Srinivasa Ramanujan: A Short, Extraordinary Life

Srinivasa Ramanujan was an Indian mathematician who taught himself advanced mathematics. He was born on 22 December 1887 in Erode, in present-day Tamil Nadu. His father worked in Kumbakonam as a clerk in a cloth merchant’s shop.

1.1 How did a clerk's son learn advanced mathematics?

At the Town High School, he came across a book by G. S. Carr. Its title was Synopsis of elementary results in pure mathematics. The book was published in 1886. It contained theorems, formulae and short proofs.

Its very concise style let him teach himself mathematics. The Dictionary of Scientific Biography says he borrowed Carr’s book in 1903. From then on, mathematics was nearly his only interest. MacTutor adds that the book was his only model of written mathematical arguments.

A paper in the Royal Society’s journal says he was, as regards mathematics, entirely self-educated. Self-taught does not mean he skipped school. In his first letter to Hardy, he wrote: “I have had no university education but I have undergone the ordinary school course.”

1.2 Was he good at every school subject?

No. A good school record won him a scholarship to the Government College in Kumbakonam. He entered it in 1904. The next year, the scholarship was not renewed. MacTutor says he gave more and more time to mathematics. He neglected his other subjects.

Later he sat the First Arts examination. Passing it would have let him enter the University of Madras.

MacTutor says he passed in mathematics but failed all his other subjects. So he failed the examination. The Dictionary of Scientific Biography links his failure to his neglect of English.

A paper on Bernoulli numbers appeared in 1911 in the Journal of the Indian Mathematical Society. After it, Ramanujan became known in the Madras area. MacTutor says this paper brought him recognition.

He began work as a clerk on 1 March 1912. The job was at the Madras Port Trust.

Professor E. W. Middlemast wrote a reference for that job. He called Ramanujan a young man of “quite exceptional capacity in mathematics and especially in work relating to numbers”.

2.1 Whom did he write to?

Ramanujan wrote to E. W. Hobson and H. F. Baker, but neither replied. In January 1913, he wrote to G. H. Hardy at Cambridge. He had seen a copy of Hardy’s 1910 book Orders of infinity.

The Dictionary of Scientific Biography says the first letter held more than 100 theorems. It says Hardy was duly impressed.

2.2 Did Hardy believe him at once?

Hardy wanted proof first. He replied on 8 February 1913. He wrote that he was exceedingly interested in the theorems. He added that he should see proofs of some of the assertions. Only then could he judge their value.

Young’s abstract describes Hardy and Littlewood studying the manuscript in Littlewood’s rooms at Trinity College. It quotes: “A single look at them is enough to show that they could only be written down by a mathematician of the highest class”.

Our own reading: Hardy’s request for proofs is how a mathematician checks a claim, not a sign that he doubted the writer.

The University of Madras gave Ramanujan a scholarship in May 1913 for two years. MacTutor says that in 1914 Hardy brought him to Trinity College, Cambridge.

The Royal Society journal abstract adds that Hardy joined others in Madras to win him a research studentship. It says Hardy also arranged his coming to Trinity in April 1914.

The Dictionary of Scientific Biography says Hardy first invited him to England. Ramanujan was a Brahman and had scruples about leaving India. A two-year fellowship at the University of Madras was arranged instead. MacTutor says the difficulty was overcome, partly by the work of E. H. Neville.

3.1 What did the journey look like?

He sailed from India on 17 March 1914. It was a calm voyage, except for three days of seasickness. He arrived in London on 14 April 1914 and was met by Neville. On 30 April he moved into rooms in Trinity College.

Food was a problem from the start. MacTutor says World War I made special foods harder to get. It says he soon had health problems.

Hardy later asked what was to be done to teach Ramanujan modern mathematics. His own summary was this: “The limitations of his knowledge were as startling as its profundity”. He also wrote that Ramanujan’s ideas of what makes a proof “were of the most shadowy description”.

MacTutor says Ramanujan’s response to Littlewood was “an avalanche of original ideas”. It adds that this made it almost impossible for Littlewood to persist in his original intention.

The Dictionary of Scientific Biography says he developed rapidly under Hardy and Littlewood. It says they also helped him publish his papers in English periodicals.

4.1 What degree did he earn in Cambridge?

On 16 March 1916, he graduated from Cambridge. His degree was a Bachelor of Arts by Research. MacTutor says the degree was called a Ph.D. from 1920.

His dissertation was on highly composite numbers. It consisted of seven of his papers published in England.

A partition of a number is a way of writing it as a sum of whole numbers. These numbers are called summands. The number itself counts as one such sum.

The Dictionary of Scientific Biography says Ramanujan studied the partition of numbers into summands. The count of partitions of n is written p(n).

Here is our own worked example. The number 4 has these partitions:

• 4
• 3 + 1
• 2 + 2
• 2 + 1 + 1
• 1 + 1 + 1 + 1

That is 5 ways, so p(4) = 5.

5.1 How did he work on it?

MacTutor calls this perhaps his most famous work. It says MacMahon had produced tables of p(n) for small numbers. Ramanujan used this numerical data to conjecture remarkable properties. He proved some using elliptic functions. Others were only proved after his death.

5.2 What did Hardy and Ramanujan prove together?

Their joint paper is Asymptotic Formulaae in Combinatory Analysis. It appeared in 1918 in the Proceedings of the London Mathematical Society.

The Dictionary of Scientific Biography calls their result remarkable. Their formula gave good approximations to the values already calculated. It also seemed to give an exact expression for p(n). Hans Rademacher later established that this was correct.

Crossref lists Rademacher’s paper On the Partition Function p(n) in the same journal, in 1938. Our own reading: a formula that fits many checked values is not yet a proof. Here, the proof of exactness came later.

The number 1729 comes from a story that Hardy told. In his words: “I remember once going to see him when he was ill at Putney.” He says he had ridden in taxi-cab number 1729. He told Ramanujan the number seemed to him “rather a dull one”.

Ramanujan replied: “No, it is a very interesting number”. He said it is “the smallest number expressible as the sum of two cubes in two different ways”. MathWorld gives the same property.

6.1 What are the two ways?

Ono and Trebat-Leder write that 1729 is the smallest natural number that is the sum of two positive cubes in two ways. The two sums are 1³ + 12³ and 9³ + 10³.

Check them: 12³ is 1,728, so 1 + 1,728 gives 1729.

Also, 9³ is 729 and 10³ is 1,000, so 729 + 1,000 gives 1729.

6.2 Did Ramanujan think of this on the spot?

Ono and Trebat-Leder say he had written it down even before coming to England. In his second notebook, he wrote it down explicitly as part of a list of examples. They quote Littlewood’s remark that “every number is a personal friend of Ramanujan”.

MathWorld adds that the property was also known as early as 1657 to F. de Bessy. Our own reading: Ramanujan’s reply drew on a number he already knew well.

6.3 What are taxicab numbers?

MathWorld says taxicab numbers get their name from the Hardy-Ramanujan number. It lists the first few as 2, 1729, 87539319 and 6963472309248. So 1729 is the second taxicab number.

Ramanujan fell seriously ill in 1917. The Dictionary of Scientific Biography says it was possibly tuberculosis. It says the rest of his stay in England was spent in several sanatoriums. In February 1918, Hardy wrote that tuberculosis had been “the provisionally accepted theory”.

7.1 What honours came in 1918?

• On 18 February 1918, he was elected a fellow of the Cambridge Philosophical Society.
• Three days later, his name appeared on the list for election as a fellow of the Royal Society of London. MacTutor calls this the greatest honour he would receive.
• On 2 May 1918, the Royal Society election was confirmed.
• On 10 October 1918, he was elected a Fellow of Trinity College, the fellowship to run for six years.

In late 1918, Hardy wrote of him: “He will return to India with a scientific standing and reputation such as no Indian has enjoyed before”.

7.2 When and where did he die?

MacTutor says he sailed to India on 27 February 1919. He arrived on 13 March. The Dictionary of Scientific Biography says he returned to Madras in April 1919. MacTutor says his health was very poor and he died the following year.

He died on 26 April 1920, aged 32 by our own count from the two dates. MacTutor places his death in Kumbakonam. The Dictionary places it at Chetput, near Madras. This lesson does not choose between them.

MacTutor says Ramanujan left unpublished notebooks filled with theorems. Mathematicians have continued to study them. Hardy passed his manuscripts on to G. N. Watson. Some had been written before 1914 and some in Ramanujan’s last year in India.

Watson published 14 papers called Theorems stated by Ramanujan. MacTutor says that, in all, Watson published nearly 30 papers inspired by Ramanujan’s work.

The Dictionary of Scientific Biography says most theorems in the notebooks were not proved in the standard sense. It says they were only made plausible.

Manmohan Singh described the notebooks in 2011. He called them photographic reproductions of unpublished work, written in Ramanujan’s own hand. He said the originals are in the University of Madras.

Good biographies can differ on small points. The table below shows the points where the sources we read give different details. It does not list every difference. It leaves out points where they only use different words.

The table states the sources side by side. This lesson does not decide between them.

Our lesson on D.R. Kaprekar covers another Indian mathematician. Read it to compare his path with Ramanujan’s.

On 26 December 2011, Prime Minister Manmohan Singh spoke in Chennai. The event marked 125 years since Ramanujan was born.

The Prime Minister said the government had decided to declare Ramanujan’s birthday, 22 December, as National Mathematics Day. He also said the government had decided to declare 2012 as National Mathematical Year.

The Hindu reported on 27 December 2011 that the Centre would mark the birthday as National Mathematics Day every year.

In the same speech, the Prime Minister honoured Robert Kanigel. Kanigel wrote a biography of Ramanujan. The Prime Minister said Hardy was responsible for Ramanujan getting the recognition that was his legitimate due.

11.1 What is the film connection?

The Man Who Knew Infinity is a biopic of Ramanujan, with Dev Patel as Ramanujan and Jeremy Irons as Hardy. IndieWire’s report is dated 14 October 2015. It says IFC Films had acquired North American rights after the film’s recent TIFF world premiere.

The UPSC Civil Services (Preliminary) Examination 2016, General Studies Paper-I, Series A, asked about this film.

The answer follows from the IndieWire report: the film is a biopic of Ramanujan. The other three names are not its subject. The stem calls the film recent. IndieWire described its TIFF world premiere as recent in October 2015.

View this question on the full 2016 paper →

12.1 Which facts are likely exam traps?

• Hardy’s first reply, dated 8 February 1913, asked for proofs.
• The first letter went out in January 1913, but he reached London on 14 April 1914.
• His Royal Society election was confirmed on 2 May 1918.
• National Mathematics Day is 22 December, Ramanujan’s birthday.
• 1729 is the smallest number that is the sum of two cubes in two ways.