Srinivasa Ramanujan: Numerical Imagination

http://www.rarityguide.com/articles/articles/181/1/Srinivasa-Ramanujan-Numerical-Imagination/Page1.html

By chronodev (Ron)

Published on 11/19/2009

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.

Introduction

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’s Life

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 him spread and his name started to become well know in the mathematics field in India. In May 1913, with the help of Rao Ramachandra, a famous Indian scholar and president of the Indian Mathematics society, Ramanujan secured a research scholarship at the University of Madras. In the same year, Ramanujan sends a letter to G. H. Hardy, professor in the University of Cambridge and a famous British Mathematician and a who has already established a name for himself in the mathematics literature. This letter ignites an interesting partnership between the two. In his letter, Ramanujan introduces himself to Hardy. He tries hard to explain why Hardy should listen to him despite Ramanujan not having University Education. Ramanujan emphasizes that while he had no University education, he still had undergone the ordinary school but then decided to strike a new path for himself, rather than trod through the conventional university curriculum. He mentions some theorems which he found himself that are not taught in the conventional university coursework. He also stresses how he has devoted himself to mathematics in all of his spare time. He states that local mathematicians of whom he had shown his investigation on divergent series had termed his results “startling”.

Ramanujan then proceeds to respond to a paper Hardy wrote titled Orders of Infinity. In particular, Ramanujan refers to this problem: Given a number x, how many prime numbers can be found that are less than x? For example, if x = 20, the answer would be 7 since there are 7 prime numbers less than 20: 2, 3, 7, 11, 13 and 17 and 19. According to Hardy, an expression for x has yet to be found. Ramanujan, however, managed to find a close enough approximation, and he asks Hardy to review it and other theorems, and help Ramanujan get his theorems published (Ramanujan mentions that he is poor and thus doesn’t have the funds to get his works out). 7This letter caught Hardy’s attention, however Hardy was still a bit skeptical at first since after all, how could a poor man from India with no academic training show such mathematical talent? Also, many of the theorems in the letter were provided with no proof.

Hardy asks Ramanujan for more explanation and proofs. Ramanujan responds with more explanations about his theorems, and asks Hardy to verify the results. From the tone of the letter I believe Ramanujan realizes that this is the opportunity he had been waiting for. Previous letters Ramanujan sent to different British mathematicians he received a negative answer, this was the only time in which a prestigious British mathematician took interest in Ramanujan’s work and asked for more information. Ramanujan asks Hardy for a chance, and also honestly confides to Hardy that his main motivation for this letter was having food on his plate, for if he is hungry it is harder for him to do the math he loves.

The two letters convince Hardy. Hardy had also consulted with some colleagues at his University and they were all impressed with Ramanujan’s mathematical talent. Hardy writes to Ramanujan that even though proofs are either missing, inaccurate or incomplete, his work appears to be right and is very remarkable. Ramanujan writes to Hardy that he sees him as a “sympathetic friend”. Hardy arranges to bring Ramanujan to England under a scholarship from the University of Cambridge.

Over the course of the next 6 years, Hardy and Ramanujan collaborated and published research together. They were a good complement to each other, for Ramanujan was an “Intuitive” mathematician who liked to find new theorems, while Hardy, according to his letters, cared a lot about “Rigorous proofs”. Hardy helped Ramanujan expand and prove many of his theorems.

At that period it also became apparent that Ramanujan is ill. In many of the letters which Ramanujan wrote during that period, he states that he was not feeling so well. He was suffering from Tuberculosis, a lung disease that was hard to treat at that period. Perhaps all of Ramanujan’s efforts and over straining himself with conjuring mathematical theorems catalyzed the deterioration of his health. On 1919 Ramanujan returned back to India/ on April 29, 1920 a letter was sent from the registrar of the University of Madras to Hardy, informing Hardy of Ramanujan’s death which occurred on April 26. Ramanujan had succumbed to his illness and died on the young age of 33.

Ramanujan seemed to be interested in theorems rather then in proofs thereof. He saw properties in numbers which others could not see. He wrote his theorems moving from one to another without devoting much to proofs. For example, in the first letter Ramanujan wrote to Hardy he included 9 pages full of different theorems.

As shown in this essay, Ramanujan was a genius with a intuition and imagination when it came to numbers. He was not heavy on proofs, but rather enjoyed writing examples and theorems. I will end this essay with a story told about him by Hardy. Ramanujan was lying sick in the hospital bed, when Hardy dropped for a visit. Hardy told Ramanujan how he took a cab to the hospital, and noticed that the cab number was “1729”. Hardy tried hard to think about any interesting properties about that number but could not do so, reaching the conclusion that 1729 is “rather a dull number” Ramanujan, immediately hearing that number, even though he was not in good health, immediately managed to come up with an interesting property for this number. He told hardy how the 1729 was not a dull number but rather a very interesting one. It is the smallest number which can be expressed as a sum of 2 cubes in 2 different ways. for example, is equal to 35, but there are no other two numbers of which the sum of cubes equals 35. However, for the number 1729, we have and both of which are equal to 1729. This is a perfect example illustrating Ramanujan’s magic and intuition with numbers.

Bibliography

1.Brendt, Bruce C. and Rankin, Robert A., Ramanujan: Letters and Commentary, Rhode Island: American Mathematical Society, 1995.

2.Clawson, Calvin C., Mathematical Mysteries: The Beauty and Magic of Numbers, New York: Perseus Books, 1999.

3.Kanigel, Robert, The Man Who Knew Infinity: A Life of the Genius Ramanujan, New York: Charles Scribner’s Sons, 1991.

4.Ranganathan, S. R., Ramanujan: The Man and the Mathematician, London: Asia Publishing House, 1967.

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’s Life

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 him spread and his name started to become well know in the mathematics field in India. In May 1913, with the help of Rao Ramachandra, a famous Indian scholar and president of the Indian Mathematics society, Ramanujan secured a research scholarship at the University of Madras. In the same year, Ramanujan sends a letter to G. H. Hardy, professor in the University of Cambridge and a famous British Mathematician and a who has already established a name for himself in the mathematics literature. This letter ignites an interesting partnership between the two. In his letter, Ramanujan introduces himself to Hardy. He tries hard to explain why Hardy should listen to him despite Ramanujan not having University Education. Ramanujan emphasizes that while he had no University education, he still had undergone the ordinary school but then decided to strike a new path for himself, rather than trod through the conventional university curriculum. He mentions some theorems which he found himself that are not taught in the conventional university coursework. He also stresses how he has devoted himself to mathematics in all of his spare time. He states that local mathematicians of whom he had shown his investigation on divergent series had termed his results “startling”.

Ramanujan then proceeds to respond to a paper Hardy wrote titled Orders of Infinity. In particular, Ramanujan refers to this problem: Given a number x, how many prime numbers can be found that are less than x? For example, if x = 20, the answer would be 7 since there are 7 prime numbers less than 20: 2, 3, 7, 11, 13 and 17 and 19. According to Hardy, an expression for x has yet to be found. Ramanujan, however, managed to find a close enough approximation, and he asks Hardy to review it and other theorems, and help Ramanujan get his theorems published (Ramanujan mentions that he is poor and thus doesn’t have the funds to get his works out). 7This letter caught Hardy’s attention, however Hardy was still a bit skeptical at first since after all, how could a poor man from India with no academic training show such mathematical talent? Also, many of the theorems in the letter were provided with no proof.

Hardy asks Ramanujan for more explanation and proofs. Ramanujan responds with more explanations about his theorems, and asks Hardy to verify the results. From the tone of the letter I believe Ramanujan realizes that this is the opportunity he had been waiting for. Previous letters Ramanujan sent to different British mathematicians he received a negative answer, this was the only time in which a prestigious British mathematician took interest in Ramanujan’s work and asked for more information. Ramanujan asks Hardy for a chance, and also honestly confides to Hardy that his main motivation for this letter was having food on his plate, for if he is hungry it is harder for him to do the math he loves.

The two letters convince Hardy. Hardy had also consulted with some colleagues at his University and they were all impressed with Ramanujan’s mathematical talent. Hardy writes to Ramanujan that even though proofs are either missing, inaccurate or incomplete, his work appears to be right and is very remarkable. Ramanujan writes to Hardy that he sees him as a “sympathetic friend”. Hardy arranges to bring Ramanujan to England under a scholarship from the University of Cambridge.

Over the course of the next 6 years, Hardy and Ramanujan collaborated and published research together. They were a good complement to each other, for Ramanujan was an “Intuitive” mathematician who liked to find new theorems, while Hardy, according to his letters, cared a lot about “Rigorous proofs”. Hardy helped Ramanujan expand and prove many of his theorems.

At that period it also became apparent that Ramanujan is ill. In many of the letters which Ramanujan wrote during that period, he states that he was not feeling so well. He was suffering from Tuberculosis, a lung disease that was hard to treat at that period. Perhaps all of Ramanujan’s efforts and over straining himself with conjuring mathematical theorems catalyzed the deterioration of his health. On 1919 Ramanujan returned back to India/ on April 29, 1920 a letter was sent from the registrar of the University of Madras to Hardy, informing Hardy of Ramanujan’s death which occurred on April 26. Ramanujan had succumbed to his illness and died on the young age of 33.

Ramanujan seemed to be interested in theorems rather then in proofs thereof. He saw properties in numbers which others could not see. He wrote his theorems moving from one to another without devoting much to proofs. For example, in the first letter Ramanujan wrote to Hardy he included 9 pages full of different theorems.

As shown in this essay, Ramanujan was a genius with a intuition and imagination when it came to numbers. He was not heavy on proofs, but rather enjoyed writing examples and theorems. I will end this essay with a story told about him by Hardy. Ramanujan was lying sick in the hospital bed, when Hardy dropped for a visit. Hardy told Ramanujan how he took a cab to the hospital, and noticed that the cab number was “1729”. Hardy tried hard to think about any interesting properties about that number but could not do so, reaching the conclusion that 1729 is “rather a dull number” Ramanujan, immediately hearing that number, even though he was not in good health, immediately managed to come up with an interesting property for this number. He told hardy how the 1729 was not a dull number but rather a very interesting one. It is the smallest number which can be expressed as a sum of 2 cubes in 2 different ways. for example, is equal to 35, but there are no other two numbers of which the sum of cubes equals 35. However, for the number 1729, we have and both of which are equal to 1729. This is a perfect example illustrating Ramanujan’s magic and intuition with numbers.

Bibliography

1.Brendt, Bruce C. and Rankin, Robert A., Ramanujan: Letters and Commentary, Rhode Island: American Mathematical Society, 1995.

2.Clawson, Calvin C., Mathematical Mysteries: The Beauty and Magic of Numbers, New York: Perseus Books, 1999.

3.Kanigel, Robert, The Man Who Knew Infinity: A Life of the Genius Ramanujan, New York: Charles Scribner’s Sons, 1991.

4.Ranganathan, S. R., Ramanujan: The Man and the Mathematician, London: Asia Publishing House, 1967.