# Mathematics at the Movies: Exploring the Life of Srinivasa Ramanujan

Although Indian mathematician Srinivasa Ramanujan may not be as well-known as other theorists, his many contributions to the field of mathematics are undoubtedly remarkable. With minimal formal education, Ramanujan made substantial advancements in number theory, continued fractions, elliptic functions, and infinite series. His contributions continue to impact present-day economics, computer development, and even the study of black holes. Ramanujan’s fascinating story – and partnership with Cambridge mathematician G.H. Hardy – is the focus of *The Man Who Knew Infinity*, a film starring Dev Patel (*Slumdog Millionaire*, *The Best Exotic Marigold Hotel*) as unlikely genius Ramanujan and Jeremy Irons as the prominent Hardy.

Srinvasa Ramanujan was born in 1887 in Erode, India. He demonstrated an early inclination towards mathematics, and studied geometric and arithmetic series as well as cubic equations in his teenage years. At age 15, Ramanujan discovered a copy of G.S. Carr’s *Synopsis of Elementary Results in Pure and Applied Mathematics* (vol. 2), a collection of theorems with brief proofs that he verified and used as inspiration for his own theorems. The following year, he received a scholarship to the University of Madras, but neglected his other subjects while pursuing mathematics and soon lost his scholarship. Despite his limited formal education and extended lack of employment, Ramanujan maintained his self-taught study of mathematics, exploring continued fractions, hypergeometric series, elliptic integrals, and the distribution of primes.

In 1911, after publishing his work in the *Journal of Indian Mathematical Society*, Ramanujan’s reputation grew in and around India. Eager to receive recognition from prominent mathematicians in England, he wrote two letters to E.W. Hobson and H.F. Baker but never received responses. In 1913, Ramanujan contacted leading English mathematician G.H. Hardy, a lecturer at Trinity College, Cambridge.

Ramanujan’s letter was wrought with fascinating, astounding formulas, leading Hardy to initially question the validity of Ramanujan’s work. Yet after discussing the formulas with colleague J.E. Littlewood, Hardy enthusiastically wrote back to Ramanujan. He secured Ramanujan a scholarship to the University of Madras and invited him to Cambridge to continue his work.

Ramanujan sailed to England in 1914, where he spent five years working with Hardy. His knowledge was both astounding and restricted. For example, while Ramanujan possessed an unequaled understanding of continued fractions, he had had minimal exposure to modern mathematical developments and a limited grasp of mathematical proofs. While in England, he continued working with elliptic integrals and hypergeometric series, in addition to the Riemann series, functional equations of the zeta function, and his own theory of divergent series. One of Ramanujan’s most prominent developments pertained to the partition of numbers (i.e. how many ways a positive integer can be written as the sum of positive integers). He also received a B.S. by Research from Cambridge. Ramanujan’s work, often in collaboration with Hardy, was published in multiple English and European journals.

In 1918, Ramanujan was elected fellow of the Royal Society, and became the first Indian fellow at Trinity Collge. Unfortunately, his years in England were tainted by dietary restrictions due to World War I and frequent illnesses, including a nasty round of tuberculosis in 1917. Ramanujan returned to India in 1919 and died the following year, at age 32. He left behind three notebooks filled with formulas, which mathematicians have continued to prove after his death. One of his last discoveries, mock theta functions, now has applications to a larger group of mock modular forms.

*The Man Who Knew Infinity* was written and directed by Matthew Brown, and is based on a biography of the same name by Robert Kanigel. The film opens in U.S. theaters today.

Lina Sorg is the associate editor of SIAM News. |