Srinivasa Ramanujan

Indian mathematician

• Born: December 22, 1887; Erode, India
• Died: April 26, 1920; Kumbakonam, India

Self-taught Indian mathematician Srinivasa Ramanujan, a famous collaborator of the English mathematician G. H. Hardy, became a Fellow of the Royal Society in 1918 in recognition of his many contributions to the study of pure mathematics, including an approximate formula for deriving the number of partitions of an integer.

Also known as: Srinivasa Aaiyangar Ramanujan; Srinivasa Ramanujan Iyengar

Primary field: Mathematics

Specialties: Number theory; mathematical analysis

Early Life

On December 22, 1887, Srinivasa Ramanujan was born in Erode, a city in the southern Indian state of Tamil Nadu, into a family struggling with poverty. Ramanujan’s father worked as a bookkeeper for a textile merchant, and his mother earned a small income as a devotional singer in a Hindu temple. By the time he was seven, Ramanujan had survived a bout of smallpox and three of his younger siblings had died as infants.

Ramanujan received little early formal education apart from attending various pials—classes held by teachers on the raised porches in front of their homes—until enrolling in Kangayan Primary School. Here he studied Tamil (his native language), English, geography, and mathematics. After receiving the highest examination scores in his district in 1897, Ramanujan attended an English-language high school. He soon surpassed his teachers’ curricula and began teaching himself fundamental theorems in trigonometry, geometry, algebra, calculus, and differential equations, with the help of an advanced textbook he had acquired. The book, A Synopsis of Elementary Results in Pure and Applied Mathematics (1880), by G. S. Carr, contained more than a thousand theorems but almost no proofs. Ramanujan catapulted his mathematical education forward by working through them from scratch.

In 1904, already working on original solutions to problems in areas such as infinite series and prime number theory, Ramanujan won a scholarship to Government College in Kumbakonam, but he lost it and dropped out because of his neglect of his other subjects. His refusal to spend time on anything other than mathematics was also his undoing at Pachaiyappa’s College, where in both 1906 and 1907, he failed the final exam that would have granted him a college degree.

Unsuccessful in his early attempts at higher education, Ramanujan devoted all his free time to mathematics, working largely in isolation. He collected his research on topics such as continued fractions and divergent series in a string of notebooks that were published after his death.

Ramanujan was married in 1909 to S. Janaki Ammal; his mother had arranged the match. For several years he subsisted on donations from friends and mathematical colleagues, but in 1912, he began supporting his obsession with mathematical research by working as a clerk with the Madras Port Trust.

Life’s Work

During his twenties, Ramanujan gained several supporters who were interested in mathematics and became invested in supporting his mathematical career. Among these men was S. Narayana Iyer, Ramanujan’s manager at the Madras Port Trust. With Iyer’s encouragement, Ramanujan sent letters of introduction to a few mathematicians in England. Impressed by the creativity of the one hundred or so theorems Ramanujan had enclosed, and amazed by what Ramanujan had already managed to do without formal mentorship, the renowned mathematician G. H. Hardy arranged for Ramanujan to travel to Trinity College at the University of Cambridge. Ramanujan arrived at Cambridge in April 1914, remaining there until 1919. The period was tremendously productive for him: He published more than thirty papers in five years. (Ramanujan’s first professional article, a Journal of the Indian Mathematical Society paper that outlined several new properties of Bernoulli numbers, had been published in 1911.)

Ramanujan’s work during his Trinity years spanned a wide range of topics in pure mathematics, including continued fractions, elliptic functions, and the theory of partitions. One of his most original lines of research concerned a type of number that he called “highly composite.” A highly composite number has more divisors than any other number smaller than itself. For example, 12 is considered a highly composite number because it has six divisors—it can be divided evenly by the numbers 1, 2, 3, 4, 6, and 12—while no smaller number has as many. Ramanujan published a list of about one hundred highly composite numbers and proved that they had various interesting properties. In particular, he showed that when a highly composite number is expressed as the product of increasing prime numbers in exponential form (for example, 144 = 2⁴ × 3² ), the exponents always decrease in order.

Not all of Ramanujan’s ideas were correct, and his association with Hardy provided the young mathematician with an opportunity to recognize his failures. While he was still in India, for example, Ramanujan had developed what he believed was a groundbreaking formula for computing the number of prime numbers between 0 and any number x. His method did generate exact or almost exact numbers of primes for values of x smaller than 1,000, and it also did well for values of x up to several million. However, Ramanujan had not checked his formula on still higher values of x.

When Hardy examined Ramanujan’s calculations, he found that they relied on a false and unproven assumption about a certain property of the Riemann zeta function, a famous infinite series that is written in terms of a complex variable (in mathematics, “complex” refers to a variable that has both a real and an imaginary component). Because of this misstep, the errors that Ramanujan’s prime number theorem generated at high x values were large. As it turned out, this was an important failure, because it highlighted the limitations of Ramanujan’s brilliant but intuitive and sometimes breezy style of solving problems in mathematics. Working with Hardy helped Ramanujan to realize the importance of writing mathematical proofs using scrupulous, detailed reasoning that never skipped a step, and he slowly began to return to his old ideas and work through them with increased rigor.

In 1916, Ramanujan was granted a bachelor of science by research (the degree now known as a PhD) by Cambridge. In 1918, he became a full Fellow of the Royal Society.

Ramanujan struggled with illness throughout his stay in England, something that eventually prevented him from working as much as he desired. He returned to India in 1919 in poor health. Ramanujan planned to accept a position he had been offered as a university professor in Madras when he recovered, but this never came to pass. He died on April 26, 1920, of tuberculosis at only thirty-two years of age.

Impact

Ramanujan is widely regarded as a rare genius whose creativity and natural grasp of the fundamental relationships between numbers would have placed him in the top tier of mathematicians in any era. Although he spent much of his short life rediscovering already-known mathematical concepts, it is hard to overestimate the impact of his original thinking in number theory and mathematical analysis. The work on highly composite numbers for which he was awarded his Cambridge degree, for example, birthed an entirely new line of investigation; no one before him had ever treated these numbers as a special class or examined their properties.

Several mathematical theorems bear Ramanujan’s name. The “Ramanujan conjecture,” for example, is a statement about the prime values of an infinite series known as the tau function, and there are two equations relating to partition theory that are known as “Rogers-Ramanujan identities” (these were independently discovered by both Ramanujan and, some years earlier, by the British mathematician Leonard James Rogers).

Ramanujan’s impact on the world of mathematics did not end with his death; four of his notebooks, containing thousands of original theorems and ideas, were published posthumously. His journals continue to directly stimulate new research as generations of mathematical minds attempt to confirm or expand on their arguments. For example, Ramanujan never wrote any articles on hypergeometric series (a series in which the ratio of any two consecutive terms can be expressed as a ratio of two polynomial functions). However, he wrote many notebook entries on the subject, particularly regarding a specific group of seventeen infinite series that behave in surprising ways and that he called “mock theta functions.” These functions have since turned out to be relevant across a broad range of mathematical and scientific disciplines, including superstring theory and the chemistry of polymers. In addition, his theory of modular and elliptic functions has proven to have applications in modern cryptography.

Bibliography

Alladi, Krishnaswami, and Frank Garvan, eds. Partitions, q-Series, and Modular Forms. New York: Springer, 2012. Print. Collection of papers and survey articles, several of which address elaborations or expansions on Ramanujan’s work, such as his method in partition congruences and the Rogers-Ramanujan identities in hypergeometric series. Highly technical and suitable for advanced undergraduates and graduate students of pure mathematics.

Berndt, Bruce C. Number Theory in the Spirit of Ramanujan. Providence, RI: American Mathematical Society, 2006. Print. Explores some of Ramanujan’s favorite topics in number theory, including hypergeometry, the Eisenstein series, and elliptic functions. Demands a modest knowledge of number theory at a level provided by an introductory undergraduate course.

---, and Robert A. Rankin, eds. Ramanujan: Essays and Surveys. Providence, RI: American Mathematical Society, 2002. Print. Varied collection of articles covering aspects of Ramanujan’s personal life, including a biography of his wife, S. Janaki Ammal, and his mathematical contributions.

Kanigel, Robert. The Man Who Knew Infinity: A Life of the Genius Ramanujan. London: Abacus, 2006. Print. A biography that gives equal weight to conveying the details of Ramanujan’s life and introducing the fundamental meaning of his work, with technical explanations accessible even for nonmathematical readers.