Ramanujan and Mathematics

Srinivasa Ramanujan

"An equation means nothing to me unless it expresses a thought of God."
-Srinivasa Ramanujan

"Every positive integer is one of Ramanujan's personal friends."
-John Littlewood

"He could remember the idiosyncrasies of numbers in an almost uncanny way. It was Littlewood who said that every positive integer was one of Ramanujan's personal friends. I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." 
-G.H. Hardy

G.H. Hardy

     Srinivasa Ramanujan (22 December 1887 – 26 April 1920) was an Indian mathematician who had a natural born talent for mathematics. Having no formal training of mathematics, Ramanujan taught himself  through textbooks starting at the age of 10. The house he lived in as a child was also a place for lodgers to stay. As a result, he bonded with individuals staying and received textbooks from them and Ramanujan studied them.  As his school career progressed, he was awarded many merits and certificates due to his outstanding performance in mathematics. He was awarded a scholarship to the best school in India but after focusing on math and not his other subjects, he would fail. It would be after dropping out of two colleges in India that he would begin to search for a job. After receiving a job as a clerk in Madras, India, making 30 rupees a month, Ramanujan began correspodence with British mathematicians.

     In 1913, his secondary school headmaster would contact three mathematicians from the University College of London to look at Ramanujan's conjectures and theorems scribbled in his notebook. Having been looked down upon by two claiming that his mathematical ability was there, but not fully developed and there were gaps in his argument, it would be G.H. Hardy that would take him under his wing. Hardy recognized that Ramanujan was a very gifted individual and brought him to Cambridge University where the two would work together.

     Ramanujan's contribution to mathematics is quite large in areas of composite numbers, the partition function, gamma functions, modular forms, divergent series, hyper geometric series, and prime number theory. Ramanujan also looked at infinite series for the calculation of pi. These series that he looked at would be responsible for creating algorithms today that calculate pi with accuracy up to 5 trillion decimal places.

     Both Hardy and Ramanujan shared an unfortunate time in their lives where both tried to commit suicide. Both men were very involved in their work and shared extreme depression. The collaboration between the two was looking at the Riemann Hypothesis and other parts of it. Some say now it has a curse.

     It would be in 1920 that Ramanujan would pass away at 32 years of age. There is a lot of speculation as to what caused his death. Some claim it was the stress of being in a new country and that during the first World War there was no vegetarian food around for him to consume. He was diagnosed with Tuberculosis and extreme vitamin deficiencies. It would be G.H. Hardy that was quoted later in his life talking about his research and collaboration with Ramanujan saying it was  "the one romantic incident in my life".

The documentary that made me want to learn more about Ramanujan and Hardy:

https://www.youtube.com/watch?v=OARGZ1xXCxs











Sources:

http://www.storyofmathematics.com/20th_hardy.html

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Ramanujan.html

https://www.youtube.com/watch?v=OARGZ1xXCxs

Descartes' Folium

René Descartes (1596 - 1650)

"I think, therefore I am"
-René Descartes
 


René Descartes (1596 - 1650) was a philosopher, mathematician, and a writer. Descartes made extremely important discoveries and transformations of mathematics, but is most noted for his philosophical work. Descartes refused to accept the previous philosophical ideologies before him and started to think more abstractly. His famous quote, "I think, therefore I am" is a proposition that translates to being able to think as a proof of existence. If we doubt our existence, we are thinking. Since we are thinking, we must exist. Descartes lived in a time where people were very skeptical. It was the time of Enlightenment and the Scientific Revolution. This meant that people were changing their traditional views about the world around them and further developing through use of scientific method. During these skeptical times, people like Descartes had to use reasoning and logic to promote new thoughts.

Mathematics being the most powerful tool to utilize reasoning and logic, Descartes was interested in trying to combine the geometrical side with the algebraic side. In doing so, he created what we know as the Cartesian plane. This is a plane created by perpendicular real numbered axes. This allowed for the creation of the sub field analytic geometry which is geometry using the Cartesian plane. This new approach to taking x and y values and plotting them as coordinates (x,y) in the plane allowed for great development in the area of calculus by Issac Newton and Gottfried Leibniz.

If creating the Cartesian coordinate system wasn't enough, Descartes also modernized notation that was easier to work with and understand. Unknown variables were labelled x,y,z and known coefficients were labelled a,b,c. Also writing (2 x 2 x 2 x 2) as 2^4 made for neater and concise writings.  He also developed the 'rule of signs' which was a technique for determining the number of positive or negative real roots of a polynomial p(x). Descartes noted that this rule only applies when p(x) is written in descending powers of x and has a non-zero constant term. The rule is as follows:

Number of positive roots of a polynomial p(x) is either:

1) The number of times the sign changes
or
2) an even integer less than that number

Number of negative roots of a polynomial p(x) is either:

1) the number of times the sign changes in the negative polynomial, that is p(-x)
or
2) an even integer less than the number

All of this brings us to Descartes' Folium pictured above. The name comes from the Latin word folium which means leaf. The folium is an algebraic curve with equation x^3 + y^3 -3axy = 0. Pictured above, it has symmetry with y = x and has an asymptote with equation x + y + a = 0.  Consequently this curve sparked discussion between Descartes and Fermat about finding tangent lines at a point on the curve. Fermat having discovered a new technique with finding tangent lines was able to solve this problem easily. Below is the folium graphed below with different values of a.

The folium does not posses any interesting or special properties except its asymptote and symmetry properties. It can be expressed in polar coordinates and can be solved implicitly. Another folium, called Kepler's folium is pictured below.

Fermat's Last Theorem

Fermat's Last Theorem: No three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two.

Conjectured by:  Pierre de Fermat in 1637

Proven by:  Andrew Wiles in 1995

Fermat's Last Theorem was a conjecture (now proven) in number theory which states that there are no integer solutions that satisfy the equation a^n + b^n = c^n for any integer n greater than two. Fermat proved specific cases for when n = 3 and when n = 4, but there was no proof by him or anyone else to prove the general case until Andrew Wiles became interested in the problem. Of course when it was conjectured in a letter by Fermat, he had written that he did have a proof for this theorem because the margin was too small to fit it!

If we begin by looking at some examples, right away we can see that this works when we remove the restriction for n greater than two.

3^2 + 4^2 = 5^2
or
5^2 + 12^2 = 13^2

But once we focus our attention for values greater than two, we cannot find any solutions. It is teasing in a way to think that there are no integer solutions for x^3 + y^3 = z^3 or for greater powers.

Even though it looks simple and elegant, it stumped mathematicians for over 300 years. Andrew Wiles first saw the problem when he was ten years old and became fascinated with it. After graduating and obtaining a bachelors degree in mathematics, he moved on to his PhD specializing in number theory. While working on his graduate research, Wiles dealt with elliptic curves which would be the foundation to proving Fermat's Last Theorem.

In order to prove Fermat's Last Theorem, Wiles needed to prove a conjecture that implied a relationship between the conjecture and Fermat's Last Theorem. This conjecture is called the Taniyama-Shimura conjecture. In effect, the conjecture says that every rational elliptic curve is a modular form in disguise. Wiles then established a correspondence between the set of elliptic curves and the set of modular elliptic curves by showing that the number of each was the same. He did this by counting Galois representations and comparing them with the number of modular forms.

In 1993, Wiles was ready to submit the proof of the Taniyama-Shimura Conjecture and upon doing so an error was found. Digging deeper this is what Wiles had said in an interview about the error:

"The error is so abstract that it can't really be described in simple terms. Even explaining it to a mathematician would require the mathematician to spend two or three months studying that part of the manuscript in great detail."

But this did not set him back at all. With the help of a former student Richard Taylor, the two had patched up the error and were ready to submit the correct proof to Princeton in 1995. Later that year, it had been recognized as a correct proof and finally after 300 years Fermat's Last Theorem had been proven true.  

A common question that is asked is whether or not this would be the same proof that Fermat claimed he had. The answer is no. Wiles had used 20th century techniques to prove the theorem while Fermat living in the 17th century did not have access to. So Wiles not only had the correct proof, but it was unique from Fermat's which was too large to fit in a margin.

This is a great example of why my passion lies within the field of mathematics. Take a problem that has been unsolved for 300 years, apply current or modern mathematics to this problem, and voilà! Unfortunately it is not that easy as any mathematician will tell you. But what I would like to illustrate is that if you have a problem that you want to solve, you can use properties, and theorems, and the like that were solved and presented to you before your time to continue and add to our ever growing bank of knowledge. This can be seen as a snowball starting at the top of an infinite hill. A long time ago, someone kicked that snowball down and it has been growing ever since.


Sources:

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Fermat.html

http://www.pbs.org/wgbh/nova/proof/

http://www.storyofmathematics.com/17th_fermat.html