Srinivasa Ramanujan, born on December 22, 1887, in Erode, India, is celebrated as one of the greatest mathematical geniuses of all time. Despite limited formal education and facing significant personal and professional challenges, Ramanujan made groundbreaking contributions to mathematical analysis, number theory, infinite series, and continued fractions. His work has had a profound and lasting impact on mathematics.
Early Life and Education
Ramanujan was born into a Brahmin family in the small town of Erode in the Tamil Nadu state of India. His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop, while his mother, Komalatammal, was a housewife and sang at a local temple. From a young age, Ramanujan displayed an extraordinary aptitude for mathematics. At the age of 10, he entered the Town Higher Secondary School in Kumbakonam, where he first encountered formal education in mathematics.
In 1903, at the age of 16, Ramanujan obtained a copy of George Shoobridge Carr’s book, “A Synopsis of Elementary Results in Pure and Applied Mathematics.” This book, a collection of about 5,000 theorems, served as his primary mathematical resource. Despite its terse and condensed format, which provided little explanation or proof, Ramanujan was able to teach himself advanced mathematics through it. This self-study laid the foundation for his future work and helped him develop his own theories and ideas.
Early Struggles
Although Ramanujan’s talent was evident, his obsession with mathematics led to academic challenges. He neglected other subjects, failing his exams at the Government College in Kumbakonam and later at Pachaiyappa’s College in Madras (now Chennai). These setbacks prevented him from obtaining a degree, which in turn hindered his ability to secure a stable job. Nevertheless, Ramanujan continued to pursue his mathematical work independently.
In 1909, Ramanujan married Janaki Ammal, who was just ten years old at the time, as was customary in India. The marriage, however, did little to alleviate his financial difficulties. To support his family, Ramanujan sought clerical work, eventually finding a modest position in the Madras Port Trust office. It was during this period that he began to correspond with prominent mathematicians, seeking recognition and validation for his work.
Breakthrough and Correspondence with G.H. Hardy
Ramanujan’s breakthrough came in 1913 when he wrote to G.H. Hardy, a leading mathematician at the University of Cambridge in England. In his letter, Ramanujan included several of his original results, many of which were novel and unconventional. Initially skeptical, Hardy soon realized that Ramanujan’s work was not only correct but also profound. He described Ramanujan as a “natural genius” and, along with his colleague J.E. Littlewood, sought to bring Ramanujan to Cambridge.
In 1914, after overcoming numerous obstacles, including financial and cultural challenges, Ramanujan traveled to England. At Cambridge, Hardy and Ramanujan collaborated on various mathematical projects. Hardy introduced Ramanujan to the rigor and structure of formal mathematics, while Ramanujan’s intuitive and original approach to problem-solving greatly impressed Hardy. Their partnership resulted in several important papers and the discovery of new theorems.
Contributions to Mathematics
Ramanujan’s contributions to mathematics were vast and diverse, covering several areas. Some of his most notable achievements include:
- Partition Theory: Ramanujan made significant contributions to the theory of partitions, which deals with the ways in which numbers can be expressed as the sum of other numbers. His work in this area, including the famous Ramanujan-Hardy partition formula, has had a lasting impact on combinatorics and number theory.
- Ramanujan Prime and Ramanujan Theta Function: Ramanujan introduced the concept of Ramanujan primes and developed the Ramanujan theta function, both of which have become fundamental in analytic number theory.
- Mock Theta Functions: One of Ramanujan’s most enigmatic contributions was his discovery of mock theta functions, which he introduced in a letter to Hardy shortly before his death. These functions were later found to play a crucial role in various areas of mathematics and theoretical physics.
- Highly Composite Numbers: Ramanujan studied highly composite numbers, which are integers with more divisors than any smaller number. His work in this area was pioneering and influenced subsequent research in number theory.
- Infinite Series and Continued Fractions: Ramanujan’s exploration of infinite series and continued fractions led to numerous new identities and theorems, many of which continue to be studied and applied in contemporary mathematics.
Personal Struggles and Health
Despite his mathematical brilliance, Ramanujan faced significant personal and health challenges. The cold and damp climate of England, combined with his strict vegetarian diet, took a toll on his health. He struggled with various illnesses, including tuberculosis and severe vitamin deficiencies. His health deteriorated rapidly, and he spent several periods in hospitals and sanatoriums.
In 1919, due to his failing health, Ramanujan returned to India. Unfortunately, his condition did not improve, and he continued to suffer from recurring illnesses. Despite his poor health, Ramanujan remained dedicated to his work, continuing to produce new mathematical results until his final days.
Legacy and Impact
Srinivasa Ramanujan passed away on April 26, 1920, at the age of 32. Although his life was tragically short, his contributions to mathematics have had a profound and lasting impact. His work laid the groundwork for numerous advancements in various fields of mathematics, including number theory, combinatorics, and mathematical analysis.
Ramanujan’s legacy is commemorated in numerous ways. In India, his birthday, December 22, is celebrated as National Mathematics Day. The Ramanujan Institute for Advanced Study in Mathematics, established in 1950 at the University of Madras, honors his contributions. Additionally, the Ramanujan Prize, awarded annually by the International Centre for Theoretical Physics (ICTP) and the Department of Science and Technology (India), recognizes outstanding work by young mathematicians from developing countries.
Ramanujan’s story is also an inspiring testament to the power of innate talent and perseverance. Despite facing immense obstacles, his passion for mathematics and his remarkable intuition enabled him to make significant discoveries that continue to influence and inspire mathematicians around the world.