Srinivasa Ramanujan (1887-1920) is perhaps one of the most original mathematicians of all time. In a career that only lasted around ten years, he produced hundreds of highly innovative results in several areas of pure mathematics, particularly number theory and analysis. After his death, his notebooks and unpublished results inspired decades of research by succeeding mathematicians, the impact of which is still being felt in mathematics today. What’s more remarkable, Ramanujan was a self-taught mathematician who derived his amazing results without a formal university-level education in mathematics and totally unaware of results that were already known in the body of mathematical knowledge. So how did his genius come about?
The Prodigy of Tamil Nadu
Ramanujan was born to a modest Brahmin family on 22 December 1887 in the town of Erode in Tamil Nadu, southern India. Performing well at primary school, he passed exams in all subjects at the age of 10 with the best scores in his school district. After beginning secondary level mathematics, by his early teens he was investigating and discovering his own independent results. Of course, having no contact with any active mathematical researchers and being essentially self-taught, he had no knowledge of contemporary research topics. So he largely pursued his own ideas, often inspired by formulae and techniques he read from compendiums such as A Synopsis of Elementary Results in Pure and Applied Mathematics by a certain G.S. Carr. Although the book was dry and contained little to motivate the results listed in it, Ramanujan was hooked, particularly by formulas concerning infinite series.
As early as 1904 when he was only 17, Ramanujan began to produce mathematical research of substantial sophistication. An example of a problem he solved around that time was the following: Find the value of
Not bad for a 17-year old, self-taught student of math.
To Cambridge, England
Ironically, his child-like fascination with infinite series led him to spend so much time thinking about math his own way that he actually flunked out of the college, not once but twice. Fortunately in January 1913 Ramanujan wrote a letter (certainly one of the most famous letters in the history of mathematics), to G.H. Hardy, a lecturer at Trinity College, Cambridge, who was one of Britain’s foremost pure mathematicians. On reading the multi-page letter crammed with dozens of intricate formulae and theorems from Ramanujan’s notebooks, Hardy could hardly believe his eyes. A close examination of its content by Hardy and his Cambridge colleague J.E. Littlewood revealed a host of amazing results, which Hardy divided into three categories. Firstly, there were theorems that, unbeknown to Ramanujan, were already known. Secondly, there were results that, while new, were interesting rather than important. Finally, there were entirely original results that were simply astonishing, such as this:
So impressed was Hardy that he took Ramanujan under his wings and helped fill the gaps in Ramanujan’s mathematical knowledge. The two men soon became research collaborators, although their style could not be more different: Hardy and Littlewood, being strict analysts, were insistent on absolute rigour and formal proofs, while Ramanujan was content to rely on intuition and inductive experimentation. When asked where he got his intuition from, Ramanujan cryptically remarked that they mostly came to him through dreams given by the goddess Namagi.
The most famous famous collaboration between Hardy and Ramanujan was their joint paper on partitions The partition number p(n) is the number of ways a positive integer could can be written as a sum of positive integers where the order of addition does not matter. So for instance, for the integer n = 4, p(4) is 4 since
This is easy enough. But what if n is large? It can be shown that p(17) = 176 and p(34) = 12,310!. Clearly, a quick and accurate way to figure out partitions for large numbers is useful, which was what Hardy and Ramanujan achieved, producing one of the most staggering formula in all of mathematics. In case you’re curious, here is the formula:
The duo then proceeded to show that their formula was capable of producing results of unprecedented accuracy. For example, n = 100, it gave an output of 190,569,291.996 compared with the actual partition number of 190,569,292, a difference of only 0.004. And their formula worked equally well for even larger numbers.
This breathtaking formula was truly a remarkable meeting of two great minds, something that was unlikely to come about without either Ramanujan’s intuition or Hardy’s analytic rigor. In an acknowledgment of Ramanujan’s brilliance, the prestigious Royal Society elected him as a Fellow in 1918, one of the youngest person ever to be elected as Fellow. Unfortunately, Ramanujan died two years later at age of 32 from a combination of illness and malnutrition.
Fortunately for math, before he passed, Ramanujan filled various notebooks and manuscripts with thousands of results and conjectures. In 1976, the American mathematician George Andrews was searching for math papers in the Wren Library in Cambridge when he stumbled on handwriting notes by Ramanujan. What Andrews found is now known as Ramanujan’s “lost notebook”, a treasure trove containing four thousand results, many of which have never been derived before. This is perhaps Ramanujan’s finest legacy for the subject he loved, something that will continue to keep mathematicians busy for a very long time.
This post is adapted from an essay by Adrian Rice, titled “Srinivasa Ramanujan (1887–1920): The Centenary of a Remarkable Mathematician”, posted on the Institute of Mathematics website, http://www.imag.org.