Thread by @SadaaShree

The Man Who Knew Infinity, Jayanti of #SrinivasaRamanujan today, one of the greatest mathematicians of modern era, a genius like none other, a true maverick.

The world of mathematics owes a great debt to India , because this is the land from which the basic theories of mathematics have originated. Aryabhatta who contributed the number-place value system and the concept of zero, as well as calculating the area of the triangle.

Bhaskara who came up the concept of the decimal system. Halayudha who provided a clear description of the Pascal’s triangle. And between 1300-1600 AD, we had the Kerala school of Mathematics and Astronomy founded by Madhava of Sangamagrama.

He was born into an Iyengar family, in the town of Erode, Tamil Nadu on Dec 22, 1887. His father was a clerk in a saree shop, while his mother was a devout Brahmin housewife. Theirs was a typical Tamil Brahmin household, pretty religious and orthodox. #SrinivasaRamanujan

Growing up under his mother’s care and guidance, he learnt about Indian tradition and puranas. His first brush with mathematics was in 1898, in Higher Secondary School. And that was when his prodigal talent came to being. #SrinivasaRamanujan

At 11 years, he exhausted the mathematical knowledge of two college students. Again in school, Ramanujan was absolutely brilliant at maths, but quite often flunked in the other subjects. #SrinivasaRamanujan

The book that influenced Ramanujan the most though was Synopsis of Pure Mathematics by G.K.Carr that had around 5000 theorems, when he was 16. .The book a collection of around 5000 theorems, stoked his curiosity, made him explore mathematics much more deeper.

The following year, Ramanujan, developed and investigated the Bernoulli numbers, calculated the Euler-Mascheroni constant up to 15 decimal places, winning him a new found respect among his peers. #SrinivasaRamanujan

His genius though would be a sort of curse for him, fact is Ramanujan was interested only in maths, he hardly cared about other subjects at school and college. So while he passed Maths with distinction, he was below par in the other subjects.

And this meant that Ramanujan failed his Fellow of Arts exam in 1906, in spite of his brilliance at Maths, and with no formal degree, he had to pursue maths research on his own, at times living in dire poverty. He made ends meet, by taking up tuition for school students.

It was a chance meeting with V Ramaswamy Iyer in 1910, would be the turning point in Ramanujan’s life. Iyer was the founder of the Indian Mathematical Society, and thanks to him, Ramanujan began to be recognized in the math circles of Madras.

He however wanted a regular job to make ends meet, Iyer felt that a genius like him should not be wasted in clerical jobs. He sent letters to his friends in Madras, one of em was R.Ramachandra Rao, the district collector of Nellore.

I was struck by the extraordinary mathematical results contained in the notebooks. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department. -R.Ramachandra Rao on #SrinivasaRamanujan

In Madras he got to meet the then Nellore district collector, R.Ramachandra Rao, who initially doubted that such a genius was original. However he had a long discussion with Ramanujan over elliptic integrals and divergent series.

Rao, who was also the Secretary for the Indian Mathematical Society, was now convinced about Ramanujan, and supported him financially. With the help from Ramaswamy Aiyer, Ramanujan managed to get his work published in Journal of the Indian Mathematical Society.

One of the first problems Ramanujan posted in the journel was to find the value of what is called the infinite nested radicals( pic1). After 6 months when none failed to answer it. he himself posted the solution, this equation was to solve infinitely nested radicals.

Ramanujan’s first formal paper for the Maths Journal was on Bernoulli Numbers and their properties. He came up with his own method of calculating Bn, based on previous Bernoulli numbers.

Basically Ramanujan’s method to calculate Bn was based on this
1) Bn is a fraction, and the numerator in it’s lowest wud be a prime number
2) Denominator of Bn contains each of factors 2, 3 only once
3) 2^n(2^n-1) Bn/ is an integer.

He however had a rather erratic style of writing, often not too clear enough, and this made his work quite hard to understand for the average reader. He had the most innovative methods to solve problems, yet his not too clear writing style made it hard to understand.

In 1912, he applied for the post of a clerk in Accounts Dept at Madras Port Trust, with a letter of recommendation from his Maths Professor, E.W.Middlemast at Presidency which ensured he got the job.

During his spare time, he would still continue his research in mathematics. He was encouraged by his boss, Sir Francis Spring and his colleague, S.Narayana Iyer. #SrinivasaRamanujan

Some of his friends like Spring, Narayana Iyer, Ramachandra Rao and Middlemast, helped him to send his work for Cambridge University, but he was rejected due to lack of formal educational qualifications. #SrinivasaRamanujan

In 1913 Middlemast, Narayana Iyer and Ramachandra Rao, presented Ramanujan’s work to British mathematicians. MJM Hill rejected him saying that though he was talented, he did not have a formal educational background, but he gave him valuable suggestions. #SrinivasaRamanujan

Encouraged by Hill’s suggestions, Ramanujan sent his works to Cambridge, two of them HF Baker, and EW Hobson did not show interest. On Jan 16, 1913, he wrote to G.H.Hardy and that would prove to be a turning point in his life.

Hardy was amazed by Ramanujan’s take on the Bauer Theorem, what really impressed him was his next work on infinite series derived from a class called hypergeometric series. This class had earlier been researched by Euler and Gauss.

By the time he finished reading Ramanujan’s work, Hardy was completely impressed, mind you he was one of the leading mathematicians of that time. He figured that his theorems must be true, because none wud really have the imagination to invent them.

Hardy requested Ramanujan to come to Cambridge, but he refused, due to his own apprehensionns. For the time being, he was given a research grant at the University of Madras, where he did pioneering work on Frullani’s 1821 integral theorem.

Hardy’s colleague Neville, again asked Ramanujan to come to Cambridge, and this time he agreed. According to a popular anecdote, his mother had a dream, in which their family goddess Namagiri, commanded her not to stand in the way of her son’s progress.

From 1913-18, Ramanujan spent 5 fruitful years in Cambridge, collaborating with Hardy and Littlewood, on many research projects. While Littlewood said that “this man was at least a Carl Gustav Jacob”, Hardy remarked that he can be compared only with Euler or Jacobi.

In March 1916, Ramanujan was awarded the Doctorate for his research on highly composite numbers. Hardy later remarked that this was one of the most unusual papers he had ever seen.

Both of them had contrasting ways of working, Hardy followed the Western model of proof and rigor, and was an atheist. Ramanujan was a devout believer and often relied on intuition and gut feeling. It was the perfect mix of East and West. #SrinivasaRamanujan

On Oct 1918, he became the first Indian to be elected as Fellow of Trinity College, Cambridge. In 1917, he became a member of the London Mathematical Society. In 1918 he became the second Indian to gain fellowship of the Royal Society, and one of it’s youngest members.

Ramanujan came up with his own variant of the Brocard’s problem, which asks to find for integer values of n and m, where n!+1=m2. Still remains an unsolved problem, as only 3 integers, 4,5 and 7 have been found as solution for this. #SrinivasaRamanujan

Sadly his poor health meant Ramanujan had to return back to India, and he passed away in Kumbakonam on April 26, 1920, he was just 32, a beautiful mind, that left the world at a very young age.

However his wife Janaki Ammal, took up the cause of her husband’s work. Though having no children, she adopted Narayanan in 1950 who later became an officer with SBI. And ensured his work reached public. #SrinivasaRamanujan

Ramanujan came up with his own variant of the Brocard’s problem, which asks to find for integer values of n and m, where n!+1=m2. Still remains an unsolved problem, as only 3 integers, 4,5 and 7 have been found as solution for this.

Another problem solved by him, the Landau- Ramanujan constant, Landau here stated it first in the Limit form, later Ramanujan approximated N(x) as an integral. Basically it states that for large X, number of positive integers below are sum of 2 squares. #SrinivasaRamanujan

Ramanujan Theta Function generalizes the Jacobi theta functions, in a much more elegant manner. And this is used a lot in the Bosonic String Theory, M-Theory and Superstring theory. #SrinivasaRamanujan

His home state of Tamil Nadu celebrates his birthday as State IT day, while Government Arts College, Kumbakonam remembers the day as Ramanujan Day. It is fitting that his birthday should be declared as National Mathematics Day by Government, a truly great Indian.

The legacy of #SrinivasaRamanujan is a separate thread by itself, but some notable ones.
The Man Who Knew Infinity a book by Robert Kanigel on his life later made into a movie.

The Steradian Trail a thriller novel by M. N. Krish uses Ramanujan's discoveries in a plot that connects mathematics, finance, economics, religion #SrinivasaRamanujan

On the Jayanti of #SrinivasaRamanujan take time to pay tribute to one of the great genius of modern India, and one of the greatest mathematicians ever.

My article on #SrinivasaRamanujan on his Jayanti, do check out and share.
historyunderyourfeet.wordpress.com/2013/12/22/the…

My podcast on #SrinivasaRamanujan on his Jayanti, do check out and share.
spotifyanchor-web.app.link/e/WDQppMmlXvb
gaana.com/song/the-man-w…

Two good movies on Ramanujan, the 2014 Tamil movie, an Indo British collaboration with Abhinay Vaddi in the lead role, Kevin McGowan as Prof Hardy, Suhasini as his mother. Was quite a good watch. #SrinivasaRamanujan

Also The Man Who Knew Infinity 2016 movie version with Dev Patel as Ramanujan and Jeremy Irons as Prof Hardy. #SrinivasaRamanujan