Ramanujan

Sreenivas Aiyengar Ramanujan: A name that ranks among some of the greatest of mathmaticians' of all time. Ramanujan, virtually unaided and untaught, growing up in the modest town of Erode, in the great south-Indian state of Tamil Nadu, was born on December 22, 1887. Ramanujan was a prodigy and his unique and anomalous friendship and partnership with G.H. Hardy was clearly illustrative of the well-known axiom “two heads are better than one.”

It is the year 1919, close to the time when Ramanujan would sadly pass away of an incurable illness. His achievements are immortal and his contributions too many to count to the mathematical world.

MA: Mr. Ramanujan, I am pleased to meet you finally after hearing about the wonderful feats that you have performed in math without the aid of a proper or extensive education of any sort. How is it that without any extensive higher education you have surpassed boundaries that will propel you into immortality with other famous mathematicians?

SR: Yes, it was very hard as a child for me and later on as an adult to procure money for my researches lacking an all-around liberal education. For me, math was the only subject in which I had increasing interest in and thus excelled.

MA: Who taught you at such a young age to love the art of math?

SR: Hmm… I think that it was more of an intrinsic yearn inside of me that propelled me to plunge into mathematics. I taught myself many of the basic mathematical concepts since we were a poor family living in a small town. I was fortunate that my family moved from Erode, my birthplace, to the city of Kumbakonam where the educational facilities existed. I actually started school in an English medium school being that India is a British colony and education is organized along the English system.

MA: I can imagine what a stir you were as a child being the first in the class and always inquisitive. Is this not so?

SR: Some are shocked when I reply no. As a child, I was considered a slow minded one as my verbal abilities did not come into play until I was three years old. But, later on when I entered school the teachers were more open to me and did not think I was as slow as others previously had. Yes, I did do well in school, especially math. Inquisitiveness was always in me as I, at times, even challenged my teachers; this they were not as open to, let me tell you.

MA: With little resources or other intellectuals around you, what gave impetus to your ideas and thoughts as well as providing a background on previous math concepts formulated by other great mathematicians?

SR: Ahh, yes this is a turning point in my mathematical journey. I was fortunate to have come across a book written by G.S. Carr, Synopsis of Results in Pure and Applied Mathematics, which greatly affected my life as it provided the knowledge that I was desperately seeking. I had actually obtained it from another college student who had it in his possession, and I am indebted to that student as well for opening me up to a world in which I thrived and loved; a world that I soon became obsessed with. Even today I relish the hours I spent with that book, the 5,000 mathematical formulas, equations, and results that were unfolding from within its covers. I think that this is the turning point where mathematics took a hold of my life and in which all other cares became nonexistent.

MA: How did this affect your education? Did it open up newer opportunities because of your broad knowledge or did it confine you in any way?

SR: One would think that the specification and honing in one particular subject would open up a house of opportunities, but not at the age at which I was at, not yet into college and still fresh. My obsession in mathematics to the point that every thing else was insignificant caused me to neglect my other subjects in the Government College in Kumbakonam where I had won a scholarship to. I could not graduate because of this and even though I attended another college in Madras, I lost the scholarship eventually. This was around 1905 and the consequent years I spent wandering and doing what I loved--mathematics.

MA: Wow. But isn't it true that four years later, in 1909, your mother forced you to start thinking about marriage and living a normal, practical life?

SR: Yes, it is customary to marry at the age I was at and for the children to be in a marriage arranged by the parents. Janaki, my wife, was approved of by my mom and her horoscope checked with mine. She was only nine years old on our wedding day, July 14, 1909. Now, I had had to come back to the flow of life and wake up from my reverie as I now had a wife to take care of and support. I went looking for a job though my love for mathematics was in no case mitigated.

MA: What doors opened to you on this “job hunting” and who helped you along the way?

SR: I went to the founder of the Indian Mathematical Society, naturally as I was trying to procure a job in that field. They helped me to get in touch with Ramachandra Rao who was a mathematician and had business connections. I told him that I wanted to find a job that would give me enough to live on and pursue mathematics. I am fortunate that Rao wanted to help me so much that he himself supported me until he could find me a fellowship. This did not work out though so he found me a job as an accounts clerk. It was around this time, 1912, in which I had produced my first paper, which was published in the Journal of the Indian Mathematical Society.

MA: Certainly by this time you had made much progress in your endeavors, on a higher level than others could follow, where did you go from here?

SR: My friends encouraged me to get in touch with mathematicians of higher abilities from England and to write for help and guidance. I selected some well known names and sent off my letters hoping that one would take time to write to a humble man from India. I was turned down often but finally got a response from the person who would turn out to be one of my greatest friends and partners, Godfrey Harold Harvey of Trinity College at Cambridge.

MA: What did Mr. Hardy think of your work and what you had done?

SR: Well, to my knowledge he was intrigued at first by my work and then convinced of its merit. I had included in the letter that I had had no university education and that I had gone on a self-created unique path into my inquiry of mathematics and its far reaches. He wrote back that he was interested by my letter and the various theorems that I had stated. He wanted to see proofs of my assertions and said that some of my results were already known, that some were interesting but had no apparent cause, and there were some that appeared to be new and important. Hardy then set on trying to bring me to England to further talk and confer with me on my ideas and mathematics.

MA: Being an orthodox Hindu Brahmin, did this not deter you from crossing the seas into England?

SR: Yes, this is true. By traveling to England, I would not be able to attend Brahmin weddings and funerals and being a vegetarian, diet would be difficult to get on with. But, my mother had a dream in which the Goddess Namagiri appeared and told her that I should have the freedom to meet my destiny. In 1914, I sailed to England after receiving a scholarship in May 1913 to fund my inquiries from the University of Madras.

MA: How were your experiences in England and what was your view on being a part of the mathematical society?

SR: It was a unique experience. I was to spend five years in England with Hardy. Because I had worked up to this time without any influence whatsoever from European mathematicians, it seemed that I was behind in some basic concepts though fluent in other areas of innovation. Hardy helped me greatly by filling in gaps that I had.

MA: In England, is it not correct that you received a formal and well-rounded education?

SR: Yes, this is true. I graduated from Cambridge with a Bachelor of Science by Research on March 16, 1916.

MA: You are most well-known for your work in number theory on partitions in combination with Hardy. Can you supply us with a little insight into this theory?

SR: The partitioning of an integer in number theory, is the breaking down of that integer into its constituent parts in as many ways as possible. An example of this is the number 4 which has 5 different ways of adding up to 4, or 4 different “partitions.” This can be expressed as p(4)=5, there are 5 partitions for the number four (2+2, 1+3, 1+1+2, 1+1+1+1, and 4+0). This problem seems very simple in that it is not difficult of adding up possibilities to get a number. But, only in principle. As the numbers increase just plain simple adding up and trial and error methods are not sufficient. For example, the number of partitions for 3, p(3), is 3 but for 50, p(50) is 204, 226. Just listing all the numbers that add up to 50, counting one every 5 seconds, would take two weeks. So, the question was, is there a formula that can speed up this laborious process into something that could be done in minutes instead of weeks and months?

MA: So the problem was there and you set out to solve it, what occurred next?

SR: Well, we worked off of previous discoveries, especially that of Euler who had developed the generating function. In theory, a generating function supplies not just a particular answer to a particular problem but all the answers to related problems; it generates the answers as fast as the numbers are plugged in. Euler helped as his generating function gave rise to a “power series,” a series of successively higher powers, each multiplied by a coefficient.

f(x) = 1 + p(1) x1 + p(2) x2 + p(3) x3 + …

MA: So, what did this equation entail and what did the coefficients do?

SR: The coefficients p(1), etc. supplied us with the answers. They would turn out to be not just any number but the p(n) desired. p(50) is 204,226 and would read as 204,226 x5.

MA: Is there any simpler rendition of this that can be seen in simpler areas of mathematics?

SR: Of course. What do you think 22 x 23 = 25 entails? 4 x 8 = 32 can be rewritten like this and the exponents add up. This is what occurs when dealing with partitions as well.

MA: Was this the end?

SR: No. Actually the way that we had used for the derivation of partitions was approximate and we dwelled on for a long time to get more accurate answers within a smaller margin of error and better results. Hardy and I did work hard on this concept and theory and I am very proud of our accomplishment.

MA: Yes, it is a very big accomplishment especially to mathematical society for years to come. You received a great honor last year (1918) by being elected as a fellow of the Royal Cambridge Philosophical Society and then just three days later for election as a fellow of the Royal Society of London. After independently having discovered numerous theorems and axioms including elliptic integrals, hypergeometric series, and functional equations of the zeta function, I feel that you have justly earned this award and am proud to say that I have had the opportunity of meeting you and discussing your various experiences and ultimately your great accomplishments.

SR: Thank you very much. Yes, I am so honored to be on the list for election as a fellow of the Royal Society of London. It is more than I have ever dreamed of. I owe so much to my fellow mathematicians and friends and will never forgive their minds or their spirit.

MA: Thank you.