राष्ट्रीय गणित वर्ष

वर्ष २०१२ को राष्ट्रीय गणित वर्ष घोषित किया गया है , इसका क्या उद्देश्य है ?

मानव – इतिहास में गणित का विकास बहुत पहले ही संभवतः एक स्वतंत्र विषय के रूप में हुआ | गणित के विकास में भारत का महत्वपूर्ण योगदान रहा है और शताब्दियों तक शीर्ष स्थान पर रहा है | पाँचवी शताब्दी में आर्यभट और उसके बाद ब्रह्मगुप्त भारत के महान गणितज्ञों में गिने जाते हैं | भारतीय गणित के इतिहास में शून्य और दशमलव पद्धति का अविष्कार पूरे मानव – इतिहास और गणित के विश्व – इतिहास में सबसे महत्वपूर्ण स्थान रखता है और संभवतः कोई अन्य आविष्कार भविष्य में इसका स्थान नहीं ले सकेगा |

                                                                                                                 
आर्यभट के बाद लगभग हजार वर्षों तक भारतीय गणित विश्व में प्रथम स्थान पर रहा | इस काल में भारत के महान गणितज्ञों में केरल के माधव अंतिम गणितज्ञ थे | गणितज्ञ माधव ने न्यूटन और लेबनीज से लगभग दो शताब्दी पूर्व कलन शास्त्र यानि कैलकुलस के मूलभूत सिद्धांतों की खोज की थी | उनका यह शोध उनके द्वारा स्थापित केरल के एक विद्यालय की चहारदीवारी में ही सिमटकर रह गया और शेष दुनिया इससे अनभिज्ञ रहा | दुर्भाग्यवश सोलहवीं शताब्दी के मध्य के बाद इस विद्यालय का अस्तित्व नहीं रहा | इसके बाद उन्नीसवीं शताब्दी तक भारत में गणित का विकास अवरूद्ध रहा , जब तक कि इस शताब्दी के अंत में महानतम भारतीय गणितज्ञ श्रीनिवास रामानुजन का अवतरण नहीं हुआ | परंतु रामानुजन के बाद कुछ गिने – चुने गणितज्ञ ही अंतर्राष्ट्रीय स्तर पर प्रसिद्ध हुए |

                  

हमारा देश जितना विशाल है , इसकी तुलना में यहाँ विश्वस्तरीय गणितज्ञों की काफी कमी है | हमारे देश में छात्र उच्चस्तरीय गणित के अध्ययन में बहुत कम ही रूचि दिखाते हैं , फलस्वरूप यहाँ गणित का गुणवत्तापूर्ण और समुचित विकास नहीं हो पा रहा है | यहाँ छात्रों की गलत आम धारणा है कि गणित के क्षेत्र में रोजगार की कोई सम्भावना नहीं है | यह धारणा कुछ वर्षों पहले तक सही थी , परंतु आज गणित में शोध और शिक्षण के क्षेत्र में अपार संभावनायें व्याप्त हैं |

भारत में गणित के विकास को प्रोत्साहित करने हेतु यह आवश्यक है कि गणितज्ञ – समुदाय इसके लिए उपायों और साधनों कि खोज करें जिससे कि  गणित में उच्च कोटि के शोध और इनके अनुप्रयोग को प्रोत्साहन मिल सके | इसी उद्देश्य हेतु केंद्र सरकार ने वैज्ञानिक गतिविविधियों को प्रोत्साहित करने हेतु एक नीति बनायी है | इसी कड़ी को आगे बढ़ाते हुए हमारे वर्तमान प्रधानमंत्री ने २६ दिसंबर २०११ को रामानुजन जयंती के अवसर पर  वर्ष २०१२ को “राष्ट्रीय गणित वर्ष” घोषित किया | इसका उद्देश्य गणित के विकास को प्रोत्साहित करने हेतु हर संभव प्रयास करना है , विश्वविद्यालयों में मूल्यांकन पद्धति में सुधार लाना हैं , प्रतिभाशाली छात्रों को प्रोत्साहित करना है ताकि उन्हें रामानुजन कि तरह कठिनाइयों का सामना न करना पड़े और वे शोध में अपना महत्वपूर्ण योगदान दे सके |

Indian Postage Stamp on Ramanujan

 Indian Postage Stamp on Ramanujan

 

 

India Post issues  second time, a postage stamp on S. Ramanujan 

 Date of Issue : 26 December 2011

India Post issued a commemorative postage stamp on 75th Birth Anniversary of Srinivasa Ramanujan  on 22 December 1962.

Rediscovering Ramanujan : Interview with Prof. Bruce C. Berndt.

Rediscovering Ramanujan 

Interview with Prof. Bruce C. Berndt.

Interviewer : Asha Krishnakumar                                                                            

The academic lineage of most eminent scholars can be traced to famous centres of learning, inspiring teachers or an intellectual milieu, but Srinivasa Ramanujan, perhaps the greatest of Indian mathematicians, had none of these advantages. He had just one year of education in a small college; he was basically self-taught. Working in isolation for most of his short life of 32 years, he had little contact with other mathematicians. 

"Many people falsely promulgate mystical powers to Ramanujan's mathematical thinking. It is not true. He has meticulously recorded every result in his three notebooks," says Dr. Bruce C. Berndt, Professor of Mathematics at the University of Illino is, whose 20 years of research on the three notebooks has been compiled into five volumes. 

Between 1903 and 1914, before Ramanujan went to Cambridge, he compiled 3,542 theorems in the notebooks. Most of the time Ramanujan provided only the results and not the proof. Berndt says: "This is perhaps because for him paper was unaffordable and so he worked on a slate and recorded the results in his notebooks without the proofs, and not because he got the results in a flash." 

Berndt is the only person who has proved each of the 3,542 theorems. He is convinced that nothing "came to" Ramanujan but every step was thought or worked out and could in all probability be found in the notebooks. Berndt recalls Ramanujan's well-known i nteraction with G.H. Hardy. Visiting Ramanujan in a Cambridge hospital where he was being treated for tuberculosis, Hardy said: "I rode here today in a taxicab whose number was 1729. This is a dull number." Ramanujan replied: "No, it is a very interestin g number; it is the smallest number expressible as a sum of two cubes in two different ways." Berndt believes that this was no flash of insight, as is commonly thought. He says that Ramanujan had recorded this result in one of his notebooks before he cam e to Cambridge. He says that this instance demonstrated Ramanujan's love for numbers and their properties. 

Although Ramanujan's mathematics may seem archaic by today's standards, in many respects he was far ahead of his time. While the thrust of 20th century mathematics has been on building general theories, Ramanujan was a master in finding particular result s which are now recognised as providing the core for the theories. His results opened up vistas for further research not only in mathematics but in other disciplines such as physics, computer science and statistics. 

After Ramanujan's death in 1920, the three notebooks and a sheaf of papers that he left behind were handed over to the University of Madras. They were sent to G.N. Watson who, along with B.M. Wilson, edited sections of the notebooks. After Watson's death in 1965, the papers, which contained results compiled by Ramanujan after his return to India from Cambridge in 1914, were handed over to Trinity College, Cambridge. In 1976, G. E. Andrews of Pennsylvania State University rediscovered the papers at the T rinity College Library. Since then these papers have been called Ramanujan's "lost notebook". According to Berndt, the lost notebook caused as much stir in the mathematical world as Beethoven's Tenth Symphony did in the world of Western classical music. 

Berndt says that the "unique circumstances surrounding Ramanujan and his mathematics" make it very difficult to assess his greatness among such mathematical giants as Newton, Gauss, Euler and Reimann. According to Berndt, Hardy had provided the following assessment of his contemporary mathematicians on a scale of 0 to 100: "On the basis of pure talent he gave himself a rating of 25, his collaborator J. E. Littlewood 30, German mathematician D. Hilbert 80, and Ramanujan 100." Berndt says that it is not R amanujan's greatness but only its measure that is in doubt. 

Besides the five volumes, Berndt has written over 100 papers on Ramanujan's works. He has guided a number of research students in this area. He now works on Ramanujan's "lost notebook" and on some other manuscripts and fragments of notes. Recently in Che nnai to give lectures on Ramanujan's works at the Indian Institute of Technology, the Institute of Mathematical Sciences and the Ramanujan Museum and Mathematical Centre, Berndt spoke to Asha Krishnakumar on his work on Ramanujan's notebooks, the broad areas in mathematics that Ramanujan had covered, the vistas his work has opened up and the application of his work in physics, statistics and communication. 

Excerpts from the interview: 

How did you get interested in Ramanujan's notebooks?

 

After my Ph.D. at the University of Wisconsin, I took my first position at the University of Glasgow (Scotland) in 1966-67. Prof. R. A. Rankin was a leader in number theory at that time. I remember being in Rankin's office in 1967 when he told me about Ramanujan's notebooks for the first time. He said: "I have a copy of the notebooks published by the Tata Institute of Fundamental Research, Bombay (Mumbai). Would you be interested in looking at it?" I said, "No, I am not interested in it."

I did not think about the notebooks for some years until early 1974 when I was on leave at the Institute for Advanced Studies in Princeton, U.S. In February that year, I was reading two papers of Emil Grosswald in which he proves some formulae from Ramanujan's notebooks. I realised I could prove these formulae as well by using a theorem I proved two years ago. I did that and then I was curious to find out whether there were other formulae in the notebooks that I could prove using my methods. So, I went to the Princeton University library and got hold of Ramanujan's notebooks published by the TIFR. I was thrilled to find out that I could actually prove some more formulae. But there were a few thousand others I could not. 

I was fascinated with the notebooks and in the next few years I wrote papers around the formulae I had proved from the notebooks. The first was a repository paper on Ramanujan's theta 2n+1 formula, for which I did a lot of historical research on other pr oofs of the formula. This I wrote for a special volume called Srinivasa Ramanujan's Memorial Volume, published by Jupiter Press in Madras (Chennai) in 1974. After that, wherever I went, I was all the time working on, and proving, the various formulae of Ramanujan's - to be precise - from Chapter 14 of the second notebook. Then I wrote a sequel to this. 

Let me jump ahead to May 1977, when I decided to try and prove all the formulae in Chapter 14. I took this on as a challenge. There were in all 87 results in this chapter. I worked on this for the next one year. I took the help of my first Ph.D. student, Ron Evans. 

After about a year of working on this, the famous mathematician George E. Andrews visited Illinois and told me that he discovered in the spring of 1976 Ramanujan's "lost notebook" along with G. N. Watson and B. M. Wilson's edited volumes on Ramanujan's t hree notebooks and some of their unpublished notes in the Trinity College Library. I then got photocopies of Ramanujan's lost notebook and all the notes of Watson and Wilson. And so I went to the beginning of the second notebook.

What does the second notebook contain?

This is the main notebook because it is the revised and enlarged version of the first. I went back to the beginning and went about working my way through it using Watson and Wilson's notes when necessary. 

How long did you work on the second notebook?

I really do not know how many years exactly. But some time in the early 1980s Walter Kaufmann-Buhler, the mathematics editor of Springer Verlag in New York, showed interest in my work and decided to publish it. That had not occurred to me till the n. I agreed and signed a contract with Springers. 

That was when I started preparing the results with a view to publishing them. I finally came out with five volumes; I had thought it would be three. It also took a much longer time than I had anticipated. 

After I completed 21 chapters of the second notebook, the 100 pages of unorganised material in the second notebook and the 33 pages in the third had a lot more material. I also found more material in the first which was not there in the second. So, I fou nd a lot of new material. It was 20 years before I eventually completed all the three notebooks. 

Why did you start with the second notebook and not the first?

I knew that the second was the revised and enlarged edition of the first. The first was in a rough form and the second, I was relatively certain, had most of the things that were there in the first and a lot more.  

What did each notebook contain?

The new results that were in the second notebook were generally among the unorganised pages of the first. And the third notebook was all unorganised. A higher percentage of the results in the unorganised parts of the second and the third were new. In oth er words, you got a higher percentage of new results as you went into the unorganised material.

What do you mean by new results?

 

Results that have not been got earlier. 

What is the percentage of new results in the notebooks?

 

Hardy estimated that over two-thirds of the work Ramanujan did in India was rediscovered. That is much too high. I found that well over half is new. It is difficult to say precisely. I would say that most results were new because we also have to consider that in the meantime, from 1920 until I started doing this work, other people discovered these things. So, I would say that at least two-thirds of the material was really new when Ramanujan died.

Ramanujan is popularly known as a number theorist. Would you give a broad idea about the results in his notebooks? What areas of mathematics do they cover?

 

You are right. To much of the mathematical world and to the public in general, Ramanujan is known as a number theorist. Hardy was a number theorist but he was also into analysis. When Ramanujan was at Cambridge with Hardy, he was naturally influenced by him (Hardy). And so most of the papers he published while he was in England were in number theory. His real great discoveries are in partition functions. 

Along with Hardy, he found a new area in mathematics called probabilistic number theory, which is still expanding. Ramanujan also wrote sequels in highly composite numbers and arithmetical functions. There are half a dozen or more of these papers that ma de Ramanujan very famous. They are still very important papers in number theory. 

However, the notebooks do not contain much of number theory. It is, broadly speaking, in analysis. I will try and break that down a little bit. I would say that the area in which Ramanujan spent most of his time, more than any other, is in elliptic funct ions (theta functions), which have strong connections with number theory. In particular, Chapters 16 to 21 of the second notebook and most of the unorganised portions of the notebooks are on theta functions. There is a certain type of theta functions ide ntity which has applications in other areas of mathematics, particularly in number theory, called modular equations. Ramanujan devoted an enormous amount of effort on refining modular equations. 

Ramanujan is also popular for his approximations to pie. Many of his approximations came with his work on elliptic functions. Ramanujan computed what are called class invariants. Even as he discovered them, they were computed by a German mathematician, H . Weber, in the late 19th and early 20th centuries. But Ramanujan was unaware of this. He computed 116 of these invariants which are much more complicated. These have applications not only in approximations to pie but in many other areas as well. 

Have you gone through every one of the 3,254 entries in the three notebooks and proved each of them, including in the unorganised material?

 

I have gone through every entry in the notebooks. If a result has already been proved in the literature, then I just wrote the entry down and said that proofs can be found in this literature and so on. But I will also discuss the relevance in history of the entry.

What are the applications of Ramanujan's discoveries in areas such as physics, communications and computer science?

 

This is a very difficult question to answer because of the way mathematics and science work. Mathematics is discovered and it is then there for others to use. And you do not always know who uses it. But I have regular contact with some physicists who I know use Ramanujan's work. They find the results very useful in their own application. 

What are the areas in physics in which Ramanujan's work is used?

 

The most famous application in physics is in the area of statistical mechanics. Among those who I know have used Ramanujan's mathematics extensively is W. Backster, the well-known physicist from Australia. He used the famous Rogers-Ramanujan identities in what is called the hard hexagon model to describe the molecular structure of a thin film. 

Many of Ramanujan's works are used but his asymptotic formulae have found the most important application; I first wrote this in 1974 from his notebook.

Then there is a particular formula of Ramanujan's involving the exponential function which has been used many times in statistics and probability.

Ramanujan had a number of conjectures in regard to this formula and one is still unproven. He made this con jecture in a problem he submitted to the Indian Mathematical Society. The asymptotic formula is used, for instance, in the popular problem: What is the minimum number of people you can have in a room so that the probability that two share a common birthd ay is more than half? I think it is 21, 22 or 23. Anyway, this problem can be generalised to many other types of similar problems. 

Have you looked at the lost notebook?

 

That is what I am working on now with Andrews. It contains about 630 results. About 60 per cent of these are of interest to Andrews. He has proved most of these results. The other 40 per cent are of great interest to me as most of them were a continuatio n of what Ramanujan considered in his other notebooks. So, I began working on them.

What are your experiences of working on Ramanujan's notebooks? Do you think Ramanujan was a freak or a genius or he had the necessary motivation to write the notebooks?

 

I think one has to be really motivated to do the kind of mathematics he was doing, through either teachers or books. We understand from Ramanujan's biographers that he was motivated in particular by two books: S. L. Loney's Plane Trigonometry and Carr's Synopsis of Elementary Results in Pure Mathematics (which was a compilation of 5,000 theorems with a few proofs) at the age of 12. How much his teachers motivated him, we really do not know as nothing about it has been recorded. Reading these book s and going through the problems must have aroused the curiosity that he had and inspired him.

He is particularly amazing because he took off from the little bit he knew and extended it so much in so many directions, leading to so many new and beautiful results. 

Did you find any results difficult to decipher in any of Ramanujan's notbooks? 

 

Oh yes. I get stuck all the time. At times I have no idea where these formulae are coming from. Earlier, Ron Evans, whom I have already mentioned as having worked on Chapter 14, helped me out a number of times. There are times I would think of a formula over for about six months or even a year, not getting anywhere. Even now there are times when we wonder how Ramanujan was ever led to the formulae. There has to be some chain of reasoning to lead him to think that there might be a theorem there. But ofte n this is missing. To begin with, the formulae look strange but over time we understand where they fit in and how important they are than they were previously thought to be.

Did you find any serious errors in Ramanujan's notebooks?

 

There are a number of misprints. I did not count the number of serious mistakes but it is an extremely small number - maybe five or ten out of over 3,000 results. Considering that Ramanujan did not have any rigorous training, it is really amazing that he made so few mistakes. 

Are the methods of mathematics teaching today motivating enough to produce geniuses like Ramanujan?

 

Some like G. E. Andrews think that much of the reforms have come about because students do not study as much. This, along with the advent of computers, has changed things. A lot of mathematics which can be done by computations, manipulations and by doing exercises in high school are now being done using calculators and computers. And the computer, I do not think, gives any motivation. 

The books on calculus reform (that is now introduced in the U.S.) include sections on using a computer. To calculate the limit of a sequence given by a formula, the book says press these numbers, x, y and z... Then there appears a string of numbers that get smaller and smaller and then you can see that is tends to zero. But that does not lead to any understanding as to why they are tending to zero. So, this reasoning, motivation and understanding of why the sequence tends to zero is not being taught. I think that is wrong. 

There seem to be two schools of thought: one which thinks that the development of concepts and ideas is important and the other, like that in India, which thinks that development of skills is important in teaching mathematics. Which do you think is more important?

 

I think you cannot have one without the other. Both must be taught. The tendency in the U.S. is to move away from skills and rely on computers. I do not think this is correct because if you have the skills and understanding, then you can see if you have made an error in punching in the computers. Andrews and I have the experience of students putting down results that are totally ridiculous because they have not understood what is going on. They do not even realise that they made mistakes while punching in the computers. So, developing skills is absolutely necessary. But on the other hand if you just go on with the skills and have no understanding of why you are doing this, you lose the motivation and it becomes just a mechanical exercise. 

However, even now there is a possibility that geniuses like Ramanujan will emerge. It is important that once you identify such children, books and material should be found for them specially. The greatest thing about number theory in which Ramanujan work ed is that you can give it to people of all ages to stimulate them. Number theory has problems that are challenging, that are not too easy, but yet they are durable and motivating. A foremost mathematician (Atle Selberg) and a great physicist (Freeman Dy son) of this century have said that they were motivated by Ramanujan's number theory when they were in their early teens. 

Srinivasa Ramanujan: A Remarkable Mathematical Genius

Srinivasa Ramanujan A Remarkable Mathematical Genius

	By: Dr Subodh Mahanti
	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 in his Discovery of India Sheer intuitive brilliance coupled to long, hard hours on his slate made up for most of his educational lapse. This ‘poor and solitary Hindu pitting his brains against the accumulated wisdom of Europe’ as Hardy called him, had rediscovered a century of mathematics and made new discoveries that would captivate mathematicians for next century. Robert Kanigel in The Man who Knew Infinity : A Life of the Genius Ramanujan
I still say to myself when I am depressed and find myself forced to listen to pompous and tiresome people, ‘Well, I have done one thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms’. Godfrey Harold Hardy “Ramanujan’s life”, as Robert Kanigel, the author of a marvellous biography of Ramanujan, wrote, “can be made to serve as parable for almost any lesson you want to draw from it.” Ramanujan’s example stirred the imagination of many–particularly that of mathematicians. Thus, Subrahmanyan Chandrasekhar (1910-95), the Indian born astrophysicist, who got Nobel Prize in 1983, said : “I think it is fair to say that almost all the mathematicians who reached distinction during the three or four decades following Ramanujan were directly or indirectly inspired by his example.” Even those who do not know about Ramanujan’s work are bound to be fascinated by his life. As Kanigel wrote: “Few can say much about his work, and yet something in the story of his struggle for the chance to pursue his work on his own terms compels the imagination, leaving Ramanujan a symbol for genius, for the obstacles it faces, for the burden it bears, for the pleasure it takes in its own existence.” Ramanujan’s life is full of strange contrasts. He had no formal training in mathematics but yet “he was a natural mathematical genius, in the class of Gauss and Euler.” Probably Ramanujan’s life has no parallel in the history of human thought. Godfrey Harold Hardy, (1877-1947), who made it possible for Ramanujan to go to Cambridge and give formal shape to his works, said in one of his lectures given at Harvard Universty (which later came out as a book entitled Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work): “I have to form myself, as I have never really formed before, and try to help you to form, some of the reasoned estimate of the most romantic figure in the recent history of mathematics, a man whose career seems full of paradoxes and contradictions, who defies all cannons by which we are accustomed to judge one another and about whom all of us will probably agree in one judgement only, that he was in some sense a very great mathematician.”  Srinivasa Ramanujan Iyengar (best known as Srinivasa Ramanujan) was born on December 22, 1887, in Erode about 400 km from Chennai, formerly known as Madras where his mother’s parents lived. After one year he was brought to his father’s town, Kumbakonam. His parents were K. Srinivasa Iyengar and Komalatammal. He passed his primary examination in 1897, scoring first in the district and then he joined the Town High School. In 1904 he entered Kumbakonam’s Government College as F.A. student. He was awarded a scholarship. However, after school, Ramanujan’s total concentration was focussed on mathematics. The result was that his formal education did not continue for long. He first failed in Kumbakonam’s Government College. He tried once again in Madras from Pachaiyappa’s College but he failed again.  While at school he came across a book entitled A Synopsis of Elementary Results in Pure and Applied Mathematics by George Shoobridge Carr. The title of the book does not reflect its contents. It was a compilation of about 5000 equations in algebra, calculus, trigonometry and analytical geometry with abridged demonstrations of the propositions. Carr had compressed a huge mass of mathematics that was known in the late nineteenth century within two volumes. Ramanujan had the first one. It was certainly not a classic. But it had its positive features. According to Kanigel, “one strength of Carr’s book was a movement, a flow to the formulas seemingly laid down one after another in artless profusion that gave the book a sly seductive logic of its own.” Thisbook had a great influence on Ramanujan’s career. However, the book itself was not very great. Thus Hardy wrote about the book: “He (Carr) is now completely forgotten, even in his college, except in so far as Ramanujan kept his name alive”. He further continued, “The book is not in any sense a great one, but Ramanujan made it famous and there is no doubt it influenced him (Ramanujan) profoundly”. We do not know how exactly Carr’s book influenced Ramanujan but it certainly gave him a direction. `It had ignited a burst of fiercely single-minded intellectual activity’. Carr did not provide elaborate demonstration or step by step proofs. He simply gave some hints to proceed in the right way. Ramanujan took it upon himself to solve all the problems in Carr’s Synopsis. And as E. H. Neville, an English mathematician, wrote : “In proving one formula, as he worked through Carr’s synopsis, he discovered many others, and he began the practice of compiling a notebook.” Between 1903 and 1914 he had three notebooks.  While Ramanujan made up his mind to pursue mathematics forgetting everything else but then he had to work under extreme hardship. He could not even buy enough paper to record the proofs of his results. Once he said to one of his friends, “when food is problem, how can I find money for paper? I may require four reams of paper every month.” In fact Ramanujan was in a very precarious situation. He had lost his scholarship. He had failed in examination. What is more, he failed to prove a good tutor in the subject which he loved most.  At this juncture, Ramanujan was helped by R. Ramachandra Rao, then Collector of Nellore. Ramchandra Rao was educated at Madras Presidency College and had joined the Provincial Civil Service in 1890. He also served as Secretary of the Indian Mathematical Society and even contributed solution to problem posed in its Journal. The Indian Mathematical Society was founded by V. Ramaswami Iyer, a middle-level Government servant, in 1906. Its Journal put Ramanujan on the world’s mathematical map. Ramaswami Iyer met Ramanujan sometime late in 1910. Ramaswami Iyer gave Ramanujan notes of introduction to his mathematical friends in Chennai (then Madras). One of them was P.V. Seshu Iyer, who earlier taught Ramanujan at the Government College. For a short period (14 months) Ramanujan worked as clerk in the Madras Port Trust which he joined on March 1, 1912. This job he got with the help of S. Narayana Iyer.  Ramanujan’s name will always be linked to Godfrey Harold Hardy, a British mathematician. It is not because Ramanujan worked with Hardy at Cambridge but it was Hardy who made it possible for Ramanujan to go to Cambridge. Hardy, widely recognised as the leading mathematician of his time, championed pure mathematics and had no interest in applied aspects. He discovered one of the fundamental results in population genetics which explains the properties of dominant, and recessive genes in large mixed population, but he regarded the work as unimportant. Encouraged by his well-wishers, Ramanujan, then 25 years old and had no formal education, wrote a letter to Hardy on January 16, 1913. The letter ran into eleven pages and it was filled with theorems in divergent series. Ramanujan did not send proofs for his theorems. He requested Hardy for his advice and to help getting his results published. Ramanujan wrote : “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 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 would request you to go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me “. The letter has become an important historical document. In fact, ‘this letter is one of the most important and exciting mathematical letters ever written’. At the first glance Hardy was not impressed with the contents of the letter. So Hardy left it aside and got himself engaged in his daily routine work. But then he could not forget about it. In the evening Hardy again started examining the theorems sent by Ramanujan. He also requested his colleague and a distinguished mathematician, John Edensor Littlewood (1885-1977) to come and examine the theorems. After examining closely they realized the importance of Ramanujan’s work. As C.P. Snow recounted, ‘before mid-night they knew and knew for certain’ that the writer of the manuscripts was a man of genius’. Everyone in Cambridge concerned with mathematics came to know about the letter. Many of them thought `at least another Jacobi in making had been found out’. Bertrand Arthur William Russell (1872-1970) wrote to Lady Ottoline Morell. “I found Hardy and Littlewood in a state of wild excitement because they believe, they have discovered a second Newton, a Hindu Clerk in Madras … He wrote to Hardy telling of some results he has got, which Hardy thinks quite wonderful.” Fortunately for Ramanujan, Hardy realised that the letter was the work of a genius. In the next three months Ramanujan received another three letters from Hardy. However, in the beginning Hardy responded cautiously. He wrote on 8 February 1913. To quote from the letter. “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 … I hope very much that you will send me as quickly as possible at any rate a few of your proofs, and follow this more at your leisure by more detailed account of your work on primer and divergent series. It seems to me quite likely that you have done a good deal of work worth publication; and if you can produce satisfactory demonstration I should be very glad to do what I can to secure it” .  In the meantime Hardy started taking steps for bringing Ramanujan to England. He contacted the Indian Office in London to this effect. Ramanujan was awarded the first research scholarship by the Madras University. This was possible by the recommendation of Gilbert Walker, then Head of the Indian Meteorological Department in Simla. Gilbert was not a pure mathematician but he was a former Fellow and mathematical lecturer at Trinity College, Cambridge. Walker, who was prevailed upon by Francis Spring to look through Ramanujan’s notebooks wrote to the Registrar of the Madras University : “The character of the work that I saw impressed me as comparable in originality with that of a Mathematical Fellow in a Cambridge College; it appears to lack, however, as might be expected in the circumstances, the completeness and precision necessary before the universal validity of the results could be accepted. I have not specialised in the branches of pure mathematics at which he worked, and could not therefore form a reliable estimate of his abilities, which might be of an order to bring him a European reputation. But it was perfectly clear to me that the University would be justified in enabling S. Ramanujan for a few years at least to spend the whole of his time on mathematics without any anxiety as to his livelihood.”  Ramanujan was not very eager to travel abroad. In fact he was quite apprehensive. However, many of his well-wishers prevailed upon him and finally Ramanujan left Madras by S.S. Navesa on March 17, 1914. Ramanujan reached Cambridge on April 18, 1914. When Ramanujan reached England he was fully abreast of the recent developments in his field. This was described by J. R. Newman in 1968: “Ramanujan arrived in England abreast and often ahead of contemporary mathematical knowledge. Thus, in a lone mighty sweep, he had succeeded in recreating in his field, through his own unaided powers, a rich half century of European mathematics. One may doubt whether so prodigious a feat had ever been accomplished in the history of thought.” Today it is simply futile to speculate about what would have happened if Ramanujan had not come in contact with Hardy. It could happen either way. But then Hardy should be given due credit for recognizing Ramanujan’s originality and helping him to carry out his work. Hardy himself was very clear about his role. “Ramanujan was”, Hardy wrote, “my discovery. I did not invent him — like other great men, he invented himself — but I was the first really competent person who had the chance to see some of his work, and I can still remember with satisfaction that I could recognize at once what I treasure I had found.” It may be noted that before writing to Hardy, Ramanujan had written to two well-known Cambridge mathematicians viz., H.F. Baker and E.W. Hobson. But both of them had expressed their inability to help Ramanujan.  Ramanujan was awarded the B.A. degree in March 1916 for his work on ‘Highly composite Numbers’ which was published as a paper in the Journal of the London Mathematical Society. He was the second Indian to become a Fellow of the Royal Society in 1918 and he became one of the youngest Fellows in the entire history of the Royal Society. He was elected “for his investigation in Elliptic Functions and the Theory of Numbers.” On 13 October 1918 he was the first Indian to be elected a Fellow of Trinity College, Cambridge. Much of Ramanujan’s mathematics comes under the heading of number theory — a purest realm of mathematics. The number theory is the abstract study of the structure of number systems and properties of positive integers. It includes various theorems about prime numbers (a prime number is an integer greater than one that has not integral factor). Number theory includes analytic number theory, originated by Leonhard Euler (1707-89); geometric theory - which uses such geometrical methods of analysis as Cartesian co-ordinates, vectors and matrices; and probabilistic number theory based on probability theory. What Ramanujan did will be fully understood by a very few. In this connection it is worthwhile to note what Hardy had to say of the work of pure mathematicians: “What we do may be small, but it has certain character of permanence and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something beyond the powers of the vast majority of men.” In spite of abstract nature of his work Ramanujan is widely known. Ramanujan was a mathematical genius in his own right on the basis of his work alone. He worked hard like any other great mathematician. He had no special, unexplained power. As Hardy, wrote: “I have often been asked whether Ramanujan had any special secret; whether his methods differed in kind from those of other mathematicians; whether there was anything really abnormal in his mode of thought. I cannot answer these questions with any confidence or conviction; but I do not believe it. My belief that all mathematicians think, at bottom, in the same kind of way, and that Ramanujan was no exception.” Of course, as Hardy observed Ramanujan “combined a power of generalization, a feeling for form and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his peculiar field, without a rival in his day. Here we do not attempt to describe what Ramanujan achieved. But let us note what Hardy had to say about the importance of Ramanujan’s work. “Opinions may differ as to the importance of Ramanujan’s work, the kind of standard by which it should be judged and the influence which it is likely to have on the mathematics of the future. It has not the simplicity and the inevitableness of the greatest work; it would be greater if it were less strange. One gift it shows which no one will deny—profound and invincible originality.” The Norwegian mathematician Atle Selberg, one of the great number theorists of this century wrote : “Ramanujan’s recognition of the multiplicative properties of the coefficients of modular forms that we now refer to as cusp forms and his conjectures formulated in this connection and their later generalization, have come to play a more central role in the mathematics of today, serving as a kind of focus for the attention of quite a large group of the best mathematicians of our time. Other discoveries like the mock-theta functions are only in the very early stages of being understood and no one can yet assess their real importance. So the final verdict is certainly not in, and it may not be in for a long time, but the estimates of Ramanujan’s nature in mathematics certainly have been growing over the years. There is doubt no about that.” Often people tend to speculate what Ramanujan would have achieved if he had not died or if his exceptional qualities were recognised at the very beginning. There are many instances of such untimely death of gifted persons, or rejection of gifted persons by the society or the rigid educational system. In mathematics we may cite the cases of Niels Henrik Abel (1809-29) and Evarista Galois (1811-32). Abel solved one of the great mathematical problems of his day - finding a general solution for a class equations called quintiles. Abel solved the problem by proving that such a solution was impossible. Galois pioneered the branch of modern mathematics known as group theory. What is important is that we should recognise the greatness of such people and take inspiration from their work.  Even after more than 80 years of the death of Ramanujan the situation is not very different as far the rigidity of the education system. Today also a ‘Ramanujan’ is not likely to get a chance to pursue his career. This situation remains very much similar as described by JBS Haldane (1982-1964), a British born geneticist and philosopher who spent last part of his life in India. Haldane said : “Today in India Ramanujan could not get even a lectureship in a rural college because he had no degree. Much less could he get a post through the Union Public Service Commission. This fact is a disgrace to India. I am aware that he was offered a chair in India after becoming a Fellow of the Royal Society. But it is scandalous that India’s great men should have to wait for foreign recognition. If Ramanujan’s work had been recognised in India as early it was in England, he might never have emigrated and might be alive today. We can cast the blame for Ramanujan’s non-recognition on the British Raj. We cannot do so when similar cases occur today...” Nehru’s statement given at the beginning is very much valid even today. And for these very reasons the story of Ramanujan should be told and retold to our younger people particularly to those who aspire to do something extraordinary but feel dejected under the prevailing circumstances. And in this connection it is worthwhile to remember what Chandrasekhar had to say: “I can recall the gladness I felt at the assurance that one brought up under circumstances similar to my own could have achieved what I could not grasp. The fact that Ramanujan’s early years were spent in a scientifically sterile atmosphere, that his life in India was not without hardships that under circumstances that appeared to most Indians as nothing short of miraculous, he had gone to Cambridge, supported by eminent mathematicians, and had returned to India with very assurance that he would be considered, in time as one of the most original mathematicians of the century — these facts were enough, more than enough, for aspiring young Indian students to break their bands of intellectual confinement and perhaps soar the way what Ramanujan had. "As someone has written “Ramanujan did mathematics for its own sake, for thrill that he got in seeing and discovering unusual relationships between various mathematical objects.” Today Ramanujan’s work has some applications in particle physics or in the calculation of pi up to a very large number of decimal places. His work on Rieman’s Zeta Function has been applied to the pyrometry, the investigations of the temperature of furnaces. His work on the Partition Numbers resulted in two applications — new fuels and fabrics like nylons. But then highlighting the importance of the application side Ramanujan’s work is really not very important.  Ramanujan died of tuberculosis in Kumbakonam on April 26, 1920. He was only 32 years old. “It was always maths ... Four days before he died he was scribbling,” said Janaki, his wife. The untimely death of Ramanujan was most unfortunate particularly so when we take into account the circumstances under which he died. As Times Magazine rightly wrote: “There is something peculiarly sad in the spectacle of genius dying young, dying with the first sweets of recognition and success tasted, but before the full recognition of powers that lie within. "The only Ramanujan Museum in the country, founded by Shri P. K. Srinivasan, a mathematics teacher, operates from March 1993 in the Avvai Academy, Royapuram, Madras. The achievement of Ramanujan was so great that those who can really grasp the work of Ramanujan ‘may doubt that so prodigious a feat had ever been accomplished in the history of thought". Further Reading Ramanujan: Twelve Lectures on the Subjects Suggested by His Life and Work by G. H. Hardy, Chelsea Publishing Co, New York, 1940. The Man Who Knew Infinity : A Life of the Genius Ramanujan by R. Kanigel, Abacus Books, London, 1992. Ramanujan’s Notebooks (Part I&II) by B.C. Berndt Springer, New York, 1985-1989. Ramanujan:The Man and the Mathematician by S.R. Ranganathan, Asia Publishing House, Bombay, 1967. Srinivasa Ramanujan : A Mathematical Genius by K. Srinivasa Rao; East West Books (Madras) Pvt. Ltd. 1998. Srinivasa Ramanujan, Suresh Ram, National Book Trust India, 1989. 7) Ganit Jagater Bismay Ramanujan by Satyabachi Sar, Gyan Bichitra Prakashani, Agartala, 2000. A well-written book in Bengali.

Two Exams Taken by Ramanujan in India

                           ''On Thursday and Friday, December 3 and 4, of 1903, Srinivasa Ramanujan, who was to become the greatest Indian mathematician in his country’s history, sat for the Matriculation Examination of Madras University . From documents recently found in the Tamil Nadu Archives, we have learned that Ramanujan obtained a Second Class place, permitting him to enter the Government College Kumbakonam in the following year with a scholarship. As is now well known, by the time he entered college, Ramanujan was totally immersed in mathematics and would not study any other subject. He thus failed his exams, except for mathematics, at the end of his first college year and lost his scholarship.''                                                            

                                                                                                                                     

This is an excerpt from the article " Two Exams Taken By Ramanujan In India" by the two authors and mathematicians Bruce C. Berndt and C.A. Reddi in which they have produced the complete account....................! 

To see the Full Article, click on the following link :                          

TWO EXAMS TAKEN BY RAMANUJAN IN INDIA

PM's Speech on the Ramanujan's Birth Anniversary

PM's Speech on the Ramanujan's Birth Anniversary

Chennai, Dec 26: The Prime Minister, Dr Manmohan Singh, addressed the 125th Birth Anniversary celebrations of noted mathematician Srinivas Ramanujan in Chennai today. Following is the text of the Prime Minister's address:

"We have gathered here today to celebrate the life and work of a great son of India and of Tamil Nadu, and one of the greatest mathematicians the world has seen. It is a pleasure to participate in this function in the memory of Srinivas Ramanujan, whose extraordinary genius so very brightly lit up the world of mathematics in the second decade of the last century. Men and women of such dazzling brilliance and deep intellect are born but rarely. Indeed, Srinivas Ramanujan´s genius was ranked by the English mathematician G. H. Hardy in the same class as giants like Euler, Gauss, Archimedes and Isaac Newton. While we rightly claim Ramanujan as one of our own, he equally belonged to all humanity like the other great men and women in any sphere of human thought. 

Before I proceed further, let me compliment the Ministry of Human Resource Development, and my colleague Kapil Sibal, for planning year long celebrations to mark the 125th birth anniversary of Ramanujan. As a tribute to the great mathematician, our government has decided to declare his birthday, that is December 22, as the National Mathematics Day and the year 2012 as a whole as the National Mathematical Year. India has a long and glorious tradition of mathematics that we need to encourage and nurture. I hope these steps will help in providing the additional impetus to the study of mathematics in our country, apart from making our people more aware of the work of Ramanujan.

                 
Mathematics seems to have acquired an independent identity as an intellectual discipline early on in human history. This identity became more sharply defined in the second half of the millennium before Christ, thanks to major developments in Greece. In this period, India too made great strides in mathematics, though in ways very different from the Greeks. In the early centuries of the Common Era, India was in fact in the lead in mathematical developments. Aryabhata in the fifth century, followed by Brahmagupta in the next are reckoned to be among the all-time great mathematicians. And we taught the world to think of zero as a number and the modern way of representing all numbers with 10 symbols. This arguably is the single most important mathematical development in all human history. 

                                                                                                                     
Indian mathematics remained in the forefront for almost a thousand years following Aryabhata. The last name in the great mathematicians we produced in this period is that of Madhava of Kerala. I understand Madhava had discovered the essentials of Calculus some two centuries before Newton and Leibnitz. His work however was not known beyond the school he had created in Kerala. That school unfortunately did not last beyond the middle of the 16th century. 

              
Intellectual activity receded into the background in the country for the next few centuries to revive only in the nineteenth century. In the early part of the nineteenth century, most of our intellectual energies were given to humanities and it is towards the end of the century that India began taking an interest in the sciences. In the second decade of the 20th century, the country could once again stake a claim to producing world class mathematics, and that was because of the work of Srinivasa Ramanujan. Ramanujan came from an economically disadvantaged background and had but a minimal training in mathematics and yet his genius overcame formidable difficulties to reach the pinnacle of greatness. The whole of India is proud of Ramanujam and many an Indian has been inspired by his shining example. Tamil Nadu of course has a special claim on him for he was a Tamilian. Along with Sir C. V. Raman and Subramanyam Chandrashekhar, he is among the three great men of science and mathematics that Tamil Nadu and India have given to the world in modern times. 

                    
The story of Ramanujan cannot be told without a mention of the Cambridge mathematician G. H. Hardy, who was responsible for Ramanujan getting the recognition that was his legitimate due. The parts that Hardy and Cambridge, which became Ramanujan´s alma-mater, played in the great mathematician's development represent the very best of the academic traditions of the West. The stories of the special relationship between G.H. Hardy and Ramanujan are a part of the folklore of mathematics.

                                                                         
The fascinating and inspiring story of Ramanujan needs to be told to the world. I am, therefore, very happy that the organizers have chosen this occasion to honour Professor Robert Kanigel who has written an excellent biography of Ramanujan and I am very happy that he is present here among us today. I greet you sir, and I understand that yours is a book that has made Ramanujan well known to the public at large all over the world, capturing vividly the atmosphere of the early nineteenth century academic world in that part of the world as well as in Britain. I am also happy that on this occasion, a new edition of the Notebooks of Ramanujan is also being released. These are photographic reproductions of unpublished mathematical work of Ramanujan, written out in his own hand in a series of notebooks during the days he spent in the city of Chennai. The originals, I am told are in the University of Madras and I congratulate the University authority for preserving them in its archives. 

            
The Government of India sets great store by science and has pursued a policy of encouraging scientific activities of diverse kinds. Given our traditions, we naturally attach special importance to mathematics. Since Ramanujan, a number of mathematicians from the country have distinguished themselves by performing at very high levels. However, it is a matter of concern that for a country of our size the number of competent mathematicians that we have is badly inadequate. Over more than the last three decades many of our young men and women with a natural ability in mathematics have not pursued the discipline at advanced levels. This has also resulted in a decline in the quality of our mathematics teachers both at the school and college levels. There is a general perception in our society that the pursuit of mathematics does not lead to attractive career opportunities. This perception must change. This perception may have been valid some years ago but today there are many new career opportunities available to mathematicians, and the teaching profession itself has become much more attractive in recent years.  

                                                      
The mathematical community has a duty to find out ways and means to address the shortage of top quality mathematicians in our country. It must reach out to the public particularly in the modern context where mathematics has tremendous influence on every kind of human behaviour. In many ways, mathematics can be regarded as the mother science. The Natural Sciences have had a long symbiotic relationship with mathematics. Life Sciences did not seem to have much use for mathematics till about a hundred years ago, but lately mathematical interventions have had a tremendous impact on Biology. Mathematics has also influenced the study of Social Sciences in a big way. The work of many of the Nobel Laureates in Economics is highly mathematical. Students, parents and people at large need to be more aware of these facts so that the study of mathematics as an academic discipline gains popularity in our country. I am happy that the activities planned in the National Mathematical Year would focus on promotion of mathematics at all levels- from schools to cutting edge research.

                                                                                              
The Ramanujan story illustrates the inadequacy of the university evaluation system in the early decades of the last century while at the same time it shows that the system displayed enough flexibility to take care of mavericks like Ramanujan. There have been many reforms since those days but there would still be talent which would elude proper evaluation. Our institutions of higher learning, therefore, must be sensitive to this problem. A genius like Ramanujan would shine bright even in the most adverse of circumstances, but we should be geared to encourage and nurture good talent which may not be of the same caliber as that of Ramanujan.

Let me end by wishing the National Mathematical Year all success. I expect the activities initiated during this year would be continued in the coming years as well, so as to help our country make it to the forefront of education and research in mathematics. That is my ardent prayer and with these words I thank you for listening to me patiently.