Ramanujan – The Man Who Knew Infinity

Srinivasa Ramanujan, born on December 22, 1887, in Erode, Tamil Nadu, India, is one of the most celebrated mathematical geniuses in history. Despite a brief life, he made monumental contributions to the field of mathematics, with discoveries that are still being explored and understood today. His work spanned number theory, infinite series, continued fractions, and complex analysis. What makes Ramanujan’s story even more compelling is that he achieved much of his mathematical insight with little formal training, relying instead on his innate talent, intuition, and passion for mathematics. His legacy continues to inspire mathematicians and scholars around the world.

Early Life and Education

Ramanujan was born into a Brahmin family in Tamil Nadu. His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop, while his mother, Komalatammal, was a homemaker who also sang in a local temple. His early life was spent in Kumbakonam, a town known for its educational and religious significance. Ramanujan’s brilliance became apparent at an early age, especially in mathematics, where he showed extraordinary talent.

He started formal schooling at the age of five and quickly excelled in mathematics. However, Ramanujan faced challenges in formal education. His interest in other subjects, like English and history, was minimal, and he often neglected them in favor of his passion for numbers. At the age of 15, he came across G. S. Carr’s A Synopsis of Elementary Results in Pure and Applied Mathematics, a book that listed thousands of mathematical theorems without much explanation. This book became a turning point in Ramanujan’s life, serving as his primary reference as he independently developed many mathematical concepts and theorems.

However, his single-minded focus on mathematics led to his failure in other subjects, and he was unable to pass his college exams. This ended his formal education and forced him into a life of poverty and struggle. Despite these hardships, Ramanujan never gave up his pursuit of mathematics.

Breakthrough and Correspondence with G.H. Hardy

Ramanujan continued to develop his mathematical ideas independently, writing down formulas and theorems in his notebooks. He worked in isolation, unaware of the modern developments in mathematics that were taking place in Europe and the United States. His approach to mathematics was highly intuitive, and he often arrived at results that took others years to prove rigorously.

In 1911, he published his first paper in the Journal of the Indian Mathematical Society. This work, on Bernoulli numbers, caught the attention of mathematicians in India. However, Ramanujan sought recognition beyond his homeland, and in 1913, he began a correspondence with G.H. Hardy, a prominent mathematician at the University of Cambridge in England. Ramanujan sent Hardy a letter filled with theorems and results he had discovered, many of which were completely unfamiliar to Hardy.

Hardy was initially skeptical of the letter because of its unconventional style and lack of formal proofs, but after carefully examining the results, he realized that he was dealing with an extraordinary mathematical mind. As Hardy later said, “I had never seen anything in the least like them before. A single look at them is enough to show they could only be written down by a mathematician of the highest class.”

Hardy arranged for Ramanujan to come to Cambridge, despite the latter’s initial reluctance due to cultural and religious reasons. Ramanujan eventually agreed, and in 1914, he traveled to England, where he began working with Hardy and other leading mathematicians.

Work in Cambridge

Ramanujan’s time at Cambridge proved to be immensely productive. He collaborated with Hardy and produced groundbreaking work in various fields of mathematics, particularly in number theory, modular forms, and partitions. Hardy helped Ramanujan refine his proofs and mathematical rigor, while Ramanujan’s intuition and creativity opened up new vistas in mathematics.

One of Ramanujan’s most famous contributions during this time was the development of the Ramanujan-Hardy number, 1729, now known as the “taxicab number.” The story behind this number illustrates Ramanujan’s intuitive genius. When Hardy visited him in the hospital and remarked that 1729 was a rather dull number, Ramanujan immediately corrected him, noting that it is the smallest number expressible as the sum of two cubes in two different ways:

[
1729 = 1^3 + 12^3 = 9^3 + 10^3
]

This anecdote is a testament to Ramanujan’s incredible ability to see patterns and relationships in numbers that were invisible to others.

During his time at Cambridge, Ramanujan also made significant advances in the field of partition theory. He developed intricate formulas to calculate the number of ways an integer could be expressed as the sum of other integers, a problem that had fascinated mathematicians for centuries. His work on modular forms and continued fractions, including the famous Ramanujan theta function, laid the groundwork for future developments in these areas, influencing fields as diverse as cryptography and string theory.

Mathematical Intuition

Ramanujan’s approach to mathematics was unlike that of most Western mathematicians of his time. His results were often based on deep intuition rather than formal derivations. This led to tensions between Ramanujan and his peers, who were more accustomed to rigorously proven theorems. While many of his results were later proven correct, some were so far ahead of their time that mathematicians only fully understood them decades later.

One of Ramanujan’s great strengths was his ability to work with infinite series and continued fractions, which are fundamental to modern mathematical analysis. His formulas for infinite series, such as his formula for pi, have found applications in diverse areas, including computational mathematics.

One of the most remarkable aspects of Ramanujan’s mathematical genius was his intuitive grasp of the infinite. He had an almost mystical relationship with numbers, believing that mathematical truths were divinely inspired. In his words, “An equation for me has no meaning unless it expresses a thought of God.” This belief in the divine nature of mathematics was rooted in his deeply religious upbringing and remained with him throughout his life.

Illness and Death

Unfortunately, Ramanujan’s health began to deteriorate during his time in England. The harsh climate, coupled with his vegetarian diet and cultural isolation, took a toll on his body. Despite his declining health, Ramanujan continued to work feverishly, producing some of his most brilliant results during this period.

In 1919, Ramanujan returned to India, but by then, his health had worsened considerably. He continued to work on mathematics until his death, producing important results even from his sickbed. On April 26, 1920, at the age of 32, Ramanujan passed away, leaving behind an extraordinary mathematical legacy.

Legacy

Though his life was tragically short, Ramanujan’s contributions to mathematics are immeasurable. His work continues to influence many areas of modern mathematics, and his notebooks, filled with theorems and conjectures, remain a treasure trove for mathematicians. Some of his discoveries have only recently been fully understood, and his influence extends beyond number theory into areas such as physics, computer science, and engineering.

In 2012, the film “The Man Who Knew Infinity,” based on the biography of the same name by Robert Kanigel, brought Ramanujan’s life and work to a broader audience, further cementing his status as one of the most remarkable mathematical minds of all time.

Conclusion

Srinivasa Ramanujan’s genius was unparalleled. His contributions to mathematics, despite the challenges he faced, have left an indelible mark on the field. His work, grounded in a unique blend of intuition and raw talent, continues to inspire mathematicians worldwide. Ramanujan’s life is a testament to the power of human potential, even in the face of adversity. His story is not just one of mathematical brilliance but also of passion, perseverance, and an unwavering belief in the beauty of numbers.