Srinivasa Ramanujan (1887-1920) was one of India’s greatest mathematical figures. He was a child prodigy, with a natural instinct for mathematics. As though he could see how everything comes together. He was particularly interested in modular functions and number theory.
This essay is divided into two sections. In the first section I will report on Ramanujan’s life. I have decided to take a unique approach, and instead of reporting solely based on biographies (and biographies about Ramanujan are a-plenty) I will be analyzing letters he wrote. These letters have been stored in the National Archives in Delhi, the Archives of the State of Tamil Nadu, and in collections of various mathematicians with whom Ramanujan corresponded. They were compiled by Bruce Berndt and Robert Rankin in their book Ramanujan, Letters and Commentary. I believe these letters will give me a “first source” not only pertaining to Ramanujan’s life but also a more profound view of his thoughts, his personality. The letters are my main source for this essay, however I do use a couple of good biographies to enhance the information I got from the letters and to fill in some gaps. And while in this biographic section I will survey some of Ramanujan’s mathematical ideas, I will not go very deep into them. This I will do in the second section in which I will focus on a few of Ramanujan’s mathematical ideas. In the last section, I will use Mathematica to compute and verify some of Ramanujan’s theorems from the second section.
Ramanujan was born on the 22nd of December, 1887 in his Maternal grandmother’s house in Erode. Erode is a small town approximately 250 miles south west of Madras (see map). At the age of 1, Ramanujan’s mother took him to her home in Kumakonam (160 miles from Madras) where her husband, Kuppuswamy Srinivasa Ayingar was a clerk in a cloth merchant shop. Ramanujan inherited his first name, “Srinivasa” from his father.1 I was curious as to what the name “Srinivasa” means. A quick online search revealed that Srinivasa means “The Adobe of Sri”, the Goddess of Fortune in Sanskrit.
From 1892 to 1898 Ramanujan studied in several different primary schools in Kumbakonam, and in 1898 he enrolled in the town high school. at that point he had begun to realize his passion for mathematics- he became more and more absorbed in the world of numbers and mathematics. This came at a price- Ramanujan started neglecting all other school subjects, and in 1905 he dropped from school and ran away to Vizagapatnam, a town 400 miles north of Madras.
The one book that really influenced Ramanujan and ignited his mathematical flames was A Synopsis of Elementary Results in Pure and Applied Mathematics by G. S. Carr. The book was mostly a collection of approximately 6,000 theorems of Algebra, Calculus, Trigonometry and Analytical Geometry. Most of the theorems had no proofs, they were just listed “as-is”. Ramanujan proceeded to test many of those theorems, and come up with new theorems of his own. One effect this book had on Ramanujan is that it molded Ramanujan’s own style of listing theorems without resorting to proving them. Contrast this to mathematicians such as Euclid who were heavy on proofs.
In 1906, Ramanujan attempted to enter the University of Madras. He took the “First Arts Examination” which was an entry exam which would have allowed him to get enrolled in the University. However Ramanujan failed the exam (all topics, except for mathematics) and was denied admission. Ramanujan kept studying mathematics on his own, including continuing to analyze the theorems from Carr’s book. In a letter applying for a job as an accountant in the local port, Ramanujan reports “I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further… I have, however, been devoting all my time to Mathematics…”
Ramanujan met with different Indian scholars and by 1912 the word about...