Aron Lum
Honors Collegium 41
Prof. R. Fioresi
December 8, 1999
Ramanujan, The Prince of Intuition
Some are born with the ability to tell a whole story with one picture and others the ability to paint a picture with a few lines of poetry. Others still can create harmonies to soothe the most savage beast, but the gift given to one particular Indian is like no other ever seen. Srinivasa Ramanujan Iyengar had the ability to see numbers as no other had. He could see patterns in them and apply them to higher mathematical theory. Bringing back pride to the once great Indian Mathematical tradition, Ramanujan’s theorems are still being investigated to this day and are being used in everything from particle physics to finding a cure for cancer.
Born in 1887 on the ninth day of the Indian month Margarirsha, December 22, Srinivasa Ramanujan Iyengar was close to his mother, Komalatammal from the start. She made the trek to her hometown to give birth to her first born as per Indian tradition and carried her son back to Kumbakonam, the town where he would spend the next twenty years of his life. Although his name has three parts, he would go only by Ramanujan his entire life, except in formal writings, where he would include the initial "S." in the front. Srinivasa was merely his father's name and Iyengar referred to the caste of South Indian Brahmins he was born into.
Ramanujan was a very silent child and his mother worried about his intelligence, that is, until he learned the 12 vowels, 18 consonants, and 216 combinations of the two in the Tamil alphabet. At the age of five, he was enrolled in school but did not particularly enjoy it. At such a tender age, he showed the sort of odd brilliance that he showed his entire life. Still on the quiet side, he contemplated questions such as "Who was the first man in the world?" and "How far is it between clouds?" Through years of success Ramanujan began to like school and earned a scholarship to Kumbakonam's Government College, but not before he fell in love with math.
Shortly before Ramanujan graduated from high school, a copy of George Carr's A Synopsis of Elementary Results in Pure and Applied Mathematics fell into his hands. It contained no form of mathematical proof as it was merely a study guide for the Tripos, essentially an entrance exam for British Colleges, but it triggered something in Ramanujan, flipped on a switch, opened his natural tendency for numbers. Ramanujan naturally curious, wondered how some of the theorems in Carr’s book were derived, and set about trying to prove them on his own, deriving a unique notation filled both with several symbols similar to those used in mainstream mathematics and symbols completely alien to any mathematician. In investigating these theorems, Ramanujan led off onto tangents and derived theorems of his own. Filling notebooks upon notebooks with his mathematical investigation, he ignored all his other subjects and received failing grades in any subject that was not mathematics. Due to his poor marks, the college threatened to take his scholarship away, and he was put on a probationary period. He managed to go to his other classes just enough to get a certificate of attendance, but that wasn’t enough. He concerned himself only with math, and upon receiving word he had lost his scholarship, Ramanujan ran away from home.
Ramanujan was never accustomed to failure and in fact could not reconcile it when it came to math. As a child in grade school, his whole class knew that he would score the highest on any arithmetic exam, and he expected nothing less. On one particular exam, he scored a 42 out of 45, a very respectable grade, but his friend K. Sarangapani Iyengar scored a 43 out of 45. Ashamed, Ramanujan refused to talk to him and ran home crying to his mother. When in high school, Ramanujan was ecstatic after finding a way to write the trigonometric functions in a way unrelated to right triangles only to find he was 150 years too late, as the great Euler had already proved this. Mortified, he stashed away his findings in the roof of his house so no one could find them.
Disappointed by his failure, Ramanujan was determined to find a scholarship that didn’t depend on him passing exams. He made his way to Pachiyappa’s college, the best college in Southern India. There, he showed his notebooks to a math professor, who, so impressed by its contents begged he school’s principal to grant Ramanujan a scholarship, this he did. He got off to a promising start at his new school. The first day of class, his professor proved an equation in two dozen steps only to be upstaged by his new student. Ramanujan proved it in four. From that day on, the professor would occasionally ask if Ramanujan had anything to offer. In addition to this, Ramanujan became very close to one of the department’s senior professor’s and would often work with him late into the night.
Despite the promising start, Ramanujan still had the problem of taking other courses. Physiology particularly bothered Ramanujan. His scores sometimes reached as low as 10, and sometimes lower. Since one needed to pass all the subjects to receive the First Arts Degree, Ramanujan had little to no chance of passing and thus no way of getting accepted to the University of Madras. In need of something to do, he went to friends and family, asking if they had any books that needed to be balanced or someone that needed to be tutored. In luck, he found a few students studying for the F.A. exam that lacked a little in the area of calculus.
Although it seemed as though tutoring others in math would flow naturally, Ramanujan would often go on philosophical tangents about math when all the student needed was to know how to integrate a certain function. In finding the height of a wall using trigonometry, one student claimed he veered off into a discussion of how high the wall would look in relation to an elephant, or how high it would seem to an ant. Ramanujan was always fascinated with how math could be applied to other areas. Not only in the obvious ways, he was fascinated with astronomy, but he also related it to spiritual areas, as he was extremely religious. In particular, Ramanujan was interested in the function 2n-1. When n=0, it evaluates to nothingness. When n=1, it evaluates to unity or God. At n=2, it evaluates to the trinity. At n=3, seven, symbolizing the Saptha Rishis in Indian religion, and so on.
Regardless, Ramanujan was without a job and did nothing but math all day. His mother could think of only one way to remedy his apparent inactivity, so she went out found a young girl for him to marry. Janaki was only nine years old when they married, so she did not move in with Ramanujan for three years. His behavior did not change at first, but a sense of urgency soon hit him. In Indian culture, there were believed to be four stages of life, in the first stage, one was a student. In the second and longest stage, one was responsible for himself and his family. In the third, one started to throw off the responsibility and seek serenity. In the fourth and final stage, one was free of all obligations and sought spiritual fulfillment. Though Ramanujan would have preferred to be in the final stage, he was in the second stage, and need to find a job. Unfortunately, Ramanujan developed a hydrocele, a swelling of the scrotal sac. Surgery was fairly simple and without risk, but the family had no money to pay for it. In 1910, a village doctor volunteered his services and performed the surgery for free.
Once healthy, Ramanujan went around Southern India selling himself to anyone that would take him. He stayed with friends as he searched for a job. He made his way from town to town, offering his notebooks as his credentials, but to no avail. Everyone he went to examined his notebooks and came to the same conclusion. Either he was a fraud and no idea what his notebooks contained or, he was a genius of incomparable caliber. Unfortunately, they all took the road of caution and opted not to offer him a position in case he was a fraud. In the face of this adversity, a lesser man may have crumbled and given up math for a less desirable position, but Ramanujan stayed loyal. He continued his mathematical investigations and worked out problems in The Journal of the Indian Mathematical Society and published a problem concerning nested radicals that no one could solve for three months. In 1911, Ramanujan published his first paper, which he called "Some Properties of Bernoulli Numbers."
The Bernoulli numbers, named for the Swiss mathematician Jacob Bernoulli, were directly related to the coefficients of the Eulerian constant when stated in an infinite series. It was probably fitting that Ramanujan’s first paper dealt with such numbers, as his favorite subject seemed to be that of infinite series as they were dominant in his famous notebooks. Ramanujan discovered that the denominators of Bernoulli numbers, as Bernoulli numbers were always wrtten as fractions, were divisible and also developed new ways of calculating Bernoulli numbers based on previous ones. Ramanujan also stated in his paper that:
"if n is even but not equal to zero,
Throughout the whole paper, he draws conclusions and connects unconnected things with amazing accuracy. Ramanujan merely relied on his intuition to draw these conclusions, but there are also some stories that he had visions in his sleep of the god Namagiri coming to him in his sleep with these equations. Later mathematicians proved that most of these equations, and many of his theorems were justified, but in this paper, property (i) does not hold for B20. But that was not nearly as important as what being published in a respected math journal meant to Ramanujan.
Upon being published, Ramanujan now had a greater chance at getting a job. With a recommendation from one of his math professor’s relating Ramanujan’s prowess with calculations, he was able to secure a position as a clerk in the accounts division of Madras Port Trust. Ramanujan could not have found a better work environment. Surrounded by men who worked in mathematical fields that had connections to other mathematicians, he could possibly get his name into the mathematical world. They all tried to get Ramanujan in touch with a mathematician that would see his genius, with little success. Fortunately, he wrote to an Englishman named Hill who gave not the normal rejection, but rather replied that Ramanujan was lacking in his formality, but should not be discouraged. And discouraged he was not. He wrote to H.F. Baker, a prominent mathematician, asking for help, only to be turned down. He wrote to E.W. Hobson, and was simliarly turned down. Ramanujan wrote a third letter asking G.H. Hardy if he would help. Hardy said yes.
Ramanujan claimed, in his letter, that, among other things, he had improved upon Gauss’ prime number theorem, that he had derived a formula that could estimate the number of prime numbers less than a certain number with ‘neglible’ error. Accompanying the letter was nine pages of Ramanujan’s calculations and derivations. Hardy had never seen anything like it before and was intrigued and astonished by Ramanujan’s claim that he had improved upon Gauss’ theorem. He replied to Ramanujan requesting more proofs and this theorem that Ramanujan referred to. Thus began a long series of correspondence between the two. Ultimately, Hardy, convinced that Ramanujan was the real thing, arranged for one of his colleagues, who was making a trip to India to meet with Ramanujan. E.H. Neville had gone to Madras to give lectures on differential geometry and was given the additional task of convincing Ramanujan to return to Cambridge with him. After seeing his work, Neville was immediately convinced that Ramanujan was a genius and tried with all his might to persuade Ramanujan to go to Cambridge to study.
Ramanujan would have liked nothing more than to study in England, home to some of the greatest mathematicians of the time, but doing so would cause irreparable harm to his social status. As a Brahmin, Ramanujan was forbidden from leaving the country. In fact, leaving India was one of the most serious offenses a Brahmin could commit. Were he to do so, his caste mates could no longer talk to him, and anyone seen talking to him would be similarly exiled. They would not be able to attend the funeral of a close friend or relative and could not go to another caste mate’s house for dinner. From this particular aspect of Indian culture does the word outcast come. So, it was not an option for Ramanujan to go to Cambridge to study.
That is, not until his mother had a dream. One night, she dreamt of her son mingling amongst the westerners when the god Namagiri intervened in the dream and implored her not to interfere with this, as it was her son’s destiny. Some stories tell that Ramanujan himself went to the temple of Namagiri himself and slept outside of it praying to the great god everyday until one night, when he had a vision similar to that of his mother. But either way, Ramanujan now had permission to go to England, and this he did.
Setting sail on March 17, 1914, Ramanujan’s journey lasted less than a month and set foot on English soil April 14, 1914. Neville took Ramanujan into his house to help ease the transition from the relatively relaxed, slow-paced life to which he was accustomed to the fast-paced, hectic London atmosphere. The Nevilles did an excellent job in making Ramanujan’s transition as smooth as possible. They met to his every need, especially his special dietary needs, as Ramanujan was a strict vegetarian. Hardy wasted little time in making sure Ramanujan could begin his studies. He and his counterpart, Littlewood, wasted little time getting the paperwork out of the way so Ramanujan could work on his mathematics. Hardy scanned Ramanujan’s notebooks carefully and classified the theorems contained within into three categories. Those that were wrong, those that were correct but already proven and those that were yet to be proven and possibly groundbreaking. Hardy desired to expand upon the third group even though Ramanujan held the second group as his pride and joy, for he could verify his results.
Nonetheless, Hardy and Littlewood extracted a few of the theorems from the last group, and by June had the beginnings of two papers and presented them to the London Mathematical Society. Coincidentallly, Hobson, one of the two to turn Ramanujan down was there though the author himself was not.
Ramanujan’s first paper in England appeared in the Quarterly Journal of Mathematics under the title "Modular Equations and Approximations to Pi." In it, Ramanujan presented an infinite series to estimate pi. While there already existed estimations of pi, Ramanujan’s converged the quickest, as only one term was needed to provide eight decimal places of accuracy. Ramanujan’s work in this area gave way to today’s fastest known algorithm for determining pi.
As far as Ramanujan’s claim that he could accurately determine the number of primes better than Gauss, he was wrong. His formula provided virtually exact numbers for n=1000. At n=9,000,000, his error was only 53. What was wrong was that Ramanujan had assumed that the Riemann zeta function had no complex zeroes. The Riemann zeta function was an infinite series in terms of a complex variable. The Riemann hypothesis, unproven at the time of Ramanujan, said that when set to zero, the complex zeros of the zeta function would lie along a line half a unit to the right of the imaginary axis, and from this the distribution of primes would come easily. What Ramanujan had done was to derive his own version of the zeta function, and ignored the most crucial complex zeroes. Using a flawed zeta function in derving his Prime Number Theorem led to an inaccurate formula. Ramanujan learned that he could not consistently rely on his intuitions, which was exactly the reason why he came to Cambridge, to learn tools so he would not have to.
Ramanujan also worked with numbers that, as Hardy had put it, "were as far from prime numbers as possible." Where prime numbers have the fewest factors possible, itself and one, highly composite numbers had more prime factors than any number less than it. For example, 12 can be divided by 1, 2, 3, 4, 6, and itself, or six factors, more than any other number less than it. The first few highly composite numbers were 2, 4, 6, 12 24, 36, 48, 60, 120 and so on. Ramanujan sought a pattern in these numbers and found that the prime factorization of the highly composite number N can be written as:
N=2a x 3b x 5c x…
Where a is always greater than or equal to b which is greater than or equal to c. He also showed that with the exception of 4 and 36, the last factor is always raised to a power of 1. Through 52 pages, he proved these and other properties in a paper published in late 1915 in the Proceedings of the London Mathematical Society. It was for this paper that he received his Bachelor of Science by Research degree from Cambridge University in March of 1916.
The subject of Hardy and Ramanujan’s possibly most famous paper dealt with another aspect of numbers, that of partitions. A partition of a number is simply the number of ways you can uniquely represent a number as a some of whole numbers. The partition function p(x) returns that number. For example, p(3)=3, because 3=1+1+1=2+1=3. The value of the partition function rises extremely quickly, p(10)=42 and p(50)=204,226. Finding p(x) was what Hardy and Ramanujan sought. To do this, they developed a method known as the circle method. They used Cauchy’s Theorem, which lies in the domain of analysis and deals with continuous rather than discrete quantities. Also, the integral at the heart of Cauchy’s Theorem couldn’t be evaluated because the "countour" over which it was to be integrated held impermissible points, so Hardy and Ramanujan took sections of the unit circle and made approximations as they went along. They employed the services of an army man-turned mathematician named McMahon to do the calculations, and it turned out that they had found a function close enough to the exact value it could be rounded with no harm. Hardy presented their preliminary findings to the Quatrieme Congres des Mathematiciens Scandinaves in Stockholm in 1916 and published their findings early the next year in the French Journal Comptes Rendus. They did not get their findings into the Proceedings of the London Mathematical Society until 1918.
In 1917, Ramanujan fell ill. It can not be said with certainty what he fell ill with, but in all probability it was tuberculosis. The colder climate, lack of sun, but most probably, World War I affected his illness. The war lead to blockades of ports and made it harder for Ramanujan to keep a healthy diet because of his vegetarianism. He remained faithful to this even though it could cost him his life because of his Brahminic upbringing. He had promised his mother that despite all odds, he would stay a strict vegetarian during his stay in England, and this he did. While Ramanujan lay ill in bed, Hardy campaigned for his induction into the Royal Society of London. Elected a fellow of the Cambridge Philosophical Society lifted Ramanujan’s spirits a little, but becoming a Fellow of the Royal Society boosted them. To have the initials F.R.S. placed after your name was possibly the greatest honor a scientist could have had at the time. In good spirits, Ramanujan’s health recovered enough that he could make the trip back India. However, he did not recover from the disease that plagued him, and died in his hometown of Kumbakonam on April 26, 1920 at the age of 32.
What Ramanujan could have accomplished had he not died in the prime of his life, we will never know. But what he did achieve during his short life has left a lasting impression not only on the world of mathematics, but on the country of India. He made Indian mathematics once again respectable, bringing back the glory that those such as Brahmagupta had once inspired. Most Indian children know Ramanujan’s story, while they don’t know exactly what his equations mean, or what bearing they have had, they know he came out of the masses and did what he was determined to do. He knocked on doors and offered his goods and eventually he was recognized. The story of Ramanujan is a lesson that everyone can learn from and one that everyone should know.