ramanujan American Mathematical Soc., 1927

Bruce C. Berndt Ramanujan’s Notebooks Part 1 Springer Verlag, 1985

Srinivasa Ramanujan Aiyangar Ramanujan: Letters and Commentary American Mathematical Soc., 1995

Godfrey Harold Hardy Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work AMS Chelsea Pub., 1 Jan 1999

B A Kupershmidt A Review of Bruce C. Berndt’s Ramanujan’s Notebooks, Parts I – V. Journal of Nonlinear Mathematical Physics, V.7, N 2, R7–R37, 2000

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Srinivasa Ramanujan

Dec 22, 1887, in Erode, Madras Presidency (now Tamil Nadu)

April 26, 1920 (at age 32) in Chetput, Madras, Madras Presidency (now Tamil Nadu)

Indian

Landau–Ramanujan constant

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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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

Srinivasa Ramanujan was born on December 22, 1887 in Erode, a city in the Tamil Nadu state of India. His father, K. Srinivasa Iyengar was a clerk while his mother, Komalatammal performed as a singer, in a temple. Even though they belonged to the Brahmins who are known to be the highest caste of Hinduism, Ramanujan’s family was very poor.

At the age of 10, in 1897, Ramanujan attended the high school in Kumbakonam Town. There he discovered his intelligence in the field of mathematics and by his independent study of books from the school library; Ramanujan increased his knowledge and skills. At age of just 12 years, he had developed understanding of trigonometry and was able to solve cubic equations and arithmetic and geometric series as well.

Among all of the mathematical literature Ramanujan went through, a book by George Shoobridge Carr , titled as A Synopsis of Elementary Results in Pure and Applied Mathematics , written in 1886, proved to be the primary medium that laid him onto the path of becoming a great mathematician. He got access to its copy in 1902 and in a short time he not only went through all of its theorems but also verified their results. He also rediscovered the work done by many famous mathematicians including Carl Friedrich Gauss and Leonhard Euler . In addition to this, many new theorems were also formulated by him.

Ramanujan completed his high school by the age of 17, in 1904. Due to his outstanding results, he was awarded scholarship for higher studies in the Government Arts College in Kumbakonam. But his inclination towards mathematics led to his failure in non-mathematical subjects and ultimately discontinuation of his scholarship. Ramanujan had to face the same situation in Pachaiyappa’s College, an affiliation of the University of Madras by losing his scholarship there.

When Ramanujan got married at the age of 22, in 1909, he got worried for his financial instability, but was still strong-willed to continue with his passion. He started independent research work in mathematics by getting enrolled in a college. He was supported by a government official and secretary of the Indian Mathematical Society , Ramachandra Rao .

In 1911, Ramanujan got his first publication with the assistance of Ramaswamy Aiyer , the founder of the Indian Mathematical Society , in the society’s journal only. This research was on Bernoulli Numbers , done independently by him in 1904. After about a year, Ramanujan started working in Madras at the Port Trust Office as a clerk alongside his research work.

After applying for British Universities in 1913, Ramanujan’s work got acknowledged by a prominent mathematician of the Cambridge University , Godfrey Harold Hardy who funded him for research in the University of Madras. In 1914, Ramanujan went to England to utilize his scholarship at Trinity College, Cambridge and work in collaboration with G. H. Hardy and J. E. Littlewood . In 1916, Ramanujan got his Bachelors in Science degree and a year later he became a fellow of the British Royal Society .

Ramanujan has contributed a lot to mathematics in his short lifespan. This includes his independent works from India as well as the researches done under the mentorship of G. H. Hardy in England. Alongside his outstanding discoveries in continued fractions , divergent series , hypergeometric series , Reimann series and elliptic integrals , his advancements in partition of numbers are quite phenomenal. Ramanujan worked on properties of partition function and in collaboration with G. H. Hardy, developed the circular method to represent an integer in the form of its partitions. This led to many developments in analytic number theory by future mathematicians.

In 1917, Ramanujan got diagnosed with tuberculosis. He returned to India in 1919 and died in 1920, at the age of 32.

About three months before his death, Ramanujan wrote his last letter to Hardy, explaining his new discovery in mathematics; the Theta Function and its 17 identities. Later, many mathematicians worked on this function, proved the identities and found new ones too.

Even though Ramanujan had got many papers published in different journals during his life, much work remained unpublished. The notes that he left behind were studied by many mathematicians after him, who verified his discoveries, and found their potential applications.

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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:

Full Name	Srinivasa Ramanujan FRS (Fellow of the Royal Society)
Father	Kuppuswamy Srinivasa Iyengar.
Mother	Komalatamma.
Born	22nd December, 1887.
Birth Place	Erode, Madras Presidency (now Tamil Nadu), India.
Died	26th April, 1920.
Cause Of Death	Tuberculosis.
Death Place	Kumbakonam, Madras Presidency, British India.
Field Of Work	Mathematics.
Contributions In Mathematics	Mathematical analysis, number theory, infinite series, continued fractions, modular forms and mock theta functions.
Education	He was a self-taught mathematician with no formal education in mathematics.
Recognitions	He was the Fellow of the Royal Society in 1918. He was awarded the Bôcher Memorial Prize in 1921 (Posthumously).

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:


	Developed numerous formulas for infinite series, including results related to hypergeometric series.
	Identified 1729 as the smallest positive integer expressible as the sum of two cubes in two ways.
	Introduced mock theta functions, expanding the theory of modular forms and number theory.
	Explored the partition function, discovering congruences that significantly impacted number theory.
	Introduced the concept of the Ramanujan prime and contributed to the tau function in modular forms.
	Advanced the study of theta functions and elliptic functions, deepening the understanding of these mathematical concepts.
	Worked towards unifying different mathematical theories, showcasing a holistic approach.
	Collaborated with G. H. Hardy, resulting in joint publications and advancements in mathematical research.

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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.


1918	Fellow of the Royal Society
1917	Adams Prize
1920	Honorary Doctorate from the University of Cambridge

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

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

What is the meaning of frs in srinivasa ramanujan.

The meaning of FRS is Fellow of Royal Society.

When did Ramanujan got FRS?

On 2nd May 1918 Ramanujan got FRS .

Why is 1729 called Ramanujan number?

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

What is Ramanujan famous for?

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

Why did Ramanujan died at 32?

At the age of 32 Ramanujan died due to tuberculosis.

What was the invention of Srinivasa Ramanujan?

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

Who was the wife of Srinivasa Ramanujan?

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

Did Srinivasa Ramanujan have Child?

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

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Srinivasa Ramanujan Biography

Birthday: December 22 , 1887 ( Capricorn )

Born In: Erode

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Died At Age: 32

Spouse/Ex-: Janaki Ammal

father: K. Srinivasa Iyengar

mother: Komalat Ammal

siblings: Sadagopan Ramanujan

Born Country: India

Poorly Educated Mathematicians

Died on: April 26 , 1920

place of death: Kumbakonam, Madras Presidency, British India

Cause of Death: Amoebiasis

education: Government Arts College, Pachaiyappa's College, Trinity College, Cambridge (BSc, 1916)

awards: ICTP Ramanujan Prize SASTRA Ramanujan Prize

You wanted to know

What is srinivasa ramanujan known for.

Srinivasa Ramanujan is known for his significant contributions to mathematical analysis, number theory, infinite series, and continued fractions.

What is Ramanujan's famous mathematical formula?

Ramanujan is famous for his formula for the partition function, which gives the number of ways a number can be expressed as a sum of positive integers.

What is the Ramanujan conjecture?

The Ramanujan conjecture is a famous unsolved problem in mathematics related to the properties of certain arithmetic functions.

How did Ramanujan independently discover new mathematical results?

Ramanujan often stated that his mathematical ideas came to him as visions from a Hindu goddess, inspiring his groundbreaking work.

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See the events in life of Srinivasa Ramanujan in Chronological Order

How To Cite

Ramanujan: The Man Who Knew Infinity

Srinivasa Ramanujan (1887-1920), the man who reshaped twentieth-century mathematics with his various contributions in several mathematical domains, including mathematical analysis, infinite series, continued fractions, number theory, and game theory is recognized as one of history's greatest mathematicians. Leaving this world at the youthful age of 32, Ramanujan made significant contributions to mathematics that only a few others could match in their lifetime. Surprisingly, he never received any formal mathematics training. Most of his mathematical discoveries were based only on intuition and were ultimately proven correct. With its humble and sometimes difficult start, his life story is just as fascinating as his incredible work. Every year, Ramanujan’s birth anniversary on December 22 is observed as National Mathematics Day.

Born in Erode, Tamil Nadu, India, Ramanujan demonstrated an exceptional intuitive grasp of mathematics at a young age. Despite being a mathematical prodigy, Ramanujan's career did not begin well. He received a college scholarship in 1904, but he quickly lost it by failing in nonmathematical subjects. Another attempt at college in Madras (now Chennai) ended in failure when he failed his First Arts exam. It was around this time that he began his famous notebooks. He drifted through poverty until 1910 when he was interviewed by R. Ramachandra Rao, secretary of the Indian Mathematical Society. Rao was initially sceptical of Ramanujan, but he eventually recognised his abilities and supported him financially.

Srinivasa Ramanujan began developing his theories in mathematics and published his first paper in 1911. He was mentored at Cambridge by GH Hardy, a well-known British mathematician who encouraged him to publish his findings in a number of papers. In 1918, Ramanujan became the second Indian to be included as a Fellow of the Royal Society.

Ramanujan’s major contributions to mathematics:

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

Infinite series for pi: In 1914, Ramanujan found a formula for infinite series for pi, which forms the basis of many algorithms used today. Finding an accurate approximation of π (pi) has been one of the most important challenges in the history of mathematics.

Game theory: Ramanujan discovered a long list of new ideas for solving many challenging mathematical problems that have given great impetus to the development of game theory. His contribution to game theory is purely based on intuition and natural talent and is unmatched to this day.

Mock theta function: He elaborated on the mock theta function, a concept in the field of modular forms of mathematics.

Ramanujan number: 1729 is known as the Ramanujan number which is the sum of the cubes of two numbers 10 and 9.

Circle Method: Ramanujan, along with GH Hardy, invented the circle method which gave the first approximations of the partition of numbers beyond 200. This method contributed significantly to solving the notorious complex problems of the 20th century, such as Waring's conjecture and other additional questions.

Theta Function: Theta function is a special function of several complex variables. German mathematician Carl Gustav Jacob Jacobi invented several closely related theta functions known as Jacobi theta functions. Theta function was studied by extensively Ramanujan who came up with the Ramanujan theta function, that generalizes the form of Jacobi theta functions and also captures general properties. Ramanujan theta function is used to determine the critical dimensions in Bosonic string theory, superstring theory, and M-theory.

Other notable contributions by Ramanujan include hypergeometric series, the Riemann series, the elliptic integrals, the theory of divergent series, and the functional equations of the zeta function.

Ramanujan‘s achievements were all about elegance, depth, and surprise beautifully intertwined. Unfortunately, Ramanujan contracted a fatal illness in England in 1918. He convalesced there for more than a year and returned to India in 1919. His condition then worsened, and he died on 26 April 1920. One might expect that a dying man would stop working and await his fate. However, Ramanujan spent his last year producing some of his most profound mathematics.

It has been more than a century, however, his mathematical discoveries are still alive and flourishing. "Ramanujan is important not just as a mathematician but because of what he tells us that the human mind can do”. "Someone with his ability is so rare and so precious that we can't afford to lose them. A genius can arise anywhere in the world. It is our good fortune that he was one of us. It is unfortunate that too little of Ramanujan’s life and work, esoteric though the latter is, seems to be known to most of us".

More Resources:

1 . The Awardees of Ramanujan Fellowships

2. Awardees of Ramanujan Fellowship for 2019-20 & 2020-21

3 . Recipients of The Ramanujan Prize

4 . Ramanujan Fellowship

5 . India and Mathematics

6 . India celebrates National Mathematics Day

7 . Mathematical Organisations

8 . Statistical Organisations

9 . Centre of Excellence in Science and Mathematics Education

10. Ramanujan's legacy: Another cryptic clue of Ramanujan solved

Subodh Kumar

[email protected]
Delhi India, 110085

Srinivasa Ramanujan | Biography, Contributions & Speech in English

Srinivasa Ramanujan Speech in English

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

Who Was Srinivasa Ramanujan?

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

Ramanujan number speciality

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

The Early Life of Srinivasa Ramanujan

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

Also Read: Important Maths Formulas for Class 8

Ramanujan’s Contribution to Mathematics

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

The Ramanujan Prime and the Ramanujan theta function

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

The Legacy of Srinivasa Ramanujan

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

Why is Ramanujan’s story so inspiring?

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

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

Also Check Out : Geometry Formulas For Class 8

How can we learn from Ramanujan’s example?

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

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

Frequently Asked Questions of Srinivasa Ramanujan

Where and when was srinivasa ramanujan born.

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

What are some of Ramanujan’s contributions to mathematics?

Ramanujan has made many contributions to mathematics, including:

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

What is Srinivasa Ramanujan famous for?

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

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A Remarkable Letter

The beginning of the story, who was hardy, the letter and its effects, a way of doing mathematics, seeing what’s important, truth versus narrative, going to cambridge, ramanujan in cambridge, the aftermath, what became of hardy, ramanujan’s mathematics, are they random facts, automating ramanujan, modern ramanujans, what if ramanujan had mathematica, who was ramanujan.

"Hardy" and "Ramanujan" in The Man Who Knew Infinity

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:

Then there are statements that suggest a kind of experimental approach to mathematics:

$Empirical math in Ramanujan's letter (Reproduced by kind permission of the Syndics of Cambridge University Library)$

But some things get more exotic, with pages of formulas like this:

Remarkable formulas in Ramanujan's letter (Reproduced by kind permission of the Syndics of Cambridge University Library)

What are these? Where do they come from? Are they even correct?

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.

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:

Modern Mathematica reproduces Ramanujan's results

And the first surprise—just as G. H. Hardy discovered back in 1913—is that, yes, the formulas are essentially all correct. But what kind of person would have made them? And how? And are they all part of some bigger picture—or in a sense just scattered random facts of mathematics?

One of the surviving math pages from Ramanujan's letter to Hardy (reproduced by kind permission of the Syndics of Cambridge University Library)

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.

$The math book Ramanujan got at age 16$

By the time Ramanujan got to college, all he wanted to do was mathematics—and he failed his other classes, and at one point ran away, causing his mother to send a missing-person letter to the newspaper:

Ramanujan moved to Madras (now Chennai ), tried different colleges, had medical problems, and continued his independent math research. In 1909, when he was 21, his mother arranged (in keeping with customs of the time) for him to marry a then-10-year-old girl named Janaki, who started living with him a couple of years later.

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.

A page from Ramanujan's first published paper

When Ramanujan’s mathematical friends didn’t succeed in getting him a scholarship, Ramanujan started looking for jobs, and wound up in March 1912 as an accounting clerk—or effectively, a human calculator—for the Port of Madras (which was then, as now, a big shipping hub). His boss, the Chief Accountant, happened to be interested in academic mathematics, and became a lifelong supporter of his. The head of the Port of Madras was a rather distinguished British civil engineer, and partly through him, Ramanujan started interacting with a network of technically oriented British expatriates. They struggled to assess him, wondering whether “he has the stuff of great mathematicians” or whether “his brains are akin to those of the calculating boy”. They wrote to a certain Professor M. J. M. Hill in London, who looked at Ramanujan’s rather outlandish statements about divergent series and declared that “Mr. Ramanujan is evidently a man with a taste for Mathematics, and with some ability, but he has got on to wrong lines.” Hill suggested some books for Ramanujan to study.

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.

G. H. Hardy was born in 1877 to schoolteacher parents based about 30 miles south of London . He was from the beginning a top student, particularly in mathematics. Even when I was growing up in England in the early 1970s, it was typical for such students to go to Winchester for high school and Cambridge for college. And that’s exactly what Hardy did. (The other, slightly more famous, track—less austere and less mathematically oriented—was Eton and Oxford , which happens to be where I went.)

Cambridge undergraduate mathematics was at the time very focused on solving ornately constructed calculus-related problems as a kind of competitive sport—with the final event being the Mathematical Tripos exams, which ranked everyone from the “Senior Wrangler” (top score) to the “Wooden Spoon” (lowest passing score). Hardy thought he should have been top, but actually came in 4th, and decided that what he really liked was the somewhat more rigorous and formal approach to mathematics that was then becoming popular in Continental Europe.

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:

The modern version of integrals from Hardy's first research paper

For a decade Hardy basically worked on the finer points of calculus, figuring out how to do different kinds of integrals and sums, and injecting greater rigor into issues like convergence and the interchange of limits.

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…”

Ramanujan responds to Hardy (Reproduced by kind permission of the Syndics of Cambridge University Library)

Ramanujan makes a point of saying that it was Hardy’s first category of results—ones that were already known—that he’s most pleased about, “For my results are verified to be true even though I may take my stand upon slender basis.” In other words, Ramanujan himself wasn’t sure if the results were correct—and he’s excited that they actually are.

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 zeros off the real axis, 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…)

It’s a weird result, to be sure. But not as crazy as it might at first seem. And in fact it’s a result that’s nowadays considered perfectly sensible for purposes of certain calculations in quantum field theory (in which, to be fair, all actual infinities are intended to cancel out at the end).

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 utilised 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.

In modern times, we’re used to seeing incredibly complicated formulas routinely generated by Mathematica . But usually they’re just intermediate steps, and aren’t what papers explicitly talk much about. For Ramanujan, though, complicated formulas were often what really told the story. And of course it’s incredibly impressive that he could derive them without computers and modern tools.

(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.

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 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):

Most mathematicians would say, “It’s an amusing coincidence that that’s so close to an integer—but so what?” But Ramanujan realized there was more to it . He found other relations (those “=” should really be ≅):

Then he began to build a theory—that involves elliptic functions, though Ramanujan didn’t know that name yet—and started coming up with new series approximations for π:

Previous approximations to π had in a sense been much more sober, though the best one before Ramanujan’s ( Machin ’s series from 1706) did involve the seemingly random number 239:

But Ramanujan’s series—bizarre and arbitrary as they might appear—had an important feature: they took far fewer terms to compute π to a given accuracy. In 1977, Bill Gosper —himself a rather Ramanujan-like figure, whom I’ve had the pleasure of knowing for more than 35 years—took the last of Ramanujan’s series from the list above, and used it to compute a record number of digits of π. There soon followed other computations, all based directly on Ramanujan’s idea—as is the method we use for computing π in Mathematica and the Wolfram Language.

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:

And then—in pretty much his only published example of geometry—he gave a peculiar geometric construction for approximately “squaring the circle” based on this formula:

An approximate "squaring of the circle" by Ramanujan

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 utilised 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 .)

What was the Ramanujan who arrived in Cambridge like? He was described as enthusiastic and eager, though diffident. He made jokes, sometimes at his own expense. He could talk about politics and philosophy as well as mathematics. He was never particularly introspective. In official settings he was polite and deferential and tried to follow local customs. His native language was Tamil , and earlier in his life he had failed English exams, but by the time he arrived in England, his English was excellent. He liked to hang out with other Indian students, sometimes going to musical events, or boating on the river. Physically, he was described as short and stout—with his main notable feature being the brightness of his eyes. He worked hard, chasing one mathematical problem after another. He kept his living space sparse, with only a few books and papers. He was sensible about practical things, for example in figuring out issues with cooking and vegetarian ingredients. And from what one can tell, he was happy to be in Cambridge.

But then on June 28, 1914—two and a half months after Ramanujan arrived in England— Archduke Ferdinand was assassinated, and on July 28, World War I began. There was an immediate effect on Cambridge. Many students were called up for military duty. Littlewood joined the war effort and ended up developing ways to compute range tables for anti-aircraft guns. Hardy wasn’t a big supporter of the war—not least because he liked German mathematics—but he volunteered for duty too, though was rejected on medical grounds.

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 :

Approximating the number of partitions for large n

But then, using ideas Ramanujan developed back in India, it progressively improves the estimate, to the point where the exact integer result can be obtained. In Ramanujan’s day, computing the exact value of PartitionsP[200] was a big deal—and the climax of his paper. But now, thanks to Ramanujan’s method, it’s instantaneous:

Cambridge was dispirited by the war—with an appalling number of its finest students dying, often within weeks, at the front lines. Trinity College’s big quad had become a war hospital. But through all of this, Ramanujan continued to do his mathematics—and with Hardy’s help continued to build his reputation.

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.

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.

Notifying Hardy of Ramanujan's death (image courtesy of the Master and Fellows of Trinity College, Cambridge)

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.

In 1940, Hardy gave all the letters he had from Ramanujan to the Cambridge University Library , but the original cover letter for what Ramanujan sent in 1913 was not among them—so now the only record we have of that is the transcription Hardy later published . Ramanujan’s three main notebooks sat for many years on top of a cabinet in the librarian’s office at the University of Madras, where they suffered damage from insects, but were never lost. His other mathematical documents passed through several hands, and some of them wound up in the incredibly messy office of a Cambridge mathematician —but when he died in 1965 they were noticed and sent to a library, where they languished until they were “rediscovered” with great excitement as Ramanujan’s lost notebook in 1976.

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:

Word clouds of the titles of Hardy's papers, before Ramanujan

Shortly before Ramanujan entered his life, Hardy had started to collaborate with John Littlewood, who he would later say was an even more important influence on his life than Ramanujan. After Ramanujan died, Hardy moved to what seemed like a better job in Oxford , and ended up staying there for 11 years before returning to Cambridge. His absence didn’t affect his collaboration with Littlewood, though—since they worked mostly by exchanging written messages, even when their rooms were less than a hundred feet apart. After 1911 Hardy rarely did mathematics without a collaborator; he worked especially with Littlewood, publishing 95 papers with him over the course of 38 years.

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.

The beginning of Hardy's A Mathematician's Apology

To be fair, however, Hardy wrote the book at a low point in his own life, when he was concerned about his health and the loss of his mathematical faculties. And perhaps that explains why he made a point of explaining that “mathematics… is a young man’s game”. (And in an article about Ramanujan, he wrote that “a mathematician is often comparatively old at 30, and his death may be less of a catastrophe than it seems.”) I don’t know if the sentiment had been expressed before—but by the 1970s it was taken as an established fact, extending to science as well as mathematics. Kids I knew would tell me I’d better get on with things, because it’d be all over by age 30.

Hardy explains that "mathematics... is a young man's game"

Is that actually true? I don’t think so. It’s hard to get clear evidence, but as one example I took the data we have on notable mathematical theorems in Wolfram|Alpha and the Wolfram Language, and make a histogram of the ages of people who proved them. It’s not a completely uniform distribution (though the peak just before 40 is probably just a theorem-selection effect associated with Fields Medals ), but particularly if one corrects for life expectancies now and in the past it’s a far cry from showing that mathematical productivity has all but dried up by age 30.

Distribution of the ages at which mathematicians proved "notable" theorems

My own feeling—as someone who’s getting older myself—is that at least up to my age , many aspects of scientific and technical productivity actually steadily increase. For a start, it really helps to know more—and certainly a lot of my best ideas have come from making connections between things I’ve learned decades apart. It also helps to have more experience and intuition about how things will work out. And if one has earlier successes, those can help provide the confidence to move forward more definitively, without second guessing. Of course, one must maintain the constitution to focus with enough intensity—and be able to concentrate for long enough—to think through complex things. I think in some ways I’ve gotten slower over the years, and in some ways faster. I’m slower because I know more about mistakes I make, and try to do things carefully enough to avoid them. But I’m faster because I know more and can shortcut many more things. Of course, for me in particular, it also helps that over the years I’ve built all sorts of automation that I’ve been able to make use of.

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 :

Fraction of mathematics papers tagged as "number theory" vs. time

Ramanujan’s interest may have been to some extent driven by the peak in the early 1900s (which would probably go even higher with earlier data). But by the 1930s, the emphasis of mathematics had shifted away from what seemed like particular results in areas like number theory and calculus, towards the greater generality and formality that seemed to exist in more algebraic areas.

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):

Using results from Ramanujan in a 1982 physics paper

Starting in the 1970s, there was a big effort—still not entirely complete—to prove results Ramanujan had given in his notebooks. And there were increasing connections being found between the particular results he’d got, and general themes emerging in number theory.

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”):

The seemingly arbitrary definition of Ramanujan's tau function

And to my mind, the most remarkable thing about Ramanujan is that he could define something as seemingly arbitrary as this, and have it turn out to be useful a century later.

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.

Are these numerical facts significant? I don’t know. Wolfram|Alpha can certainly generate lots of similar facts , but without Ramanujan-like insight, it’s hard to tell which, if any, are significant.

Wolfram|Alpha finds possible formulas for a number

Over the years I’ve received countless communications a bit like this one. Number theory is a common topic. So are relativity and gravitation theory. And particularly in recent years, AI and consciousness have been popular too. The nice thing about letters related to math is that there’s typically something immediately concrete in them: some specific formula, or fact, or theorem. In Hardy’s day it was hard to check such things; today it’s a lot easier. But—as in the case of the almost integer above—there’s then the question of whether what’s being said is somehow “interesting”, or whether it’s just a “random uninteresting fact”.

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 knowledgebase 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.

Posted in: Historical Perspectives , Mathematics

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39 comments

Always been a fan of your software but this is the first time I’m reading your blog. Definitely one of the most comprehensive, exhaustively fact-based and interesting blogs I’ve read. The casual reader may find it a bit lengthy, but your personal inputs and speculations are amazing. The unique perspective of not being a mathematician yourself but being extensively involved with the subject really makes this a unique read. Best two hours I’ve spent this week reading this and related material. Thank you, sir!

“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.” Hardy was a top class mathematician. I don’t understand this statement.

Interesting, I had no idea this was such a big deal. I’m pretty sure I have seen Ramanujan’s notes in my great uncle’s papers. Made no sense to me. If they are potentially interesting, I could raid some attics.

Very nice article. “A Mathematician’s Apology” was one of those things that induced me to major in Math some 40 years ago, and I always enjoy coming across articles about Hardy. I look forward to seeing this movie.

Kudos – As an avid Mathematica user, this is an excellent read!

Thank you!, Brilliant. If experimental mathematics becomes more default way of exploring first , then I am sure we will make faster progress.

Wonderful article, thank you for this!

Wow great stuff. My Math teacher in college was Indian and boy did he know his stuff. It’s amazing how someone back then would master so much with no leads to go off from, pure genius.

Wonderful article. Complements Kanigel’s account by focusing on the math.

Fascinating and brilliant article, carefully constructed and very well documented in its essence. Thank you. Wishing for the best for ‘The Man Who Knew Infinity’ which will be bound to be well received!

Nice read sir. You forgot to mention his near misses on Fermat’s last theorem, I think that was realized only few months ago after almost a century 🙂

I am just a simple Dutch math-teacher, but I have read your article with great pleasure. It is very inspiring and I hope to share it with some of my best pupils.

Wonderful article. Thank you.

Mr Mark Littlewood will do a great service if he could raid the attics and unearth whatever papers of his own great uncle and Ramanujan and he could lay his hands on. The world of mathematics (and science) will be grateful.

Fascinating read! I’m anticipating breakthroughs from Experimental Mathematics in the coming years…

What’s the big deal with calculating P(200)? Just use Euler’s pentagonal number recursion. This should be a two day paper-and pencil job for a grad student in 1914.

I actually went to college because of Ramanujan. I read the book “The Man Who Knew Infinity”

…and I couldn’t understand some of the advanced math in the book so I went to college to learn and I became side-tracked with computers and physics and I came up with the complete theory of everything….

Everyone will realize I am correct in 40 or 50 years — we will all be long dead or too old to care– that’s the way the cookie crumbles.

You were my idol while I was doing my PhD and stuff like TeX, LaTeX and Mathematica were just evolving. This article is wonderful and thank you for adding your thoughts on (arguably) one of the greatest mathematicians the wold has ever seen.

Thank you for this very impressive article, Sir ! “Ramanujan’s brief life and death are symbolic of conditions in India. Of our millions how few get any education at all; how many live on the verge of starvation” (Jawaharlal Nehru)

I know next to nothing about mathematics but I was intrigued by the movie and wanted to find out more about Ramanujan. I really enjoyed the article even though a lot of it was way over my head! Your writing certainly kept my interest. Thanks

Thanks sir for this post. It is very very interesting. I like it. And perhaps first post in my internet-life, I read a post all at a time.

This article would have been so much better with no mentions of Hardy. Hardy clearly attached his name to many of the Ramanujan’s works and hence the apology which only came later after the damage is done. Ramanujan is one of the, sorry the best Mathematician of the entire world, no western mathematician can even touch the dust of his feet, such was his skills. Apparently India failed to properly recognize Ramanujan as they are caught up in their own turmoil by using English as their language for education. Anyway, I am still waiting for an article which lists ALL the Ramanujan’s works without attaching westerners to HIS works

I did empirical studies like Ramujan to fill in time while studying at university. As stated, most findings are mere curiosities. The real highly-advanced skill is to recognize the useful ones.

FROM COCHIN,INDIA

It is pretty hot here in India and therefore we tend to avoid physical work . We like to sit in the shade and brood about the nature of things. Sometimes we do accomplish much with a paper and pencil and of course knowledge and ideas. Remember Bose_Einstein Statistics, Remember SubramaniaChandrasekhar of Black-Holes fame. . He was nearly zeroised by Eddington. But we are a patient breed. We still learn our multiplication tables. And we do respect scholarship. Do read how the Russian Perelman did it solo. Hats off to all the mathematicians who struggling it alone. Einstein did it solo.

he really knows how mathematics works. my inspiring person. so love him

I want to thank you for the clarity and passion of your article. I found you online after seeing the movie (Dev Patel, etc.) on the plane earlier today. I am definitely not mathematically experienced – I love math, but I’m a humanities guy by training – but I am inspired and lifted emotionally both by Ramanujan’s life and your clear, comprehensive writing.

Wonderful blog!! Love the detail, the facts, and the love for mathematics. Thank you, thank you, thank you!

Thank you for the revealing exposition. Ramanujan had a short and brilliant life. It is a shame he died so early, much like Mozart. I wonder what got him started into mathematical explorations.

Lovely, inspriring post. Ramanujan has such a profound, unshakeable, intuitive grasp of mathematics – yet his fascination for it was deeper still “An equation means nothing to me unless it expresses a thought of God” – beautiful!

A very scholarly and engaging article on Ramanujan and Ramanujan-inspired perspective on mathematics and computation. I thoroughly enjoyed reading it. …. Interesting term “experimental mathematics”.

Great insight into the brilliant collaboration and mind! Thank You very much.

nice reading. Greatly written.

That section on his death made me weep, literally.

Excellent read! You have done a great service to Ramanujan’s life and work with this article! Thank you!

I recently saw this movie’s explanation on a youtube channel, it was very motivational yet emotional too.

Thx, interesting read.

A very scholarly and engaging article on Ramanujan and Ramanujan-inspired perspective on mathematics and computation. I thoroughly enjoyed reading it. Proud to be Indian

Wow,a whole collection of Ramanujan. I had never seen article like this before…

Excellent read! You have done a great service to Ramanujan’s life and work with this article! Love from India

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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.

Get the huge list of more than 500 Essay Topics and Ideas

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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Ramanujan | 10 Major Contributions And Achievements

Srinivasa Ramanujan FRS (1887 – 1920) was a self-taught Indian mathematical genius who made numerous contributions in several mathematical fields including mathematical analysis, infinite series, continued fractions, number theory and game theory . Ramanujan provided solutions to mathematical problems that were then considered unsolvable. Moreover, some of his work was so ahead of his time that mathematicians are still understanding its relevance . In 1914, Ramanujan found a formula for computing π (pi) that is currently the basis for the fastest algorithms used to calculate π. The circle method , which he developed with G. H. Hardy , constitute a large area of current mathematical research. Moreover, Ramanujan discovered K3 surfaces which play key roles today in string theory and quantum physics; while his mock modular forms are being used in an effort to unlock the secret of black holes. Know more about the achievements of Srinivasa Ramanujan through his 10 major contributions to mathematics.

#1 HE WAS THE SECOND INDIAN TO BE ELECTED A FELLOW OF THE ROYAL SOCIETY

A self-taught genius, Ramanujan moved to England in March 1914 after his talent was recognized by British mathematician G. H. Hardy . In 1916, Ramanujan was awarded a Bachelor of Science by Research degree (later named Ph.D.) by Cambridge even though he was not an undergraduate. The Ph.D. was awarded in recognition of his work on ‘Highly composite numbers’ . In 1918, Ramanujan became one of the youngest Fellows of the Royal Society and only the second Indian member . The same year he was elected a Fellow of Trinity College, Cambridge , the first Indian to be so honored . During his short lifespan of 32 years, Ramanujan independently compiled around 3,900 results . Apart from the below mentioned achievements his contributions include developing the relationship between partial sums and hyper-geometric series ; independently discovering Bernoulli numbers and using these numbers to formulated the value of Euler’s constant up to 15 decimal places ; discovering the Ramanujan prime number and the Landau–Ramanujan constant ; and coming up with Ramanujan’s sum and the Ramanujan’s master theorem.

#2 THE FASTEST ALGORITHMS FOR CALCULATION OF PI ARE BASED ON HIS SERIES

Finding an accurate approximation of π (pi) has been one of the most important challenges in the history of mathematics. In 1914, Srinivasa Ramanujan found a formula for computing pi that converges rapidly . His formula computes a further eight decimal places of π with each term in the series . It was in 1989, that Chudnovsky brothers computed π to over 1 billion decimal places on a supercomputer using a variation of Ramanujan’s infinite series of π. This was a world record for computing the most digits of pi . Moreover, the Ramanujan series is currently the basis for the fastest algorithms used to calculate π.

#3 RAMANUJAN CONJECTURE PLAYED A KEY ROLE IN THE FAMOUS LANGLANDS PROGRAM

In 1916 , Ramanujan published his paper titled “On certain arithmetical functions” . In the paper, Ramanujan investigated the properties of Fourier coefficients of modular forms . Though the theory of modular forms was not even developed then , he came up with three fundamental conjectures that served as a guiding force for its development . His first two conjectures helped develop the Hecke theory , which was formulated 20 years after his paper, in 1936, by German mathematician Erich Hecke . However, it was his last conjecture, known as the Ramanujan conjecture , that created a sensation in in 20th century mathematics . It played a pivotal role in the Langlands program , which began in 1970 through the proposal of American-Canadian mathematician Robert Langlands . The Langlands program aims to relate representation theory and algebraic number theory , two seemingly different fields of mathematics . It is widely viewed as the single biggest project in modern mathematical research . “On certain arithmetical functions” by Ramanujan thus effectively changed the course of 20th century mathematics .

#4 HE DEVELOPED THE INFLUENTIAL CIRCLE METHOD IN PARTITION NUMBER THEORY

A partition for a positive integer n is the number of ways the integer can be expressed as a sum of positive integers . For example p(4) = 5 . That means 4 can be expressed as a sum of positive integers in 5 ways: 4, 3+1, 2+2, 2+1+1 and 1+1+1 +1. Ramanujan, along with G. H. Hardy, invented the circle method which gave the first approximations of the partition of numbers beyond 200 . This method was largely responsible for major advances in the 20th century of notoriously difficult problems such as Waring’s conjecture and other additive questions. The circle method is now one of the central tools of analytic number theory . Moreover, circle method and its refinements constitute a large area of current mathematical research.

#5 HE DISCOVERED THE THREE RAMANUJAN’S CONGRUENCES

Related to the Partition Theory of Numbers, Ramanujan also came up with three remarkable congruences for the partition function p(n) . They are p(5n+4) = 0(mod 5); p(7n+4) = 0(mod 7); p(11n+6) = 0(mod 11) . For example, the first congruence means that if an integer is 4 more than a multiple of 5, then number of its partitions is a multiple of 5 . The study of Ramanujan type congruence is a popular research topic of number theory. It was in 2011, that a conceptual explanation for Ramanujan’s congruences was finally discovered . Ramanujan’s work on partition theory has applications in a number of areas including particle physics (particularly quantum field theory) and probability .

#6 NUMBER 1729 IS NAMED HARDY–RAMANUJAN NUMBER

In a famous incident British mathematician G. H. Hardy while visiting Ramanujan had ridden in a taxi cab with the number 1729 . He remarked to Ramanujan that the number “seemed to me rather a dull one, and that I hoped it was not an unfavorable omen” . “No,” Ramanujan replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.” The two different ways are: 1729 = 1 3 + 12 3 = 9 3 + 10 3 . 1729 is now known as the Hardy–Ramanujan number . Moreover, numbers that are the smallest number that can be expressed as the sum of two cubes in n distinct ways are now referred to as taxicab numbers due to the incident. The relevance of 1729 has recently come to light as it was part of a much larger theory that Ramanujan was developing . Theorems have been established in theory of elliptic curves that involve this fascinating number.

#7 HE DID GROUNDBREAKING RESEARCH RELATED TO FERMAT’S LAST THEOREM

In 2013 famous Japanese American Mathematician Ken Ono , along with Sarah Trebat-Leder , found an equation by Ramanujan had clearly showed that he had been working on Fermat’s last theorem, one of the most notable and difficult to prove theorems in the history of mathematics. In 1637, French mathematician Pierre de Fermat had asserted that: if n is a whole number greater than 2 , then there are no positive whole number triples x, y and z , such that x n + y n = z n . This means that there are no numbers which satisfy the equations: x 3 + y 3 = z 3 ; x 4 + y 4 = z 4 ; and so on . The equation of Ramanujan illustrates that he had found an infinite family of positive whole number triples x, y and z that very nearly, but not quite, satisfy Fermat’s equation for n=3 . They are off only by plus or minus one . Among them is 1729 , which misses the mark by 1 for x=9, y=10 and z=12 . Moving forward, Ramanujan also considered the equations of the form: y 2 =x 3 + ax + b . If you plot the points (x,y) for this equation you get an elliptic curve . Elliptic curves played a key role when English mathematician Sir Andrew Wiles finally proved Fermat’s last theorem in 1994, a feat described as a “stunning advance” in mathematics.

#8 RAMANUJAN WAS THE FIRST TO DISCOVER K3 SURFACES

Ken Ono also found that Ramanujan went on to discover an object more complicated than elliptic curves. When it was re-discovered in 1958 by Andre Weil , it was named K3 surface . Thus it has come to light that Ramanujan was using 1729 and elliptic curves to develop formulas for a K3 surface. “Elliptic curves and K3 surfaces form an important next frontier in mathematics and Ramanujan gave remarkable examples illustrating some of their features that we didn’t know before.” Moreover, K3 surfaces play key roles today in string theory and quantum physics . Like, string theory suggests that the world consists of more than the three dimensions that we can see . These extra dimensions are rolled up tightly in tiny little spaces too small for us to perceive . These tiny spaces have a particular geometric structure. Calabi–Yau manifold is a class of geometric objects that have similar structure and one of the simplest classes of Calabi-Yau manifolds comes from K3 surfaces.

#9 HIS THETA FUNCTION LIES AT THE HEART OF STRING THEORY IN PHYSICS

In mathematics, theta functions are special functions of several complex variables . German Mathematician Carl Gustav Jacob Jacobi came up with several closely related theta functions known as Jacobi theta functions . Theta functions were studied extensively by Ramanujan. He came up with the Ramanujan theta function , which generalizes the form of Jacobi theta functions while also capturing their general properties . In particular, the Jacobi triple product takes on an elegant form when written in terms of the Ramanujan theta function . Ramanujan theta function has several important applications. It is used to determine the critical dimensions in Bosonic string theory, superstring theory and M-theory .

#10 HIS MOCK MODULAR FORMS MAY UNLOCK THE SECRET OF BLACK HOLES

In a 1920 letter to Hardy, Ramanujan described several new functions that behaved differently from known theta functions , or modular forms , and yet closely mimicked them. These were the first ever examples of mock modular forms . More than 80 years later, in 2002 , a description for these functions was provided by Sander Zwegers . Further, Ramanujan predicted that his mock modular forms corresponded to ordinary modular forms producing similar outputs for roots of 1 . Ken Ono ultimately showed that a mock modular form could be computed just as Ramanujan predicted . It was found as the output of mock modular forms shoot off to enormous numbers, the corresponding ordinary modular form expand at a similar rate and thus their difference is a relatively small number. Expansion of mock modular forms is now used to compute the entropy, or level of disorder, of black holes. Thus even through black holes were virtually unknown during his time, Ramanujan was able to do mathematics which may unlock their secret.

4 thoughts on “Ramanujan | 10 Major Contributions And Achievements”

A major method for computation of Feynman integrals is the bracket integration method, a direct result from his Master Theorem ( https://en.wikipedia.org/wiki/Ramanujan%27s_master_theorem )

What is plus and minus infinity, he used in his theta function? Infinity in two opposite directions?

very useful information but no that much recognition

awesome pic. loved it

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Srinivasa Ramanujan’s Contributions in Mathematics

Srinivasa Ramanujan is considered to be one of the geniuses in the field of mathematics. He was born on 22nd December 1887, in a small village of Tamil Nadu during British rule in India. His birthday is celebrated as national mathematics day. In high school, he used to do very well in all subjects. In 1990, he started working on his mathematics in geometry and arithmetic series. Although he had no official training in mathematics, even then, he was able to solve problems that were considered unsolvable. He published his first paper in 1911. In January 1913, Ramanujan began a postal conversation with an English mathematician, G.H. Hardy at the University of Cambridge, England and wrote a letter after having seen a copy of his book Orders of infinity . He found Ramanujan’s work to be extraordinary and arranged for him to travel to Cambridge in 1914. As Ramanujan was an orthodox Brahmin, a vegetarian, his religion might have restricted him to travel. This difficulty of Ramanujan was solved partly by E H Neville, a colleague of Hardy. Hardy after analysing the works of Ramanujan, said,

Ramanujan had produced groundbreaking new theorems, including some that defeated me completely.I had never seen anything in the least like them before.’

At the age of 32, he died of Tuberculosis. In his short life span, he independently found 3900 results. He worked on real analysis, number theory, infinite series, and continued fractions. Some of his other works such as Ramanujan number, Ramanujan prime, Ramanujan theta function, partition formulae, mock theta function, and many more opened new areas for research in the field of mathematics. He worked out the Riemann series, the elliptic integrals, hypergeometric series, the functional equations of the zeta function, and his theory of divergent series, in which he found a value for the sum of such series, using a technique he invented, that came to be called Ramanujan summation. In England, Ramanujan made further researches, especially in the partition of numbers, i.e, the number of ways in which a positive integer can be expressed as the sum of positive integers. Some of his results are still under research. His journal, Ramanujan Journal, was established to keep a record of all his notebooks and results, both published and unpublished, in the field of mathematics. As late as 2012, researchers studied even the small comments in his book, as they do not want to miss any results or identities given by him, that remained unsuspected until a century after his death. From his last letters in 1920 that he wrote to Hardy, it was evident that he was still working on new ideas and theorems of mathematics. In 1976, mathematicians found the ‘lost notebook’, that contained the works of Ramanujan from the last year of his life. Ramanujan devoted all his mathematical intelligence to his family goddess Namagir Thayar. He once said, “An equation for me has no meaning unless it expresses a thought of God.” Now, we will discuss in detail all his contributions to mathematics.

1. Infinite series of π

William Shanks, a 19th-century British mathematician tried calculating the value of infinite series of π. In 1873, he calculated the value of π to 707 decimal places. Ramanujan, in 1914, published ‘Modular equations and approximations to π’, which contained not only one, but 17 different series, that will converge very fastly to π, after calculating just fewer terms of the series.

2. Ramanujan number

The number 1729 is known as the Ramanujan number or Hardy-Ramanujan number. It is the smallest natural number that can be expressed as the sum of two cubes, in two different ways, i.e., 1729 = 1 3 + 12 3 = 9 3 + 10 3 . There is a small story behind the discovery of this number. When Ramanujan was under treatment, G.H. Hardy once visited him in the hospital and had a conversation in which he mentioned,

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

This is how the Ramanujan number came into existence. Later on, more properties of this number were discovered.

3. Ramanujan Prime

Ramanujan published a two-page paper on the proof of Bertrand’s postulate. At the end of the last page, he mentioned a result, π(x) – π(x/2) ≥ 1, 2, 3, 4, 5,….., for all x≥ 2, 11, 17, 29, 41,…. respectively, where π(x) is the prime counting function, equals to the number of primes equal or less than x. The nth Ramanujan prime number is the least integer {R}_{n} , for which there are at least n primes between x and x/2, for all x ≥ {R}_{n} . The first five Ramanujan primes are 2, 11, 17, 29, 41

4. Ramanujan Theta Function

Ramanujan theta function is the generalised form of the Jacob theta function. In particular, Jacobi triple product can be beautifully represented by the Ramanujan theta function. The Ramanujan theta function is given below.

for |ab|<1.With the help of Ramanujan theta function, Jacobi triple product can be represented as,

5. Mock Theta Function

Ramanujan in his last letter to G.H. Hardy and in his ‘lost notebook’, gave the first example of mock theta function. A mock theta function is a mock modular form( the holomorphic part of a harmonic weak Maass form), of weight 1/2. His last letter to Hardy contained 17 examples of mock theta functions, and some more examples were mentioned in his ‘lost notebook.’ Ramanujan gave an order to his mock theta function. Before the attempts of Zwegers, the order of mock theta function was 3, 5, 6, 7, 8, 10.

6. Partition

Partition or integer partition of an integer ‘n’ is a way of writing ‘n’ as a sum of positive integers. Partitions that differ only in the order of summands are considered as the same partitions. Each summand in the partition is called a part. The number of partitions of an integer ‘n’ is denoted by p(n). For example, integer 4 has 5 partitions as given below.

Here partition 1+3 is the same as 3+1 and 1+2+1 is the same as 1+1+2 and p(4)=5. Partitions can also be visualised with the help of the Young diagram and Ferrers diagram.

7. Ramanujan Magic Squares

In his school days, he used to enjoy solving magic squares. Magic squares are the cells in 3 rows and 3 columns, filled with numbers starting from 1 to 9. The numbers in the cells are arranged in such a way that the sum of numbers in each row is equal to the sum of numbers in each column is equal to the sum of numbers in each diagonal. Ramanujan gave a general formula for solving the magic square of dimension 3×3,

where A, B, C and P, Q, R are in arithmetic progression. The following formula was also given by him.

8. Ramanujan Congruences

Ramanujan obtained three congruences when m is a whole number, p (5 m + 4) ≡ 0 (mod 5), p (7 m + 5) ≡ 0 (mod 7), p (11 m + 6) ≡ 0 (mod 11). Hardy and E.M. Wright wrote,

he was ﬁrst to led the conjecture and then to prove, three striking arithmetic properties associated with the moduli 5, 7 and 11.”

9. Highly composite numbers

Composite numbers are the numbers that have factors other than 1 and the number itself. Ramanujan raised an interesting question that if ‘n’ is a composite number then what properties make a number highly composite. Ramanujan’s definition of Highly composite numbers,

A natural number is a highly composite number if d ( m ) < d ( n ) for all m < n.”

He also published a paper on highly composite numbers in 1915. According to him, there were infinitely many highly composite numbers.

10. Symmetric Equation by Ramanujan

Ramanujan noticed symmetry in Diophantine’s equation, {x}^{y} = {y}^{x} . He proved that there exists only one integer solution to this equation, i.e., x=4, y=2, and an infinite number of rational solutions, for example, {(27/8)}^{(9/4)} = {(9/4)}^{(27/8)} .

11. Ramanujan-Nagell Equation

Ramanujan-Nagell Equation is the equation of type {2}^{n} – 7 = {x}^{2} . It is an example of Diophantine equation. In 1913, Ramanujan claimed that this equation had only 1²+7 = 2³, 3²+7 = {2}^{4} , 5²+7 = {2}^{5} , 11²+7 = {2}^{7} , 181²+7 = {2}^{15} integral solutions. This conjecture was later on proved by Trygve Navell and is widely used in coding theory.

12. On Certain Arithmetical Functions

Ramanujan published a paper “On certain arithmetic functions” in 1916, in which he discussed the properties of Fourier coefficients of modular forms. Though the concept of modular forms was not even developed then, he gave three fundamental conjectures. In 1936, after 20 years of his published paper, a Greman mathematician Erich Hecke developed the Hecke theory with the help of his first two conjectures. His last conjecture played a vital role in the Langlands program (a program that relates representation theory and algebraic number theory). “On certain arithmetical functions” by Ramanujan was very effective in creating a sensation in 2oth century mathematics.

13. On Fermat’s Last Theorem

In 2013, mathematicians found some evidence that revealed Ramanujan was working on Fermat’s last theorem. Pierre de Fermat mentioned that,

if n is a whole number greater than 2, then there are no positive whole number triples x, y and z, such that x n + y n = z n .”

Ramanujan claimed that he had found an infinite family of whole numbers that will satisfy (approximately, not exactly) Fermat’s equation for n=3. He gave the example of the number 1729, which do not fits into the equation just by the mark of 1, for x=9, y=10, z=12. Ramanujan also worked on the equations of the form, y 2 = x 3 + ax + b. An elliptic curve is obtained, when the points (x,y) of this equation are plotted. These elliptic curves were of great significance and were used by Sir Andrew Wiles while he was proving Fermat’s last theorem in 1994.

14. Roger-Ramanujan Identities

In 1894, these identities were discovered and proved by Leonard James Rogers. Nearby 1913, Ramanujan rediscovered these identities. He had no proof but found Roger’s paper in 1917. Then they both united and gave a joint new proof.

15. Roger-Ramanujan Continued Fractions

Roger discovered continued fractions in 1894, which were later rediscovered by Ramanujan in 1912.

Ramanujan found various results concerning R(q), for example, R( {e}^{-2π} ) is given below in the picture and he also calculated R( {e}^{-2π√n} ) for n= 4, 9, 16, 64

16. Ramanujan’s Master Theorem

Ramanujan’s Master Theorem provides an analytic expression for the Mellin transform of an analytical function. This theorem is used by Ramanujan to calculate definite integrals and infinite series. The result of the theorem is given in the picture below.

17. Properties of Bernoulli Numbers

In 1904, Ramanujan independently studied and rediscovered Bernoulli numbers. In 1911, he wrote his first article on this topic. Bernoulli numbers {B}_{n} are the sequence of rational numbers, that appear in the Taylor series expansion of tangent and hyperbolic tangent functions. One of the properties that he discussed states that, the denominator of all Bernoulli numbers are divisible by six. Based on previous Bernoulli numbers, he also suggested a method to calculate Bernoulli numbers. According to the method proposed by him, if n is even but not equal to zero,

B n is a fraction and the numerator of B n / n in its lowest terms is a prime number.
The denominator of B n contains each of the factors 2 and 3 once and only once.
2 n (2 n − 1) B n / n is an integer and 2(2 n − 1)B n consequently is an odd integer.

18. Euler Mascheroni Constant

Ramanujan calculated the Euler Mascheroni constant also known as the Euler constant, up to 15 decimal places. It is the limiting difference between the harmonic series and the natural logarithm. Later on, a value up to 50 decimal places was calculated and is equal to, 0.57721566490153286060651209008240243104215933593992…..

γ denotes the Euler constant

19. Ramanujan Summation

Ramanujan, in one of his books, stated that, if we add up all natural numbers starting from 1 up to infinity, then the sum will be a finite number, i.e., 1+2+3+……….+∞= -1⁄12

20. Ramanujan Puzzles

The first puzzle was to prove the equation of infinite nested radical. In 1911, Ramanujan sent the RHS of this equation to a mathematical journal as a puzzle. The puzzle and its solution are elaborated in the video below.

The next puzzle is to find the value of the Golden ratio(Φ), which is equal to the infinite continued fraction given in the picture below.

The continued fraction in the black box is the same as that in the outer red box. Setting this equal to x, we get Φ = 1 + 1/x, which yields x 2 – x – 1 = 0. The solutions of this quadratic equation are ( 1 +√5 )⁄2 and ( 1 −√5 )⁄ 2. Neglecting the negative solution, the value of Φ is ( 1 +√5 )⁄2

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Srinivasa Ramanujan Biography

Ramanujan’s early years.

Srinivasa Ramanujan was one of the most famous mathematical wizards who made important contributions to the field of advanced mathematics. Srinivasa Ramanujan was born on 22 December, 1887, to a poor Brahmin family in Erode, a small village in Tamil Nadu, India.

He grew up in Kumbakonam town, near Chennai, where his father was employed as a clerk in a cloth merchant’s shop. He was an exceptionally good student and won a number of merit certificates and awards. He loved Mathematics more than any other subject.

Once, when he was just in his middle school classes, he mathematically calculated the approximate length of the equator. He also very clearly knew the values of the square root of two and value of pi!

Srinivasa Ramanujan – Education and work

At the age of 16, he got a scholarship for his first year at the Government College in his hometown. His deep interest in Mathematics led him to neglect other subjects because of which he was not able to clear his examinations and had to forgo his scholarship. After dropping out of college, he had to struggle a lot to earn his living.
However, it did not dampen Ramanujan’s spirits and he continued to work on problems and theorems. He bought a book authored by G. S. Carr which contained over 5000 problems. He worked and reworked all the problems and theorems and made new discoveries. He also found a job as an accounts clerk in the office of the Madras Port Trust.
Then, he got in touch with V. Ramaswamy Aiyer, the founder of the Indian Mathematical Society. With his help, Ramanujan got his paper on Bernoulli numbers published in the ‘Journal of the Indian Mathematical Society’ in 1911. Soon, he became a quite popular in Chennai for his prowess in Mathematics.
In 1913, he casually wrote to the well-known Cambridge mathematician, G. H. Hardy, and told him about his work. Hardy was mighty impressed with Ramanujan’s works and assisted him in getting a grant from Trinity College, Cambridge.
Ramanujan moved abroad and started to work in collaboration with Hardy, but his health started failing. Despite poor health, he remained engrossed in his research and study of newer vistas in mathematics. In 1916, he graduated from Cambridge with a Bachelor of Science by Research.
In 1920, he moved back to India and left for his heavenly abode.

What is Srinivasa Ramanujan famous for?

Despite having almost no formal training in Mathematics , Ramanujan’s knowledge of the subject-matter was astounding. Without the knowledge of the modern developments in the subject, he had made some important contributions to the field of mathematical analysis, number theory, game theory, infinite series and continued fractions.
He was a luminary who rose to great heights from a humble background and followed his heart against the odds in his way. His innovative ideas and vision still serve as a great resource for modern mathematicians.

The Man Who Knew Infinity

In the honour of Ramanujan, December 22 is now celebrated as the National Mathematics Day in India. His biography titled ‘The Man Who Knew Infinity‘ was published in 1991 and a movie based on him starring Dev Patel was also shown at the 2015 Toronto Film Festival.

Famous quotes by Srinivasa Ramanujan

An equation means nothing to me unless it expresses a thought of God.
I have not trodden through a conventional 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.”
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19 Facts About Tim Walz, Harris’s Pick for Vice President

Mr. Walz, the governor of Minnesota, worked as a high school social studies teacher and football coach, served in the Army National Guard and chooses Diet Mountain Dew over alcohol.

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Gov. Tim Walz of Minnesota, in a gray T-shirt and baseball cap, speaks at a Kamala Harris event in St. Paul, Minn., last month.

By Simon J. Levien and Maggie Astor

Published Aug. 6, 2024 Updated Aug. 9, 2024

Until recently, Gov. Tim Walz of Minnesota was a virtual unknown outside of the Midwest, even among Democrats. But his stock rose fast in the days after President Biden withdrew from the race, clearing a path for Ms. Harris to replace him and pick Mr. Walz as her No. 2.

Here’s a closer look at the Democrats’ new choice for vice president.

1. He is a (very recent) social media darling . Mr. Walz has enjoyed a groundswell of support online from users commenting on his Midwestern “dad vibes” and appealing ordinariness.

2. He started the whole “weird” thing. It was Mr. Walz who labeled former President Donald J. Trump and his running mate, Senator JD Vance of Ohio, “weird” on cable television just a couple of weeks ago. The description soon became a Democratic talking point.

3. He named a highway after Prince and signed the bill in purple ink. “I think we can lay to rest that this is the coolest bill signing we’ll ever do,” he said as he put his name on legislation declaring a stretch of Highway 5 the “Prince Rogers Nelson Memorial Highway” after the musician who had lived in Minnesota.

4. He reminds you of your high school history teacher for a reason. Mr. Walz taught high school social studies and geography — first in Alliance, Neb., and then in Mankato, Minn. — before entering politics.

5. He taught in China in 1989 and speaks some Mandarin. He went to China for a year after graduating from college and taught English there through a program affiliated with Harvard University.

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Srinivasa Ramanujan
Srinivasa Ramanujan (born December 22, 1887, Erode, India—died April 26, 1920, Kumbakonam) was an Indian mathematician whose contributions to the theory of numbers include pioneering discoveries of the properties of the partition function.. When he was 15 years old, he obtained a copy of George Shoobridge Carr's Synopsis of Elementary Results in Pure and Applied Mathematics, 2 vol. (1880 ...
Srinivasa Ramanujan
Srinivasa Ramanujan [a] (22 December 1887 - 26 April 1920) was an Indian mathematician.Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable.. Ramanujan initially developed his own mathematical ...
Srinivasa Ramanujan (1887-1920)
Birth -. Srinivasa Ramanujan was born on 22nd December 1887 in the south Indian town of Tamil Nad, named Erode. His father, Kuppuswamy Srinivasa Iyengar worked as a clerk in a saree shop and his mother, Komalatamma was a housewife. Since a very early age, he had a keen interest in mathematics and had already become a child prodigy.
Srinivasa Ramanujan
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 ...
Srinivasa Ramanujan (1887
Biography. Srinivasa Ramanujan was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series. Ramanujan was born in his grandmother's house in Erode, a small village about 400 km southwest of Madras (now Chennai).
Biography of Srinivasa Ramanujan, Mathematical Genius
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 ...
Srinivasa Ramanujan
Lived 1887 - 1920. Srinivasa Ramanujan was a largely self-taught pure mathematician. Hindered by poverty and ill-health, his highly original work has considerably enriched number theory. More recently his discoveries have been applied to physics, where his theta function lies at the heart of string theory. Advertisements Beginnings Srinivasa Ramanujan was born on December 22,
Srinivasa Ramanujan Facts & Biography
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.
Srinivasa Ramanujan
Srinivasa Ramanujan (1887-1920) was an Indian mathematician who made great and original contributions to many mathematical fields, including complex analysis, number theory, infinite series, and continued fractions. He was "discovered" by G. H. Hardy and J. E. Littlewood, two world-class mathematicians at Cambridge, and enjoyed an extremely fruitful period of collaboration with them ...
Srinivasa Ramanujan
Srinivasa Ramanujan was born on December 22, 1887 in Erode, a city in the Tamil Nadu state of India. His father, K. Srinivasa Iyengar was a clerk while his mother, Komalatammal performed as a singer, in a temple. Even though they belonged to the Brahmins who are known to be the highest caste of Hinduism, Ramanujan's family was very poor. At ...
Srinivasa Ramanujan Biography: Education, Contribution, Interesting Facts
The Hardy-Ramanujan Number (1729): 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.
Srinivasa Ramanujan
Srinivasa Ramanujan was born on December 22, 1887 in his grandmother's house in a small village called Erode. Erode is around 400 km from Chennai, the capital of the Indian state of Tamil Nadu. Ramanujan's father, Kuppuswamy Srinivasa Iyengar, was a clerk in a cloth merchant's shop; while his mother, Komalatammal, was a housewife and sang ...
Srinivasa Ramanujan Biography
Childhood & Early Life. Srinivasa Ramanujan was born on 22 December 1887, in Erode, Madras Presidency, British India, to K. Srinivasa Iyengar and his wife Komalatammal. His family was a humble one and his father worked as a clerk in a sari shop. His mother gave birth to several children after Ramanujan, but none survived infancy.
Ramanujan, Srinivasa (1887-1920) -- from Eric Weisstein's World of
Ramanujan, Srinivasa (1887-1920) Indian mathematician who was self-taught and had an uncanny mathematical manipulative ability. Ramanujan was unable to pass his school examinations in India, and could only obtain a clerk's position in the city of Madras. However, he continued to pursue his own mathematics, and sent letters to three ...
Srinivasa Ramanujan
The Man Who Knew Infinity is a biography of Srinivasa Ramanujan by Robert Kanigel. It was published in 1991 and later was turned into a movie starring Dev Patel as Srinivasa Ramanujan and Jeremy ...
Ramanujan: The Man Who Knew Infinity
Srinivasa Ramanujan (1887-1920), the man who reshaped twentieth-century mathematics with his various contributions in several mathematical domains, including mathematical analysis, infinite series, continued fractions, number theory, and game theory is recognized as one of history's greatest mathematicians. Leaving this world at the youthful age of 32, Ramanujan made significant contributions ...
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 ...
Who Was Ramanujan?—Stephen Wolfram Writings
The Beginning of the Story. 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.
Essay On Srinivasa Ramanujan in English for Students
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.
Ramanujan
Srinivasa Ramanujan FRS (1887 - 1920) was a self-taught Indian mathematical genius who made numerous contributions in several mathematical fields including mathematical analysis, infinite series, continued fractions, number theory and game theory.Ramanujan provided solutions to mathematical problems that were then considered unsolvable. Moreover, some of his work was so ahead of his time ...
Srinivas Ramanujan: Biography, Age, Wife, contribution to mathematics
Ramanujan number is a natural number that can be represented in two different ways by the sum of the cubes of two numbers. Example, {9 3 +10 3 =1 3 +12 3 =1729} Ramanujan numbers are 1729, 4104, 20683, 39312, 40033 etc. Death of Srinivasa Ramanujan. At 32, Ramanujan died at Kumbakonam, India, on April 26, 1920.
Srinivasa Ramanujan's Contributions in Mathematics
Ramanujan Congruences. Ramanujan obtained three congruences when m is a whole number, p (5m+ 4) ≡ 0 (mod 5), p (7m+ 5) ≡ 0 (mod 7), p (11m+ 6) ≡ 0 (mod 11). Hardy and E.M. Wright wrote, he was ﬁrst to led the conjecture and then to prove, three striking arithmetic properties associated with the moduli 5, 7 and 11.".
Srinivasa Ramanujan
Srinivasa Ramanujan was born on 22 December, 1887, to a poor Brahmin family in Erode, a small village in Tamil Nadu, India. He grew up in Kumbakonam town, near Chennai, where his father was employed as a clerk in a cloth merchant's shop. He was an exceptionally good student and won a number of merit certificates and awards.
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