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Srinivasa Ramanujan,

a Mathematician Brilliant

Beyond Comparison

Had he emerged in a city of advanced learning from a family of noted mathematicians his accomplishments would still have been stunning, but he was born into a poor family of no notable professional attainments in a part of the world where hardly anyone could understand even the nature of his talents. The obstacles to his achieving his goal of becoming a professional mathematician were by reasonable assessment insurmountable. Yet he did, and people around the world still marvel at attainments of Srinavasa Ramanujan.

Ramanujan was born on December 22, 1887 in Erode, a minor city in Madras State (now Tamil Nadu) in South India. His father was a clerk in a fabric store. The family was Brahmin and Ramanujan's maternal grandfather was a minor official in a local court. Thus the family was poor but socially respectable. Ramanujan was born in that grandfather's house in Erode. Soon after his birth his mother and father moved with him to a house in the town of Kumbakonam and Ramanujan grew up there.

The family name Ramanujan means person who contains a particle of the god Rama. The personal name Srinivasa roughly means Prosperous, more literally one who abides in wealth.

Ramanujan's family sometimes took in student boarders and it was through these that Ramanujan was first introduced to formal mathematics. One of the boarders lent him a trigonometry text when he was twelve and Ramanujan by himself mastered it within a year.

When Ramanujan was sixteen and still in high school an elderly friend who knew of his precocious mathematical talent gave him George Carr's Synopsis of elementary results in pure and applied mathematics. This two-volume encyclopedic tome contained six thousand theorems on all fields of mathematics. As Ramanujan read and worked his way through these theorems he discovered that he could derive results that were not in Carr. This was the beginnings of Ramanujan's mathematical productions and set the tone for his mathematical career.



Srinivasa Ramanujan

The City of Madras,

Now Called Chennai

The British East India Company began establishing a presence in the Coromandel Coast of the Bay of Bengal in South India in the 17th century. The Company obtained permission from local rulers to build a fort there. The Company chose a site north of the mouth of the Cooum River where a fishing village called Madraspatnam was located. The fort was called Fort St. George.

There were other villages such as Mylapore in the area which were temple towns. The area was a low lying plain no more than 23 feet above sea level. Rice was the major product but there was cotton growing and weavers to convert the cotton into cloth. The East India Company encouraged the weavers and merchants to settle north of the fort. Fort St. George was occupied by British merchants and became known as White Town. The area where the Tamil merchants and weavers settled became known as Black Town. Later the area was named George Town.

The city that grew up around Fort St. George became known as Madras after the fishing village of Madraspatnam that was originally on the site. The Tamil speakers however called the city Chennaipatnam and later Chennai.

The city grew and the British took control of the region and all of India. The region was for a long period of time known as Madras State. After independence it became known as Tamil Nadu.

After the East India Company establish complete control of the region some British merchants built garden houses in districts southwest and west of the fort (Nungambakkam, Adyar and Kilpauk).

Some of the districts of early Madras

Over time there developed ethno-religious-linguistic enclaves in the city. Mylapore and Triplicane became Brahmin. Chepauk to the south of the island formed by the branching of the Cooum River became Muslim. Royapuram to the north of the fort beyond Black Town (George Town) became an area of Christian settlement. West of Royapuram in an area known as Washermanpet became an area where weavers lived. Weavers also lived in the Chintadripet area to the southwest of the fort and beyond the island in the Cooum River.

In modern times an industrial park was created at Guindy, about four miles southwest of the fort beyond the Adyar River.

(To be continued.)





Demographics of Madras (Chennai)

The Linguistic Composition

of the Population of Madras,

1961 Language Proportion Tamil 72% Telugu 14% Malayam 3% Urdu 5% Other 6%

Tamil, Telugu and Malayam are languages in the Dravidian family. Urdu is a language closely related to Hindi but with elements of Persian and Arabic incorporated. For more on the nature of Urdu see Urdu. About 17% of the population speak English as a second language. The population is predominantly Hindu in religion, but there are small Muslim and Christian minorities.

Poverty and the Mathematical

Education of Ramanujan

As mentioned previously Ramanujan's family was poor but respectable. He was able to go to the local high school where he received training in elementary mathematics. Through college students boarding with his family he was introduced to more advanced mathematics and began learning on his own. When he completed high school he took a competitive examination in which he earned such high marks that he was given a scholarship to a local college, the Government College at Kumbakonan. There his mathematical development proceeded well but he was not much interested in the other subjects. In part this was a matter of him being so fascinated with mathematics that he did not want to spend his time thinking about other academic subject. Also some of the subjects were positively distasteful to Ramanujan, in particular psychology, which was probably really physiology. His course in psychology involved the dissection of frogs. Ramanujan was a devout Hindu and was appalled at what he perceived as senseless and immoral cruelty. When the examinations came at the end of the year he did very well in mathematics but he failed the other subjects and therefore lost his scholarship.

Ramanujan was able to complete a year at another college, Pachiappa College in 1906, thanks to another scholarship. (That scholarship was cut in half when another student applied for a scholarship and the only one available was divided between the two of them.) However in the examinations at the end of 1907 he again failed due to poor performance in the subjects other than mathematics.

Despite his academic failure Ramanujan threw himself into the pursuit of new results in mathematics. He worked long and hard. A friend of Ramanujan known as Sandow related the following conversation with Ramanujan:

Sandow: Ramanju, they all call you a genius. Ramanujan: What! Me, a genius! Look at my elbow, it will tell you the story. Sandow: What is all this, Ramanju? Why is it so rough and black? Ramanujan: My elbow has become rough and black in making a genius of me! Night and day I do my calculations on slate. It is too slow to look for a rag to wipe it with. I wipe the slate almost every few minutes with my elbow. Sandow: So, you are a mountain of industry. Why use a slate when you have to do so much calculation? Why not use paper? Ramanujan: When food itself is a problem, how can I find money for paper? I may require four reams of paper every month.

Sandow goes on to inquire how Ramanujan gets money for food. Ramanujan was a devout Hindu and believed in the gods and goddesses of Hinduism. The story he relates shows how Ramanujan's culture through its members came to his aid; i.e., people of Tamil culture helped him. To the extent that the gods and goddesses are just personalizations of aspects of that culture maybe the gods and goddesses he believed in did come to his aid.

Ramanujan was under pressure from his parents to find some means to support himself. In 1907 Ramanujan went for an interview with a man who was a deputy collector of revenue with the Madras Civil Service. His name was Ramaswamy Ayyar and he was a founder of the Indian Mathematical Society. In the interview Ramanujan told Ayyar of his interest in mathematics and showed him one of his notebooks containing the results Ramanujan had found. When Ramanujan told Ayyar that he wanted a job as a clerk Ayyar said that Ramanujan might lose his mathematical talent working as a clerk. Ramanujan was distraught, but Ayyar said he would find a way to help him without having him work as a clerk. Ayyar gave Ramanujan a letter of recommendation that he was to take to another Professor Ayyar (Seshu Ayyar), who happened to have been Ramanujan's instructor in the past.

It was not until December of 1910 that Ramanujan followed through on Ramasway Ayyar's instructions. When Ramanujan took that letter of recommendation to Professor Seshu Ayyar, Professor Ayyar passed him on with a note of introduction to Ramachandra Rao who was a government official and the president of the Indian Mathematical Society. Rao asked to see Ramanujan's notebook. After skimming through it for a few minutes Rao realized that Ramanujan had found results unknown to other mathematicians. Rao then found where Ramanujan was living. It was in the district in Madras near the beach known as Triplicane. Rao informed Ramanujan that he would receive some money each month to take care of his expenses. This enabled Ramanujan to do his mathematical researches day and night. He took breaks for an evening walk on the beach and for sleep.

In February of 1912 Ramanujan became insecure enough about his future that he secured a clerical position with the Madras Port Trust Office. His pay was 25 rupees per month.

Ramanujan worked at the Port Trust Office for only a year. During that time he seems to have found time to work on mathematics because a few sheets of this work accidently got sent to the Port Trust supervisor's office along with some official papers.

In S.R. Ranganathan's book Ramanujan: The Man and the Mathematician there is no mention of the story that Ramanujan operated two adding machines at once, one with each hand. This story perhaps emerged from someone for whom mathematics meant only arithmetic and could only imagine in this way what a mathematical genius could do.

During Ramanujan's period of work with the Port Trust Office a friend of his saw him gathering up pieces of packing paper. Ramanujan told him that it was for doing his mathematics. When Ramanujan did not have plain paper he used written upon paper and wrote over the writing in red ink.

The Health of Ramanujan

Probably the most misleading aspect of the short biographies of Ramanujan is that they leave the impression that the climate and lack of suitable food in England for the vegetarian Ramanujan destroyed his health. Those factors were not good for his health but his health was precarious long before he journeyed to England. His health may have been weak due to health conditions in South India and due to genetic factors. His mother bore several children who did not survive infancy. His generally poor health may have contributed to his contraction of tuberculosis.

A student who knew Ramanujan at Pachaiyappa College in 1906-1907, Radhakrishna Ayyar, had some interesting observations of Ramanujan. Ayyar observed that Ramanujan was fair and soft of features as one would expect of someone who was not physically active. Ayyar said that when Ramanujan was concentrating on some mental task the pupils of his eyes would disappear. Ayyar and others noted that Ramanujan's eyes appeared intensely penetrating. However Ramanujan was not withdrawn and aloof. He was a genial conversationalist and liked to talk about religion and philosophy as well as mathematics. He was a quite devout Hindu and considered the god Narashima of special significance for him.

In 1909 Ayyar was living in the Triplicane district of Madras. One day Ramanujan arrived at Ayyar's residence by horse-cart. He was very ill and his landlord in George Town (another district of Madras) asked him to go live with friends who could take care of him. Ayyar gave Ramanujan his bed and tried to take of him, but Ramanujan was a difficult patient refusing what his caretakers thought would be helpful. Ayyar took Ramanujan to a doctor who said Ramanujan should be sent home to his family because he would need constant care. When Ayyar put Ramanujan on the train for Kumbakonam Ramanujan entrusted Ayyar with two notebooks and gave him instructions on what to do with them if he, Ramanujan, died. Fortunately Ramanujan recovered. By 1911 Ramanujan had returned to Madras. Ayyar noted that at that time he had a hand in putting Ramanujan in contact with Ramachandra Rao who granted Ramanujan a monthly stipend of 20 to 25 rupees per month that enabled Ramanujan to work on his mathematical research undisturbed.

Ramanujan's Family and Friends

The image of Ramanujan's personality that comes from the reminiscences of those who knew him is one of a genial, shy and modest person who was devoutly religious as well as being devoted to mathematics. To his mother he was a devoted and obedient son, countering her only on rare occasions. To his school mates he was someone who could entertain them with recitations of religious philosophy but could also tell them about scientific matters such as astronomy.

An anecdote related by one friend indicates an unusually genial personality. Ramanujan was at one point in his college career sharing a room with two other students who were brothers. Ramanujan and one brother were staying up late talking about astronomy. The other brother, peeved at having his sleep disturbed, dumped a pot of water on Ramanujan's head, saying that the dousing was to cool off Ramanujan's overheated brain. Instead of reacting with violent retribution Ramanujan announced that he had just been bathed in the Ganges River.

In 1909 Ramanujan's mother found him a wife. She was only nine years old at the time and needed to stay with her family until puberty. Even when she came to Ramanujan's family's house, his mother would not let the couple sleep together in the same bed for a number of years. They never had children and it is possible that the marriage was never consummated. Nevertheless Ramanujan was clearly quite fond of his wife, Janaki. When Ramanujan returned to India from England his mother met him without bringing Janaki along and Ramanujan was upset at this. Later at home, because of his ill health, Ramanujan's mother asked him to send Janaki back to her family and Ramanujan refused to do this.

Ramanujan's Journey over

the Deep, Dark Sea to England

The coterie of fans of Ramanujan who had worked to find him funds to live on soon began to seek ways to send him to England where he could be in contact with mathematicians who were on his level. They urged him to send examples of his work to mathematicians in Britain.

A letter was written that contained a selection of about one hundred theorems which Ramanujan had discovered. The text of the letter may well have been drafted by some supporters in Madras because it did not display the usual extreme modesty of Ramanujan himself about his work. Copies of the letter were sent to three mathematicians in England. Two of the recipients did not deign to reply and perhaps did not even read the letter. The other recipient was George Hardy of Cambridge University, one of the top mathematicians of his day. Hardy and a colleague, John Edensor Littlewood, worked through the purported theorems to verify their validity. Not all were correct but most were. This exercise convinced Hardy that Ramanujan was a superlative mathematician. Hardy then set to find a way to bring Ramanujan to Cambridge. In February of 1913 Hardy wrote a letter to Ramanujan expressing his interest in Ramanujan's work. Hardy also started pulling strings to get Ramanujan support for his work. Authorities at the University of Madras provided Ramanujan with a research scholarship which paid him at a rate more than twice what he had earned as a clerk. It was no simple matter for the University to do this because Ramanujan had not received a college degree.

Bringing Ramanujan out of India was also no simple matter. There is in Hindu theology a prohibition against crossing the deep, dark sea and Ramanujan was a devout Hindu. The first entreaty by Hardy to Ramanujan for him to come to Cambridge was rejected. Probably this rejection was because its acceptance would have not only involved his going against his own religious principles but also it would involve disobeying his devoutly religious mother.

The Goddess Namagiri

Ramanujan and a friend, Narayana Ayyar, spent three consecutive days praying to the Goddess Namagiri for her to provide guidance. The Goddess Namagiri had a special significance in the family. Ramanujan's mother said that Ramanujan was born after her parents prayed to Namagiri to bless her with a son. After Ramanujan and Ayyar prayed for the guidance of Namagiri, Ramanujan's mother had a dream in which she saw her son sitting amongst a group of Europeans with a big halo surrounding him. This convinced her that it was alright for her son to travel to England.

In England Hardy began a campaign to bring Ramanujan to Cambridge. An English scholar named Neville who was going to India for a visit was persuaded to visit Ramanujan in Madras and encourage him to come to England.

Once Ramanujan's journey to England was arranged the coterie of his admiring friends started preparing him for the trip. They thought, unfortunately as it turned out, that Ramanujan needed an English style of dress. One friend took Ramanujan in the sidecar of his motorcycle to various stores to obtain the clothing. His friends prevailed upon Ramanujan to get his hair cut in European style and abandoned his turban for a hat. All of this change of style made Ramanujan uncomfortable and the discomfort continued during his sojourn in England and was unnecessary. The people he had contact with in England would have been perfectly comfortable with Ramanujan wearing a turban instead of a hat.

Although Ramanujan reluctantly acquiesced to a European style of dress he was adamant about not changing the food he would eat. As a Hindu he would eat no meat, eggs or fats that involved the killing of animals.

Ramanujan left Madras in March of 1914. Ramanujan's friends in India arranged for him to be provided with vegetarian meals on his ship voyage to England. Ramanujan however did not trust the purity of meals prepared by non-Hindus. For this reason in England he cooked his own meals. However, since finding vegetables in the wintertime in England was difficult and took time Ramanujan's did not get the nourishment that he needed. The situation was even worse during World War I. Even with ingredients available Ramanujan nourishment would have suffered from his intense preoccupation with mathematical research. The end result was that his health deteriorated and he contracted tuberculosis.

Ramanujan was a rare creature touched by inspiration and not really competent to take care of himself. When Ramanujan complained about being cold at night in his bed his friends had to tell him that the problem was that he was sleeping on top of the bed, as he did in India, and should have been sleeping under the blankets on his bed.

There are other examples of Ramanujan's difficulty at dealing with the elements of ordinary life. Once when he cooked a meal for friends he was so distraught at one guest not wanting a third helping of a dish that he immediately left his apartment with his guest sitting there and was not seen for several days. Another problem that Ramanujan had was that he was teased by students at Cambridge University because he was so shy.

By the scientific establishment of Britain Ramanujan was lionized. He was awarded a Bachelor of Arts (B.A.) degree in 1916 purely for his mathematical studies. In December of 1917 he was elected to the London Mathematical Society. He was inducted into the Royal Society as a Fellow in 1918 at the age of 30, an extremely young age for such an honor and he was one of the few non-Britons to be so honored. There were some who were reluctant to heap such honors on a foreigner and a young one at that. Some referred to Ramanujan as the Hindoo Calculator. Near the end of 1918 he became a Fellow of Trinity College of the University of Cambridge. He was the first Indian to be so honored.

By 1919 Ramanujan's health had deteriorated to the point where he could no longer function in England and needed the care and comfort of his family in Madras. He returned to India in 1919. His sickness was not curable and he died in 1920 at the age of 32. His death is usually attributed to tuberculosis but there is evidence that he was also suffering from a liver disease.

(To be continued.)

















The Mathematical Style of Ramanujan

Ramanujan's style of doing mathematics developed from his introduction to higher mathematics through trigonometry and Carr's volume of six thousand theorems. In trigonometry one can derive multitudes of formulas (identities). Ramanujan looked for such formulas as he saw in Carr's volume. Ramanujan generated formulas which he felt to be true on the basis of intuition and the checking of some special cases. He generally did not provide a rigorous proof of his results. Generally he was not strong in establishing such rigorous proofs. He was reluctant for rigorous proof to take precedence over his intuition for the truth of his propositions.

Ramanujan for example looked for the limits of infinite series. One such series became the preferred way of computing the mathematical constant π.

For k=0 the result is 3.14159273, for k=1 it is 3.141592654. The value of π to 14 decimal places is 3.141592653589793, so Ramanujan's formula provided a result accurate to 9 places on the second step. Altogether Ramanujan had 17 series formulas for the reciprocal of π. There is no way anyone could have created such a formula without a touch of genius.

Ramanujan had a special interest in continued fractions; i.e., effectively infinite fractional constructions. For examples of the evaluation of infinite series and continued fractions see Series and Fractions.

Later Ramanujan evaluated definite integrals. In modern times computer software such as Maple and Mathematica have been created to do such evaluations. These software packages do symbolic computation by manipulating strings of symbols until a configuration is found that corresponds to a known result.

Ramanujan and Asperger's Syndrome

In 1943 Leo Kanner published a description of a pattern of behavior that seems to be an identifiable subgroup of the human population. This pattern included

excessive withdrawal of contact with other people which included noncommunication either through remaining mute or using language in a way that did not involve communication with others

an obsession with the preservation of the sameness of their environment

a fascination with physical objects

the retention of facial expressions that do not reveal any inner emotion

This pattern became known as autism. Later it was recognized that there is not just a single pattern but instead a collection or spectrum of related disorders. It was also recognized that although many autistic people are of very limited functionality that can be characterized as retarded there are some who are of normal or even above normal abilities.

In the same year, 1943, that Kanner identified autism, Hans Asperger of Austria identified another related population group that have some characteristics related to the autistic but generally are able to function quite well in human society but have a different mode of operation than the general population. Asperger's study was lost track of for about a half century. It was rediscovered and given prominence when psychologist found that Asperger's work had relevance for understanding some people of extraordinary ability and creativity. Those people became known as Asperger's Syndrome people.

The National Autistic Society of the United Kingdom provides the following estimates of proportions of the various autistic patterns in the general population.

Group Proportion

in Population Classic Autism

(Learning Disabilities

With IQ Under 70 5/10,000 Other Autistic

Spectrum Disorders 15/10,000 Asperger's Syndrome

Or High Functioning

Autism 36/10,000 Other Related

Syndrome 35/10,000 Source: Michael Fitzgerald, Autism and Creativity, Brunner-Routledge, 2004

Some scholars believe that Srinivasa Ramanujan had Asperger's Syndrome. The nature of Asperger's Syndrome and its relation to the spectrum of autism has not been yet decided. High functioning autistic people and Asperger's Syndrome have in common a social impairment but Asperger's Syndrome people do not seem to have any impairment of their language skills although they may tend to talk at people rather than converse with them. Asperger people may have notable abilities and yet lack a certain common sense about the mundane elements of life.

In the case of Ramanujan there were elements of his early life that fit the pattern of Asperger's Syndrome such as his beginning to speak relatively late. His neglecting of the nonmathematical subjects in his college work that led to his loss of his desperately needed scholarship could be an example of a lack of common sense, or just poor judgment. When in England he did not realize that he needed to sleep under the blankets on a bed would be another example of a lack of common sense. His neglect of his health would be another more serious example of a lack of common sense.