Srinivasa Ramanujan (1887-1920)- The Mathematical Genius

Introduction: Srinivasa Ramanujan, an Indian mathematician, played a pivotal role in shaping the landscape of mathematics in the early 20th century. His exceptional talents and distinctive insights have solidified his esteemed position in the realm of mathematics. This piece carries considerable importance for individuals aspiring to excel in the UPSC IAS exam. Elevate your likelihood of success by joining UPSC Coaching.

In this article, we talk about Srinivasa Ramanujan’s early life, his amazing journey in math, the time he spent in Cambridge, England, the cool math stuff he did, and the lasting mark he made.

The Early Life of Srinivasa Ramanujan

• Born on December 22, 1887, in Erode, Tamil Nadu, India, Ramanujan came from a low-income family with his father working as a clerk.
• Even as a child, Ramanujan displayed a remarkable aptitude and passion for mathematics, solving complex problems for enjoyment without any formal training in the subject.
• Relying on self-study, he borrowed math books from the local library, immersing himself in their content due to his fascination with numbers and patterns.
• At the age of 11, Ramanujan encountered Carr’s “A Synopsis of Elementary Results in Pure Mathematics,” a book that exposed him to advanced mathematical concepts and significantly influenced his mathematical journey.
• Despite his excellence in mathematics, Ramanujan struggled in conventional schooling, focusing more on math than other subjects and barely passing examinations.
• At 16, he discovered the groundbreaking work of mathematical analysts like Euler, Gauss, and Jacobi, introducing him to areas like infinite series, continued fractions, and elliptic integrals where he would later make significant contributions.
• Despite his innate genius and self-taught mathematical skills, Ramanujan lacked formal training in advanced mathematics during his early years in India, posing challenges for his future endeavours.
• Ramanujan developed his mathematical ideas in isolation during his youth, and it wasn’t until he turned 22 that his genius gained recognition beyond India.

Death of Srinivasa Ramanujan

Srinivasa Ramanujan died on April 26, 1920, at the age of 32. The cause of his death was complications related to tuberculosis, a bacterial infection that primarily affects the lungs. Ramanujan had been suffering from health issues, and tuberculosis was a contributing factor to his premature death. Despite his brilliance in mathematics, his life was cut short by health challenges.

The Mathematical Journey of Srinivasa Ramanujan

1. Ramanujan, a self-taught mathematical genius in India, developed his skills through extensive self-study, exploring advanced concepts in books.
2. Keeping detailed notebooks, he recorded numerous original mathematical identities and theorems by the age of 20.
3. In 1912, Ramanujan wrote to G. H. Hardy, an English mathematician, sharing his theorems. Recognizing Ramanujan’s talent, Hardy arranged for his journey to England.
4. At 26, in 1914, Ramanujan travelled to England, collaborating with Hardy and other mathematicians to verify and guide his work.
5. In England, he made significant contributions to mathematical analysis, number theory, infinite series, and continued fractions, pioneering the study of mock theta functions.
6. Despite his achievements, Ramanujan’s health declined due to malnutrition and illness, compounded by gaps in formal training and understanding of rigorous proofs.
7. Before his death at 32 in 1920, Ramanujan proved nearly 3,900 theorems, many of which were groundbreaking. His notebooks contained numerous unproven claims and conjectures.
8. After his death, mathematicians have worked to prove theorems from Ramanujan’s notebooks, with around 2,000 results confirmed correct. Some conjectures remain unproven over a century later.
9. Revered as one of history’s greatest mathematical geniuses, Ramanujan’s intuitive leaps and remarkable results continue to influence the direction of mathematics research.

Srinivasa Ramanujan Contributions

• Between 1914 and 1914, while Ramanujan stayed in England, he and Hardy collaborated on more than a dozen research papers.
• Over three years, Ramanujan authored approximately 30 research papers.
• Hardy and Ramanujan introduced a novel approach, now known as the circle method, to establish an asymptotic formula for a specific function.
• Ramanujan’s inaugural published paper, a 17-page piece on Bernoulli numbers, appeared in 1911 in the Journal of the Indian Mathematical Society.
• A noteworthy outcome of the collaboration between Hardy and Ramanujan was a formula for the number p(n), representing partitions of a number ‘n.’

Achievements of Srinivasa Ramanujan

• At 12 years old, he read advanced books on Plane Trigonometry and Pure and Applied Mathematics, well beyond what high school students typically study.
• In 1916, he earned a Bachelor of Science degree “by research” from Cambridge University.
• In 1918, he became the first Indian to be honoured as a Fellow of the Royal Society.
• The Ramanujan Journal, launched in 1997, publishes work influenced by Ramanujan in various mathematical areas.
• The year 2012 was declared the National Mathematical Year in India, commemorating the 125th birth year of the great mathematician.
• Since 2021, December 22, his birth anniversary, is celebrated as National Mathematicians Day in India.
• The goal is to emphasize the importance of math, especially for young people, to inspire their interest and understanding of its significance for the country’s future.

Life in Cambridge, England

• In 1914, at the age of 26, Ramanujan travelled to England with a scholarship arranged by G.H. Hardy at Trinity College, Cambridge.
• Initially, Ramanujan faced challenges adapting to the cold, wet English weather, different cultures, and unfamiliar food, affecting his health.
• Despite these struggles, he excelled intellectually and mathematically at Cambridge, collaborating with Hardy and other renowned mathematicians.
• Ramanujan made significant contributions to the theory of partitions, elliptic functions, continued fractions, and infinite series, and he pioneered the study of mock theta functions.
• His innovative insights and intuition astounded colleagues, with Hardy describing him as a “magical” mathematician.
• Ramanujan published approximately 35 mathematical papers in England, showcasing groundbreaking and ahead-of-his-time results.
• Despite his success, he faced challenges in formal mathematical training, as his approach differed fundamentally from rigorous methods.
• Ramanujan’s health deteriorated due to malnutrition and illness, leading to his untimely death in 1920 at the age of 32.
• Despite his short life, he proved nearly 4,000 theorems, leaving a lasting legacy in number theory and mathematical analysis.
• Ramanujan is revered as one of the greatest mathematical minds of the 20th century, with his unproven claims continuing to inspire further research.

Ramanujan’s Work in Mathematics

Ramanujan significantly contributed to various branches of mathematics:

• Number Theory
  – Identified properties of numbers and proved theorems, particularly focusing on number sets and special numbers like primes.
  – Many of his findings in number theory are still relevant today, especially in fields such as cryptography.
• Series
  – Explored various infinite series types, including geometric and arithmetic progressions.
  – Discovered series representations for fundamental constants like pi and the exponential function.
  – Developed unique methods for summing series, surpassing the capabilities of other mathematicians.
• Continued Fractions
  – Examined the iterative process of expressing fractions as successive fractions.
  – Utilized continued fractions to derive expressions for significant constants such as e and pi.
  – Discovered intriguing connections between continued fractions and modular forms.
• Modular Functions
  – Gained insights into number patterns described by modular functions, which exhibit repetitive patterns when numbers are increased by specific amounts.
  – Formulated innovative applications of modular functions in various mathematical contexts.
• Elliptic Functions
  – Studied functions enabling the mapping of a circle onto another curve.
  – Applied the properties of elliptic functions to series, integrals, and modular forms.
  – Many of his groundbreaking results in elliptic functions remain renowned and influential in contemporary mathematics.

Legacy of Srinivasa Ramanujan

• Made significant contributions to mathematical analysis, number theory, infinite series, and continued fractions.
• Pioneered the study of mock theta functions.
• Despite his untimely death at age 32, proved nearly 4,000 theorems, many of which were original and groundbreaking.
• Notebooks contain hundreds of unproven claims and conjectures, with mathematicians still working to prove them over a century later.
• Approximately 2,000 of his conjectures have been proven correct to date.
• Regarded as one of the greatest mathematical geniuses, shaping the direction of research in mathematics.
• A self-taught genius with humble beginnings and a lack of formal training, Ramanujan’s story inspires the potential of the human mind.