Last week saw the general release of the movie, *The Man Who Knew Infinity*, about the Indian mathematical genius Srinivasa Ramanujan. This is the 100-year-old story of a humble Indian clerk, played by Dev Patel, who was invited to study at Trinity College, Cambridge, the alma mater of Isaac Newton.

There he fell under the tutelage of the great number theorist GH Hardy (Jeremy Irons).

Hardy had recognised Ramanujan’s genius after receiving samples of his work that even he did not understand. With his logical mind, Hardy felt the equations and formulae he had derived had to be brilliant, as no one could have had the imagination to invent them. However, Ramanujan was largely self-taught in mathematics and lacked the rigour of proof on which Hardy insisted in order to seek the truth. Nevertheless, he made great contributions to Mathematics in the areas of number theory and analysis, despite working for many years in India and isolated from the Euro-centric mathematical community.

The maths in the movie focuses on the unsolved Partition theory that is simple to understand but difficult to solve.

For example, 4 can be partitioned in 5 different ways as follows;

**4, 3+1, 2+2, 2+1+1, and 1+1+1+1**

So we say *p*(4) = 5

Ramanujan discovered a formula that would find the number of partitions of 200 (Answer; *p*(200) = 3,972,999,029,388).

It was said about him that *‘Every positive integer is one of Ramanujan's personal friends’*. The number 1729 is known as the Hardy–Ramanujan number after a famous anecdote by Hardy following a hospital visit to see Ramanujan.

*“I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways”*

The two different ways are;

1729 = 1^{3} + 12^{3}

and 1729 = 9^{3} + 10^{3}.

These are now referred to as Taxi Cab numbers [Ta(n)] and so far only six are known. Here are the first three;

Ta(1) = 2 = 1^{3} + 1^{3}

Ta(2) = 1729 = 1^{3} + 12^{3} = 9^{3} + 10^{3}

Ta(3) = 87539319 = 167^{3} + 436^{3} = 228^{3} + 423^{3} = 255^{3} + 414^{3}

Sadly, Ramanujan became ill in England and within a year of returning to India, he died at the age of 32. During his short life, he independently compiled nearly 3,900 results. One formula he produced was an infinite series to evaluate π. Here, the series gets closer to the value of π the more terms that are taken. The first term in the series is ^{(9801√2)}/_{4412} which is already correct to 6 decimal places.

On his deathbed, Ramanujan produced a formula containing several new mathematical functions that came to him in a dream. Over 90 years later, researchers from the University of Florida reported that this idea could be used to help explain the behaviour of black holes. It will come as no surprise that Ramanujan’s most famous quote was, *“An equation means nothing to me unless it expresses a thought of God”*.

To recognise his contribution to Mathematics, the Indian Government declared that Ramanujan's birthday (22 December) should be celebrated annually as National Mathematics Day, and 2012 (the 125th anniversary of his birth) became National Mathematics Year. The *magic square* above has his birth date, 22/12/1887 along the top row.

The movie is great entertainment (I give it four pentagrams) and does not require an advanced degree in mathematics to enjoy it. There are a host of contrasting subjects highlighted with racial, cultural, social, religious and even intellectual issues explored all with a backdrop of the outbreak of World War One. It is based on the book by the same name and follows the true story of Ramanujan fairly accurately. There are a couple of liberties taken in the film with one being the apple tree in Trinity College where Isaac Newton supposedly watched an apple fall and ‘invented’ gravity. The tree is actually at Newton’s family home in Woolsthorpe Manor, Lincolnshire (and is still standing). Furthermore, some of the Cambridge scenes are filmed in Oxford.

As a footnote to this tale, the great GH Hardy was once asked by Paul Erdös what his own greatest contribution to mathematics had been during his lifetime. Hardy replied *“The discovery of Ramanujan”.*

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