Amazing life of S. Ramanujan and one of his famous mathematical demonstration.
1+2+3+4+5+6+7+8+ etc… = — 1/12
To answer to all comments about my last article about Ramanujan, I would like to complete with a little resume of his life and the mathematical demonstration about the above equation (source : Royal Cambridge Society Scientific Publications and from this remarquable article by Stephen Wolfram).
Srinavasa Ramanujan Iyengar, popularly known as S. Ramanujan is considered to be one of the best mathematicians of all time in India. He was born in a little town, named Erode on 22nd December, 1887. Erode is located about 500 kilometers southwest of Madras, in the state of Tamil Nadu.
S. Ramanujan’s father, Srinavasa was a petty clerk in a saree shop in Kumbhakonam that is about 320 kilometers southwest of Madras. S. Ramanujan was brought up in a small hut-like house on the Sarangapani Sannidhi Street of this little known town.
The most unique thing about Ramanujan was that he was very fond of mathematics and curious about the world, right from his childhood. At school, he would often mentally solve a mathematical question before the teacher could do that on the blackboard. Sometimes, he would also offer an alternative method of solving the same question or give an easier or quicker way of doing the same, to the teacher. He also used to finish his mathematics paper within half the time, during examinations.
Because of this strange nature and his extraordinary talent, the teachers of S. Ramanujan’s school liked him very much and they also praised him openly in class. Ramanujan was bothered about several questions like ‘What is the highest truth in mathematics?’ right from his early age. He would often quiz his mathematics teachers on the same. Ramanujan got admitted in the Town High School in Kumbhakonam in January 1898. Ramanujan’s genius was revealed for the first time during a mathematics class in the Town High School, where the students were being taught the simple sums of division. It was in this class, where Ramanujan started to develop a life-long fascination for zero and infinity, as anything that is divided by zero is infinite. He could also often rattle off the value of pi (n), e, square root of two etc., up to any places of decimals, at any time.
S. Ramanujan received the prestigious K. Ranganatha Rao Prize for standing first in Mathematics in a school examination, during his childhood. After that, he also became successful to win the Junior Subrahmanyam Scholarship in a school competitive examination for standing first in Mathematics and English. While winning those prizes, S. Ramanujan was also introduced to S. L. Loney’s book Trigonometry, through one of the college boarders in his house. The book was recommended as a textbook in colleges at that time. Though Ramanujan was only thirteen years old then, he did not take much time to master the subject. However, he faced some problems while studying the other subjects like English, Greek and Roman History, and Physiology, in the higher classes. He could not pay due attention to those subjects for his great love for Mathematics and hence, he started to score badly in those subjects.
Ramanujan became more and more interested in Mathematics after he came across George Shoobridge Carr’s book titled “A Synopsis of Elementary Results in Pure and Applied Mathematics”, published in 1886. The book contained about 6,000 problems in algebra, trigonometry, calculus and analytical geometry and it did not discuss any of the problems adequately or give step-by-step proofs. Rather on some occasions it simply gave hints to solve the problems. The book had a huge effect on Ramanujan and it actually changed his outlook and life forever. He made every problem mentioned in the book, a research project for him. It was this book that led Ramanujan to such a condition, where he could not think of anything else but mathematics, whether he was sleeping or awake.
As S. Ramanujan did not have the proper training and guidance in higher mathematics, he used to look at every problem given in the book Synopsis, with ‘fresh’ eyes. Though the problems described in Carr’s book had already been solved more than 150 years ago in Europe, Ramanujan started to solve them in his own ingenious way. As a result, though sometimes he solved them just in the way they were solved earlier by European mathematicians, he occasionally solved them through altogether novel methods. He also often used to go far beyond into new realms of mathematics, where no European mathematician had gone before, in seeking solutions of the problems. He also made wonderful discoveries in mathematics, during the course of time, almost unknowingly. He also had the habit of hiding some of the mathematical calculations, when he came to know that what he had discovered was already a part of a higher mathematics text-book.
He got his first job in the Madras Port Trust and few years later, Ramanujan and his supporters contacted a number of British professors, but only one was receptive — an eminent pure mathematician at the University of Cambridge — Godfrey Harold Hardy, known to everyone as G. H. Hardy, who received a letter from Ramanujan in January 1913. By this time, Ramanujan had reached the age of 25. Professor Hardy puzzled over the nine pages of mathematical notes Ramanujan had sent. They seemed rather incredible. Could it be that one of his colleagues was playing a trick on him?
Hardy reviewed the papers with J. E. Littlewood, another eminent Cambridge mathematician, telling Littlewood they had been written by either a crank or a genius, but he wasn’t quite sure which. After spending two and a half hours poring over the outlandishly original work, the mathematicians came to a conclusion. “I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written by a mathematician of the highest class. They must be true because, if they were not true, no one would have the imagination to invent them.”
Ramanujan arrived in Cambridge in April 1914, three months before the outbreak of World War 1. Within days he had begun work with Hardy and Littlewood. Two years later, he was awarded the equivalent of a Ph.D. for his work — a mere formality.
Ramanujan’s prodigious mathematical output amazed Hardy and Littlewood. The notebooks he had brought from India were filled with thousands of identities, equations and theorems which he had discovered for himself in the years 1903–1914. Some had been discovered by earlier mathematicians; some, through inexperience, were mistaken; many were entirely new.
Ramanujan had very little formal training in mathematics, and indeed large areas of mathematics were unknown to him. Yet in the areas familiar to him and in which he enjoyed working, his output of new results was phenomenal. In 1918 Ramanujan became the first Indian Mathematician to be elected a Fellow of the British Royal Society.
In July 1909 Ramanujan married S. Janaki Ammal, who was then just 10 years old. The marriage had been arranged by Ramanujan’s mother. The couple began sharing a home in 1912.
When Ramanujan left to study at the University of Cambridge, his wife moved in with Ramanujan’s parents. Ramanujan’s scholarship was sufficient for his needs in Cambridge and the family’s needs in Kumbakonam.
For his first three years in Cambridge, Ramanujan was very happy. His health, however, had always been rather poor. The winter weather in England, much colder than anything he had ever imagined, made him ill for a time.
In 1917 he was diagnosed with tuberculosis and worryingly low vitamin levels. He spent months being cared for in sanitariums and nursing homes. In February 1919 his health seemed to have recovered sufficiently for him to return to India, but sadly he would only live for about a year on his return. Srinivasa Ramanujan died aged 32 in Madras on April 26, 1920. His death was most likely caused by hepatic amoebiasis caused by liver parasites common in Madras. His body was cremated. Sadly, some of Ramanujan’s Brahmin relatives refused to attend his funeral because he had traveled overseas.
Demonstration :
Take the following example: 1 + 2 + 3 + 4 + 5 + 6 + 7 … and so on. How much is this sum?
I think any schoolboy supposed to answer “infinity”. Well yes, but no. The mathematicians have managed to prove that this huge sum is worth … -1/12!
As a warm up, let’s start with a slightly simpler sum:
1–1 + 1–1 + 1–1 + …
How much is this sum? The cleverest will notice that the value of this sum oscillates between 0 and 1 as one adds terms to it. If you really want to assign an “average” value to this infinite sum, you can type between the two and choose 1/2.
Well one can in fact rigorously demonstrate that this sum is worth 1/2. Here is the idea: let’s call for this sum, so we ask
A = 1–1 + 1–1 + 1 — …
We can then observe that
A = 1–1 + 1–1 + 1 — … = 1 — (1–1 + 1–1 + 1 — …)
but we recognize that the term in parentheses is none other than itself, so we have equality
A = 1 — A
and you can easily solve this equation to find A = 1/2.
So let’s go to the next level.
Now if we consider the sum
B = 1–2 + 3–4 + 5–6 + 7 — …
It is still an oscillating sum, but this time the oscillations become bigger and bigger! This time we notice that
B = 1 — (2–3 + 4–5 + 6–7 + …)
and breaking down into two pieces the term in parentheses we have
B = 1 — (1–2 + 3–4 + 5–6 + 7 — …) — (1–1 + 1–1 + 1 — …)
Now here we recognize in the first parenthesis the sum B from which we started, and in the other parenthesis the sum A which we have evaluated in the preceding paragraph. So we have
B = 1 — B — A
Since we calculate that A is 1/2, we draw B = 1 — B — 1/2 and so B = 1/4. You see that with simple arithmetic operations, we can assign a well-defined value to this infinite oscillating sum !
Let’s come to our monstrous sum, and call it S.
S = 1 + 2 + 3 + 4 + 5 + 6 + …
This time, the sum does not oscillate: it goes straight to infinity at high speed ! And yet here is what we can do: take the sum S and withdraw the sum B
S — B = (1 + 2 + 3 + 4 + 5 + 6 + …) — (1–2 + 3–4 + 5–6 + …)
You see that the odd terms are offsetting each other and the even terms are doubled, so we have
S — B = 2 * (2 + 4 + 6 + 8 + …) = 4 * (1 + 2 + 3 + 4 + …)
and here on the right we recognize in parentheses our sum S ! So we have
S = B + 4S
or S = -B / 3. As we have seen that B = 1/4, we arrive at the long-awaited result
S = — 1/12
This is it !
It may seem shocking to you, you can look for the flaw, or imagine that you can demonstrate anything like that by fiddling with infinite amounts. Well no, if we respect some basic rules, whatever the way we do it, we find that if we want to assign a finite value to this monstrous sum S, then -1/12 is the only value that sticks.
It is perfectly understandable why some Cambridge researchers virally rejected Ramanujan’s theorems, yet most of these equations have been demonstrated both mathematically and physically. I will not argue here, this is not my role and Quora is not the right website to expose it in the best conditions. On the other hand, for those interested, I invite you to go deeper into the many publications already approved by the scientific community.
I found this demonstration on video as well >
More on IMDB
If you liked this, click the💚 below so other people will see this here on Medium.