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Iyengar, Srinivasa Ramanujan

Category: Genius

Srinivisa Ramanujan Iyengar (22 December 1887 – 26 April 1920) was a legendary Indian mathematician – a genius -  who, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions.

He was born and raised in Erode, Tamil Nadu, India.  His upbringing was very simple and unprepossessing and his schooling very basic.  But despite this, even at an early age he showed a natural ability in mathematics and was, as a consequence,  given books on advanced trigonometry by S L Loney.  By age 13, had had  discovered theorems of his own.   By age 17, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal places. So in effect, without the help of his school he was schooling himself.  His peers commented that they "rarely understood him" and "stood in respectful awe" of him.

It was not just his peers who stood in respectful awe.  He graduated from Town High in 1904, and was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, who clearly believed he was dealing with a prodigy.  He received a scholarship to study at Government College in Kumbakonam,  known as the "Cambridge of South India." 

What emerges from this time however, is that Ramanujan was only gifted in mathematics, he failed in all the other subjects and tragically lost his scholarship as a consequence.  This finding is reinforced by subsequent events.  He enrolled at Pachaiyappa's College in Madras and again excelled in mathematics, but performed poorly in other subjects such as physiology. Ramanujan failed his Fine Arts  degree exam in December 1906 and again a year later.

All of the subsequent events show that his genius was pure ‘inspiration’ – spiritually inspired, he seemed unable to explain where his ideas came from and at times equally unable to explain them to others, for example

Journal of the Indian Mathematical Society -  editor M. T. Narayana Iyengar
Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.

These two factors even led some to doubt his abilities, some assumed he had simply taken others work and that his research was not original.

For a time it looked as though he was destined to be a clerk, and that his interest in  mathematics would have to be a hobby.   But in 1912-1913, he sent samples of his theorems to three academics at the University of Cambridge. G. H. Hardy, seeing the brilliance of his work,  invited him to study at Cambridge. 

Hardy originally viewed Ramanujan's manuscripts as a possible "fraud." Hardy knew some of Ramanujan's formulas, but others "seemed scarcely possible to believe."  After Hardy saw Ramanujan's theorems on continued fractions on the last page of the manuscripts, he commented that the "[theorems] defeated me completely; I had never seen anything in the least like them before." He also came to the conclusion that Ramanujan's theorems "must be true, because, if they were not true, no one would have the imagination to invent them.”

Although invited to the UK, Ramanujan initially refused to leave his country because of his parents’ opposition.   But he was clearly torn between the demands of his parents and his heart. He sent a letter packed with theorems to Hardy, writing, "I have found a friend in you who views my labour sympathetically."

Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to try to convince Ramanujan to come to England, but by this time his mother had had a vivid dream in which the family goddess Namagiri commanded her "to stand no longer between her son and the fulfilment of his life's purpose."  Ramanujan needed no convincing and his parents' opposition was withdrawn.

Ramanujan arrived in London on 14 April 1914.  Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems to be found in his notebooks. Hardy saw that some were wrong, some were already discovered and the rest were new breakthroughs.

What is perhaps intriguing is that Ramanujan appears to have ‘discovered’ theorems that had already been discovered independently but were unknown to him.  One theorem, for example,  had already been determined by a mathematician named Bauer. But Ramanujan didn’t know this.  Whilst Ramanujan was pursuing a research scholarship at the University of Madras, he also anticipated the work of a Polish mathematician who published his work shortly after.  This speaks of a spiritual ability beyond just inspiration, reinforcing the feeling that Ramanujan’s gift of spiritual experience was both that of inspiration and an ability to tap into the common consciousness.

Ramanujan spent nearly five years in Cambridge collaborating with Hardy and published a part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas, Ramanujan was described as one who “relied very strongly on his intuition”.   But the combination was ideal.  Ramanujan left a deep impression on Hardy. He said he "can compare him only with [Leonhard] Euler or Jacobi."

A view of his house in Kumbakonam Thanjavur

Ramanujan was described by his colleagues as a shy, quiet, polite, courteous and dignified man who lived a spartan life. He was vegetarian.  He credited all his inspiration to his family goddess, Namagiri, the goddess who appeared to his mother in a dream.  He said "An equation for me has no meaning, unless it represents a thought of God."  Described as a ‘devout Hindu’, this is not strictly speaking correct.  Hardy said that Ramanujan believed all religions equally true – the sign of a mystic.

In mathematics, there is a distinction between having an insight and having a proof. Ramanujan had repeated insights, but struggled with the proofs, which is where his collaboration with Hardy must have helped.  Ramanujan suggested a plethora of formulae that could then be investigated in depth later. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself created papers exploring material from Ramanujan's work as did G. N. Watson, B. M. Wilson, and Bruce Berndt.

G. H. Hardy
The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly-periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was...".

An example of his inspirational ability can be given by a story of when he was sharing a room with P. C. Mahalanobis.  Mahalanobis posed what is called a  ‘bivariate problem with multiple solutions’. Ramanujan thought about it and gave the solution as a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded.  He asked him how he came to this answer and Ramanujan said.

"It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction? I asked myself. Then the answer came to my mind".

Ramanujan's wife Janaki Ammal .

Ramanujan was awarded a B.A. degree by research (this degree was later renamed PhD) in March 1916.  On 13 October 1918, he became the first Indian to be elected a Fellow of Trinity College, Cambridge.

Was Ramanujan naturally spiritually inclined, or was there an additional cause for his bursts of inspiration?

Probably both.

He was undoubtedly spiritually gifted, but he also had a precarious life from a health point of view.  He was plagued by health problems almost throughout his entire short life. 

In December 1889, whilst still a child, he contracted smallpox.  After college, he lived in extreme poverty and was often near the point of starvation.  So he was malnourished.  He also suffered from any number of other health episodes. 

When he got to England, he was diagnosed with tuberculosis and a severe vitamin deficiency and was later confined to a sanatorium. Ramanujan returned to India in 1919, and died soon thereafter at the age of 32.

Scene from a film Ramanujan made about his life

But a 1994 analysis of Ramanujan's medical records and symptoms by Dr. D.A.B. Young concluded that it was much more likely he had amoebiasis. This is supported by the fact that Ramanujan had spent time in Madras, where the disease was widespread. He had two episodes of dysentery before he left India. It was never properly treated.

So it is a real possibility that illness contributed to his genius.

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