Born this day in 1887, the self-taught Indian mathematician, Srinivasa Ramanujan made huge contributions to several areas of math, often with methods that were completely novel. In 1918, he became one of the youngest Fellows of the Royal Society, and only the second Fellow from India.
Ramanujan grew up in Tamil Nadu in southern Indies where as a child, he quickly devoured all the mathematical books available at his school in Kumbakonam. By the age of 13, he was already proving theorems of his own. Working alone and in extreme poverty, he rediscovered several famous mathematical results for himself, including several relating to diverging series and formulae for solving algebraic equations. He wrote his work in a highly idiosyncratic style, though even at his most brilliant, he never fully mastered the notion of a rigorous mathematical proof. Despite this, he produced a stream of highly original research, which he claimed was revealed to him in dreams by his family goddess, Namagiri.
To get a glimpse of the mind of this genius, I will present a beautiful formula derived by Ramanujan in collaboration with the brilliant Cambridge mathematician, Godfrey Harald Hardy (1877-1947) who took Ramanujan under his wings. The formula, named after them, concerns the partitions of a number, which refers to the number of ways a number can be written as a collection of smaller numbers. Partitions is today and active area of mathematical research as it has many practical applications in science and engineering.
The Hardy-Ramanujan Partition Formula
How many ways are there to write the number 4 as a collection of smaller numbers? A little experimentation reveals five possibilities: 1+1+1+1; 1+3, 1+1+2; 2+2; and 4 itself. A partition is a way of splitting a set of objects into smaller subsets and in the case of the number 4, the partition is four, not counting the four itself.
This example is considered “trivial”. But as n becomes large, it is no longer an easy matter to get even an approximate answer. For example, the partition number of n=10 is 42, while 100 has more than 190 million partitions! Hence a formula for calculating partition numbers is handy. It could also be beautiful. This is where the Hardy-Ramanujan formula comes in, Their formula is given by
Although the formula looks complicated, the answer can be very quickly calculated for any n very quickly using a computer. Moreover, although it is “only” an approximation, the accuracy of the approximation improves drastically as n becomes large. For example, for n=1,000, the exact answer is 24,061,467,864,032,622,473,692,149,727,991partitions, while the Hardy-Ramanujan formula gives 2.4402 × 1031, which is remarkably close to the exact answer. For its concise appearance and its accuracy for large n (i.e., the most problematic cases), the Hardy-Ramanujan formula gets my vote as one of the most beautiful equation in number theory and indeed all of mathematics.
References:
[1] G. H. Hardy and S. Ramanujan. Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. (2), 17:75–115, 1918.
[2] H. Rademacher. On the partition function p(n). Proc. London Math. Soc. (2), 43:241–254, 1937. Rademacher extended the Hardy-Ramanujan result to produce an exact expression for p(n) where n is any number. His formula, however, entails adding together an infinite series and hence is not of great practical use for counting partitions. Richard Elwes (2013), Maths in 100 Key Breakthroughs, Quercus: UK (p. 401)