National Mathematics Day and Srinivasa Ramanujan

Why India marks 22 December

At a ceremony at the University of Madras in Chennai on 26 December 2011, held for the 125th anniversary of Ramanujan's birth, the Prime Minister announced that 22 December would be National Mathematics Day and that 2012 would be National Mathematics Year. Since then schools, colleges and institutes across India hold lectures, quizzes, exhibitions and olympiad-style competitions on or around the day. It is an observance, not a public holiday.

The day has two aims: to honour Ramanujan and to encourage students to take up mathematics. Kumbakonam, the town where he grew up, has special weight: the SASTRA Ramanujan Prize, awarded each year since 2005 to a young mathematician working in areas he influenced, is presented there around his birthday, and his family home in Sarangapani Street is a small museum.

Ramanujan's life in key dates

The letter to Hardy

By 1912 Ramanujan had filled notebooks with results but had no degree and no one in India who could fully judge them. He wrote to several Cambridge mathematicians. Two did not help. The third, Godfrey Harold Hardy, read the letter of 16 January 1913 over breakfast, thought at first it might be a hoax, then went through it that evening with his colleague J. E. Littlewood and decided it was the work of a mathematician of the highest class. The letter opens:

The theorems came without proofs. Some were already known, a few were wrong (his claim of a near-exact formula for counting primes was one), and many were new and remarkable. Hardy later said some of the formulas “defeated me completely”: he had never seen anything like them. He arranged for Ramanujan to come to Cambridge, and the two worked together for five years. Their 1918 paper on partitions, the number of ways a whole number can be written as a sum of positive whole numbers, gave a formula of startling accuracy.

The partnership was not easy. Hardy, an atheist, wanted rigorous proofs; Ramanujan, a devout Brahmin, worked by intuition and later said that ideas came to him in dreams from the family goddess Namagiri, according to friends' accounts. In England he struggled to keep a strict vegetarian diet during wartime shortages, and he was often cold and lonely. A friend recalled him saying that an equation had no meaning for him unless it expressed a thought of God, though that remark is known only from a memoir published decades later.

The story of 1729

The most famous Ramanujan story comes from Hardy himself, in his obituary of Ramanujan (1921). When Ramanujan was ill in a nursing home in Putney, Hardy visited him and mentioned that his taxi had the number 1729, which seemed to him rather dull. Ramanujan answered at once that it was a very interesting number: the smallest number that can be written as the sum of two cubes in two different ways.

• 1729 = 1³ + 12³ = 1 + 1728
• 1729 = 9³ + 10³ = 729 + 1000

Mathematicians now call 1729 the Hardy and Ramanujan number, and the smallest numbers that are sums of two cubes in n different ways are called taxicab numbers. The point of the story is not that Ramanujan memorised a fact. Results about sums of cubes appear in his notebooks from years earlier, so he had met 1729 as part of his own work.

The notebooks and the lost notebook

Ramanujan left three notebooks, mostly written in India before 1914, containing about 3,000 to 4,000 results with little or no proof. A facsimile edition was published by the Tata Institute of Fundamental Research in Bombay in 1957. Proving or checking every entry took the American mathematician Bruce Berndt and his collaborators about twenty years; Ramanujan's Notebooks, Parts I to V, appeared between 1985 and 1998.

In 1976 George Andrews found a bundle of more than 100 pages of Ramanujan's work from his last year in India among the papers of G. N. Watson in the library of Trinity College, Cambridge. This “lost notebook” is full of mock theta functions, a new kind of function he described in his last letter to Hardy in January 1920. Their full meaning was only understood in the 2000s, and they are now used in areas from number theory to the physics of black holes.

Activities for National Mathematics Day

• Taxicab hunt: students list cubes up to 12³ and search for numbers that are sums of two cubes in two ways. Can they confirm 1729 is the smallest?
• Partitions: count the ways to write 1, 2, 3, 4 and 5 as sums (4 has five: 4, 3+1, 2+2, 2+1+1, 1+1+1+1). Then try 6 and 7 and compare with the known values, 11 and 15.
• Highly composite numbers: which numbers up to 100 have more divisors than any smaller number? (1, 2, 4, 6, 12, 24, 36, 48, 60.) This is the topic of his Cambridge degree.
• Read the letter: give students the opening of the 16 January 1913 letter and ask what Ramanujan does to persuade a stranger to read on.
• Interview Ramanujan: groups prepare questions about Kumbakonam, the letter, Cambridge and his illness, and practise them by voice with an AI recreation of Srinivasa Ramanujan on Yuga. It is built from his letters and papers but it is not Ramanujan and can be wrong about mathematics too, so check every formula against a published source.

Our questions to ask Srinivasa Ramanujan list has more, Indian scientists in history places him among others, and Pi Day and Archimedes shows how his 1914 series for pi are still used by computers.