The number pi or π (approximately 3.14159265) is well known as the ratio of the circumference of a circle to its diameter. Although π is an irrational number, meaning that it cannot be expressed exactly as a fraction, it is possible to express the number as an infinite series.
One of the simplest such series is π = 4 – (4/3) + (4/5) – (4/7) + (4/9) – (4/11)... The standard techniques of calculus can be used to prove that this series converges to π. Unfortunately, the convergence is very slow, meaning that one needs to write down a large number of terms to approximate π with any degree of accuracy.
The Indian mathematician Srinivasa Ramanujan (1887-1920) found some approximations to π that are much better than the above series. The formula for the infinite series at the bottom of the picture is due to Ramanujan. It converges so quickly that each successive term in the series computes a further eight decimal places of π. To give you some idea of how accurate the formula is, the approximation given by just one term is 9801/(sqrt(8)x1103), which works out as about 3.14159273001. This is accurate to eight significant figures, and has the first six decimal places correct!
This is a very impressive approximation from a mathematician who worked before the era of computers. Perhaps not surprisingly, Ramanujan's contemporaries were curious about where he got his ideas. The answer is quite interesting: while dreaming, he received visions of scrolls of complex mathematical content from his family goddess, Mahalakshmi of Namakkal.
Although he died at the age of 32, Ramanujan left behind a large number of mathematical results, and some of the best modern methods for computing π are based on his work. Ramanujan did not write up proofs for many of his results, although most of them turned out to be both correct and original. However, he left behind four famous notebooks of rough ideas, one of which was lost until 1976. These notebooks have inspired many papers by later mathematicians attempting to prove Ramanujan's results.
The Wikipedia page on approximations to π: http://en.wikipedia.org/wiki/Approximations_of_%CF%80
The Wikipedia page on Srinivasa Ramanujan: http://en.wikipedia.org/wiki/Srinivasa_Ramanujan
A popular post by from Halloween 2013, showing the first infinite series I mentioned, carved into a pumpkin: https://plus.google.com/102744407669548081722/posts/frJVPykwpWV
A popular post by me from August 2013 about π, featuring the digital art of Cristian Ilies Vasile: https://plus.google.com/101584889282878921052/posts/8AFefDCfV4h
(Disclaimer: I am from the UK, where March 14th is 14.3, not 3.14. Call me irrational, but I don't think that pi day is a real thing.)