Well, it was such an apparent mathematical law that it kindled no notable curiosity in the students. They nodded their heads blandly in a matter-of-fact agreement to their teacher’s mathematical proposition. “Any number divided by itself gives one. Say, if four bananas are divided among four children, then all of them would get one banana each”, the statement was quite obvious and trite to the third form class. But there was one student whose seemingly dumb question could dumbfound any devoted mathematician. He raised his hand incredulously and shot his doubt, “Anything divided by itself is one, I agree, but how about zero divided by zero? If there were zero children and there were zero bananas, will then each child still gets a banana?”. The teacher would have been stumped, we know not; neither is that important. Important is that a young kid raised a question on the indeterminate form, mathematically speaking. Zero divided by zero, 0/0 is not equal to one. Such indeterminate forms when they appear in the context of certain scientific and engineering calculations, L’Hospital’s rule or other such rules are used to evaluate a limit for the expression.
The boy was Srinivasa Ramanujan! The name would invoke enthralling emotions among mathematicians from around the globe and, the rest lesser math beings; would carelessly brush it off as nothing, simply admitting, ‘Ah a genius!’. Born in 1887, his birthday December 22 is celebrated as National Mathematics Day in India, in honor of the birth anniversary of the mathematical genius. ‘The strange kind of paradise’, India enjoys an unparalleled legacy when it comes to the realm of numbers. Starting with the visionary seers and sages of Vedic age, remarkable mathematical discoveries and pioneering innovations have been made. Much of this ancient knowledge relating to geometry and arithmetic which were essential for building the Yajña vedis or firesacrifice altars, are preserved in later treatises called Śulbasūtras. Each of these firesacrificial altars were unique in shapes; falcon shaped Garuda chiti, turtle shaped Kurma chiti and so on. Baudhāyana, Āpasthamba and Katyāyana were some of the most prominent authors of the Śulbasūtras, which describe both exact geometric constructions of irrational numbers like square root two as well as their rational approximations. These were speculated to be composed in the pre -historic period (800 BCE or earlier). The contribution of decimal system, invention of zero as a digit and place holder and numerous other hefty mathematical contributions followed for the next two and a half millennia. Aryabhatta, Brahmagupta, Varāhamihira, Bhāskara, Nīlakanda Sōmayāji, Mādhava and the list of gems of mathematicians goes on. Ramanujan would be the crest-jewel! A premier choice for celebrating National Mathematics Day.
In the temple town known as Kumbakonam, of Tanjore district in Tamil Nadu, was born Ramanujan to Kuppuswamy Srinivasa Iyengar and wife Komalatāmmāl. His father worked as a clerk in a silk saree shop and earned a modest income of around twenty rupees a month. As a toddler Ramanujan was infected with the deadly smallpox and had quite hard times with regard to health. Like a sardonic law of Nature, many exceptional scholars being perfect misfits in a formal educational institution; Ramanujan would have approved of this law, his own life being another solid proof.
His mother Komalatāmmāl instilled values in him and brought him up with lofty ideals of Sanatana Dharma. It is from her that he learned the stories of their family deity Nāmagiri Thaayaar of Nāmakkal. This Goddess Nāmagiri played a towering role in his life. Ramanujan, after having stunned the entire realm of math world with his enigmatic ideas and mind blowing equations, claimed that it was the Goddess Nāmagiri who revealed these profound mathematical ideas to him!
Ramanujan got enrolled in the famous Town High School at Kumbakonam after having earned a part scholarship owing his stellar performance. The classroom discussion on indeterminate forms, zero divided by zero incident happened in this school. With an astronomical pace, he progressed in his learnings on Mathematics. Borrowing advanced study books from his seniors, he expanded his horizon of knowledge. In 1904, he enrolled in a two year Fine Arts course at Government College at Kumbakonam. He would spend a lot of time in the Sarangapani temple near his house. He would work continuously on Math and fall asleep on the stone floors of this temple. Goddess Nāmagiri appeared in his dreams and solved equations for him!
Two books particularly propelled his learning trajectory during this period. One was the textbook on Trigonometry by S.L. Loney and the other was ‘A Synopsis of Elementary Results in Pure and Applied Mathematics’ by George Shoobridge Carr. He also gained access to the Mathematical Gazette of London and tried solving the published unsolved problems. Only mathematics interested him and so he failed in other subjects, whose outcome would be the end of scholarship which meant the end of his studies in the college. Dropped out, in 1906, he moved to another college at Chennai, named ‘Pachaiyapas college’. He formed connections with a few mathematical teachers and solved problems and continued his math journey, ; however as usual, the other subjects in fact failed to keep his attention and consequently yet another dropout from college ensued. Failing twice without any formal degree he struggled along with his family to make the both ends meet. Sitting on the verandah of his small house, he would sit with a slate and chalk and construct equations after equations. Hypergeometric series, Continued Fractions and Singular Moduli and what not! From 1907 to 1914, Ramanujan produced five notebooks full of equations and these invaluable documents have inspired mathematicians ever since, helping solve various conundrums and inspiring new fields of math. In the meantime, he got married to Janaki Ammāl.
“Dear Sir, 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 special investigations of divergent series in general and results I get are termed by the local mathematicians as “startling” ….I would request you to go through the enclosed papers……..”. This historic letter from Ramanujan reached Godfrey Herald Hardy, the most famous English Mathematician of the time then in fields of Number Theory and Mathematical Analysis. He has already published around 60 academic papers and written 3 books by that time. Hardy went through the letter and the nine page of equations attached along with the letter, thoroughly. Several equations on Number theory, Definite Integrals and Infinite Series were present. A surprised Hardy showed these manuscripts to his associates and friends and this created ripples of sensations amongst the mathematical researchers of Cambridge. G. H. Hardy invited S. Ramanujan to the University of Cambridge and opened the doors of collaborated contributions in the next 5 years to come. Working together, Hardy commented, “I have never met Ramanujan’s equal. I can compare him with Euler and Jacobi.” Euler and Jacobi were two of the greatest mathematicians’ the world had ever seen. Jointly Hardy and Ramanujan, published many interesting papers.
The renowned Statistician best remembered for a statistical measure, the Mahalanobis distance, Prasanta Chandra Mahalanobis was staying in Cambridge during this time and became friends with Ramanujan. There was a local London publication named ‘The Strand Magazine’, which used to run a regular column called “Perplexities”. In the December of 1914, this column featured a puzzle titled “Puzzles at a Village Inn”, which Mahalanobis, brought to the attention of his friend Ramanujan. The puzzle was this. … The house of a person’s Belgian friend is in a long street numbered on this side one, two, three and so on. And that all the numbers on one side of him added up exactly the same as all the numbers on the other side of him. Funny thing that! He said that he knew there were more than fifty houses on that side of the street, but not so many as five hundred? Mahalanobis is supposed to have figured out the problem in a few minutes by a trial and error method. He then posed the puzzle to Ramanujan, who was cooking a meal when he heard this question. “Ramanujan, here is a problem for you”. “Please tell me”. Mahalanobis read out the problem. “Please take down the solution”, Ramanujan blurted out. Amazed at the spontaneous answer, Mahalanobis asked how he did it. The reply was that as soon as he heard the problem he knew that there should be a continued fraction involved. And that when he asked himself which one it was, the answer apparently came to his mind.
The Rogers-Ramanujan identities which are continued fractions appear in many places in science: birth-death processes, statistical mechanics, algebraic geometry and of course number theory. G. H. Hardy in one of his lectures on Ramanujan’s work mentioned that Ramanujan’s investigations into hypergeometric series and continued fractions “suited him exactly and here he was unquestionably one of the greatest masters”. Rogers-Ramanujan continued fractions provides a convenient way of understanding the dynamics of queue formation. Physical queues like traffic on a road and subtle queues like when we Google and wait a fraction of a second while their server responds to millions of other search queries made before that of ours.
Ramanujan used to work oftentimes stretching himself to the limits for more than 24 hours continuously. His health started deteriorating and by 1917, he was diagnosed with tuberculosis and he was admitted to a hospital. The university of Madras offered Ramanujan a Professorship with an annual income of 400 Rupees. This was a princely sum in those times. However, tuberculosis never left him and the end of the Mathematical Prodigy came on April 20, 1920, now believed to have been caused by hepatic amoebiasis. He was only 32!
He has developed singular ways of calculating pi with extraordinary efficiency. His approach was later used in computer algorithms yielding millions of digits of pi.
During his short life, Ramanujan independently compiled nearly 3,900 results. Most of it were completely novel ideas; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new horizons of work and inspired a vast amount of further research. Of the many works on Ramanujan, none is better known than “The Man who Knew Infinity” by Robert Kanigel. It has been translated into more than a dozen languages, including seven Indian languages and inspired a Hollywood movie of the same title.
In his recently published autobiography ‘My Search for Ramanujan: How I learned to Count’, Ken Ono recounts how both he and his father, Takashi Ono, have been inspired in their lives by Ramanujan. For Ken Ono, Ramanujan has been a guiding influence in his mathematical work too. In 1988 Ken watched the “Letters from an Indian Clerk”, a documentary about Ramanujan which initiated him into following the studies of the genius. In an old paper by Ramanujan, on quadratic forms, Ramanujan asked for a rule that determines which odd numbers are not in the form x2+y2+10z2 where x,y,z are integers. Ken Ono and Kannan Soundararajan studied this and proved subject to General Riemann hypothesis that 2719 is the largest odd integer which is not represented. Later when Ken visited the Sarangapani temple at Kumbakonam, he was stupefied to see the number 2719 etched in charcoal in the temple. It is worth mentioning that this number is a permuted form of the famous Ramanujan Taxicab number 1729. Ramanujan continues to both inspire and surprise mathematicians and math lovers alike.
Mathematics has always been developed two ways. First, for the simple beauty of mathematics as opposed to utilitarian purposes. Second, for the utilitarian and application purposes, in order to enhance engineering and technological advancements. Ramanujan belonged to the former panel. For Ramanujan
Mathematics was not just a relation between numbers. His deep relation to the subject can be read from his statement, “An equation to me has no meaning, unless it expresses a thought of God!”. And though he ascribed all the credits of his discoveries to the Goddess Nāmagiri, perhaps he himself was that God. “The God of Math”!
Assistant Professor, International Centre for Spiritual Studies, Amrita University