The Man Who Knew Infinity

We've met Hardy and Ramanujan and their taxi-cab [1729] before.  It's in the air since that 2014 piece because of the 2015 filum [trailer] about their relationship starring Jeremy Irons and Dev Patel; which I see I was citing a mere 3 weeks ago. The great thing about having an increasingly crappy short-term memory, aka a two-week event horizon, is that I can all over-excited all over again. Everyone's favourite explainer of maths is on a Ramanujan jag at the moment putting out a succession of short vids on Numberphile. James Grime is wondering how the young Indian genius came up with his wonderful [as in full of wonder and possibility] ideas. Because we scientists harrumph cannot accept Rananujan's explanation that they were delivered into his head by god. Let's start here with episode II with Ramanujan's formula for Pi. Grime notes that mathematics is full of wildly unexpected 'coincidences' - connexions between different parts of the mathematical universe. Gottfried Leibniz pulled this rabbit out of his hat head:

Why how wha'? An infinite series of the reciprocals of odd integers gives us the ratio between the diameter of a circle and its circumference?  It's only an approximation of course and not a very good one.  A little bit better that 22/7 which was known to the ancient Greeks.  Even that was good enough for many practical purposes being accurate to 4 parts in 10,000. That's about the width of one slip of mortar between the bricks that made up the curved ends of the stadium at Olympia [28.5m x 212.5m]; so the mason wasn't going to curse his quantity surveyor about it.

One of the teaser formulae that Ramanujan sent from Madras/Chennai to Professor Hardy in Cambridge was another approximation for Pi

Whaaaa? It's all very well having a feeling for numbers such as 1729, but where would the peculiar relationship among such a clatter of numbers as Pi, 9801, 1103 and sqrt(8) come from?  Ramanujan went on to p a t i e n t l y explain, that his formula was only an approximation, although rather better than that of Leibniz.  If Hardy, or the world wanted more accuracy, then they should realise that the simple formula was only the first term in a series from which successively more accurate values of Pi could be derived:

B'godde, Holmes, why didn't I think of that?
Because, Watson, you are rather less bright than the silver cap of my pipe <puff> . . . <puff>.

While we're on the subject of Pi, why not print out the first million digits and have a look at them on an airfield? And now for something completely different: Mayday's talk by Grime about Ramanujan's Summation.

Jeremy and Dev talk about the film and other matters on WBUR: listen up it's sort of heart-warming.