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

Why Math is the “Language of the Universe”:

In response to : http://www.fromquarkstoquasars.com/why-math-is-the-language-of-the-universe/

Why Math is the "Language of the Universe":

First, lets start off with a few quotes

“Mathematics is the language with which God has written the universe.”
― Galileo Galilei

"Number rules the universe."
―Pythagoras

"Go down deep enough into anything and you will find mathematics."
―Dean Schlicter

 "Mathematics directs the flow of the universe, lurks behind its shapes and curves and holds the reins of everything from tiny atoms, to the biggest stars."
― Edward Frenkel

“The interaction between math and physics is a two-way process, with each of the two subjects drawing from and inspiring the other. At different times, one of them may take the lead in developing a particular idea, only to yield to the other subject as focus shifts. But altogether, the two interact in a virtuous circle of mutual influence.”
― Edward Frenkel


The latter quote especially emphasizes the distinction between physics and mathematics. Both influence each other, but more importantly is being able to describe the physical with math. The fact that we as humans are smart enough to take what we saw thousands of years ago with just our eyes and create this abstract idea called mathematics is astonishing. Euclid was only armed with a straightedge and a compass in his book Elements in which he laid the foundations to geometry that we still use today. Throughout history it seems that every once in a while, someone sees the power and the freedom of mathematics and utilizes it to the fullest extent. These geniuses that had the ability to see what math could do are foundational to the abstract idea itself. Without the groundwork set before us, there would be nothing to climb.

This leads to the idea of mathematics in different cultures here on Earth and far far away from our home here on Earth. That is, can mathematics be different here on Earth in other places and is there a universally accepted mathematics? First we will examine the Earthly mathematics. No matter where you go on this planet, 2 + 2 will always equal 4. It does not matter if you are in Michigan or Australia, this will always be true. Granted things in different places are expressed with different symbols and objects, but the validly and logic of mathematics will still be there. When we start to ponder about alien lifeforms further away from us then we can ever our imagine is when things get interesting.

We have developed this notion of gravity which is about 9.80665 m/s2 on this planet, but what happens if we go three hundred million light years away from here? Nobody has ever traveled this far and knows what is out there. The physical laws we have established here may be different in other places. I do not think that the way we express these quantities is different from other beings, but the numbers may be slightly off.  We can argue about the numbers and the physics, but we cannot argue about how it is explained and expressed.

Its been said many times how much math is around us. Lurking and stalking behind, and in front of us, we sometimes forget how much and how great math is. Easily overlooked, we would not be where we are today without the advancements. The only way we advance through this realm of numbers is to keep building from what we already know, and understanding that we could not of done it without the tools and discoveries that were presented to us by the great minds before us.

Mi Casa..Su Casa

What is the role of the house of wisdom in math history?

     Bayt al-Hikma meaning the house of wisdom was built in Baghdad around 810 AD and is an institution in the Islamic Golden Age responsible for bringing together great thinkers to form and collect ideas. Founded by Caliph Harun al-Rashid and passed down to his son al-Ma'mun in 813, the house of wisdom is a central figure of the Islamic Golden Age. For the next four hundred years scholars would start the process of taking the knowledge at that time from around the world and translating it into Arabic.  The role of the House of Wisdom was the same as the Library of Alexandria in Egypt in the sense that these were very large institutions created to harvest the knowledge of what we as Humans had found out up to that point. When we have all of the resources that our ancestors had provided for us in one place, we can then begin to form new ideas and build off our previous ones. The House of Wisdom was this place with these resources at our fingertips.The House of Wisdom was responsible not only for great mathematics, but great findings in other fields like astronomy, medicine, chemistry, geography, and many more. Sadly, in  1258 Mongol armies led by Hulagu Khan would destroy the house of wisdom in Baghdad, Iraq. This was a bloody siege and it is said that the Tigris river ran black from the ink from the enormous quantities of books that were thrown it to the river and red from the blood of all the scientists and scholars who were killed.

     To name a few significant House of Wisdom attendants known for their mathematics would be not at all difficult. The first one that comes to mind is Muhammad Al-Khwarimi who in the 9th Century was responsible for the strongly advocating the use of the Hindu numeral system (0,1,2,...,9) and also responsible for ideas in algebra relating to reducing, equality, and solving polynomials up to second degree. Another significant mathematician would have to be Muhammad Al-Karaji from the 10th Century who did extended work in algebra introducing the theory of algebraic calculus and also being the first to prove an idea using what we know today as mathematical induction. Thābit ibn Qurra was also a major contributor in the house of wisdom with his ideas in astronomy, physiscs, and mathematics. Thābit ibn Qurra's contribution to the area of mathematics includes finding an equation for amicable numbers, working on number theory and using ratios to describe geometrical quantities, and he also used the idea of exponential series to solve a chess board problem. Another hugely important figure in the house of wisdom is Omar Khayyám. Omar was not just limited to mathematics, he also contributed largely to astronomy, poetry, and philosophy. His work in mathematics is one of the best in the house of wisdom. He is responsible for classifying all cubic equations to a method (25 types), he also worked on the triangular array of binomial coefficients now known as Pascal's triangle, and generalized the method of completing the square to solve all quadratic equations, and did work in geometry on the theory of proportions. This is only a sample of the work that Omar Khayyám did. He is definitely the top dog of mathematics for the house along with Al-Khwarimi.

     My favorite thing that spawns from all this mathematics that is being compiled and processed at this time is the art. Specifically, the tessellations that are used in mosques. One must understand that in Islam, the depiction of any religious figure is strictly forbidden which means no kind of being should be placed on the walls of worship or anywhere for that matter. What this meant is that people had to be creative enough to paint beautiful things without beings. As a result, these tessellations that involve a lot of geometry are used. The aesthetics behind these figures is the symmetry. Why they look so appealing is because of the symmetry. Most humans appreciate symmetry and we like these kinds of objects. Maybe the reason why we like a circle so much is because of its infinite symmetries.

My favorite tessellation: http://www.thelck.com/patterns/tenPointStar.html
(ten-pointed star)

Sources:

http://www.amnh.org/exhibitions/past-exhibitions/traveling-the-silk-road/take-a-journey/baghdad/house-of-wisdom

http://archive.thedailystar.net/newDesign/news-details.php?nid=234148

http://www.theguardian.com/education/2004/sep/23/research.highereducation1

History of Math

How did Greek mathematics develop?

Why is Euclid important?


      Greek mathematics developed from reason and logic. Egyptians would not ask the question why? where Greeks were interested in how something works down to the core. Egyptians might of said it works because it does but Greeks wanted to dig deeper. This was best illustrated in the first chapter of the book I am reading "Journey Through Genius" by William Dunham. This type of thinking became prominent in the Greek culture at that time influencing many great mathematicians, inventors, astronomers, engineers, and many others. Greek mathematics developed from this non-rigorous, taken for granted, idea to a much more extreme proof-orientated way of mathematics. It was around this time that this free thinking or 'awakening' happened and many great individuals conjectured, proved, and invented things that we heavily rely on to this day.This awakening was responsible for great individuals like Thales, Pythagoras, and Euclid who showed that humans were starting to think outside of the box. We will briefly examine each individual.



























      Thales (624-547) is regarded to be the first Greek mathematician. He was the founder of the Ionian school of philosophy in Miletus, and the teacher of Anaximander. Thales went to Egypt and studied with the priests where he learned mathematics and brought it back to Greece. Upon studying in Egypt Thales did geometrical research which allowed him with the use of triangles to calculate the distance from the shore of ships at sea. Using the same technique Thales was also able to determine the height of pyramids in Egypt. Thales is credited with the following five theorems of geometry:

1. A circle is bisected by its diameter.
2. Angles at the base of any isosceles triangle are equal.
3. If two straight lines intersect, the opposite angles formed are equal.
4. If one triangle has two angles and one side equal to another triangle, the two triangles are equal in all respects. (See Congruence)
5. Any angle inscribed in a semicircle is a right angle. This is known as Thales' Theorem.

     Pythagoras (569-500) is considered the first pure mathematician of our time. Pythagoras was interested in the study of numbers along with angles, triangles, areas, proportions, polygons, and polyhedra. Pythagoras also played a seven string lyre which allowed him to study the relationship between music and mathematics. He noted that the vibrating strings of the lyre sounded when the lengths of the strings were proportional to whole numbers like 2:1, 3:2, 4:3, and so on. His most famous theorem, the Pythagorean Theorem is what he is most remembered by. He is credited with the following six results:

1. The sum of the angles of a triangle is equal to two right angles.
2. The theorem of Pythagoras - for a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. The Babylonians understood this 1000 years earlier, but Pythagoras proved it.
3. Constructing figures of a given area and geometrical algebra. For example they solved various equations by geometrical means.
4. The discovery of irrational numbers is attributed to the Pythagoreans, but seems unlikely to have been the idea of Pythagoras because it does not align with his philosophy the all things are numbers, since number to him meant the ratio of two whole numbers.
5. The five regular solids (tetrahedron, cube, octahedron, icosahedron, dodecahedron). It is believed that Pythagoras knew how to construct the first three but not last two.
6. Pythagoras taught that Earth was a sphere in the center of the Kosmos (Universe), that the planets, stars, and the universe were spherical because the sphere was the most perfect solid figure. He also taught that the paths of the planets were circular. Pythagoras recognized that the morning star was the same as the evening star, Venus.

     Euclid (330-260) is often referred to as the "Father of Geometry". He taught mathematics in Alexandria, Egypt at the Alexandria library. He also wrote the most enduring mathematical work of all time which is Elements. This is a thirteen volume work that is a comprehensive compilation of all geometric knowledge known up to that point in time. The book is based off the works of Thales, Pythagoras, Plato, Eudoxus, Aristotle, Menaechmus, and others. It was used for over 2000 years. Euclid wrote Elements at the Alexandria library which covered plane geometry, solid geometry, arithmetic, and number theory. The organization of Elements is interesting because Euclid organized the known geometrical ideas, starting with simple definitions, axioms, formed statements called theorems, and set forth methods for logical proofs. He began with accepted mathematical truths, axioms and postulates, and demonstrated logically 467 propositions in plane and solid geometry. Elements would be the most widely used textbook of all time and would appear in classrooms until the twentieth century. It has sold more copies than any other book besides the bible.

     Personally with this research of Greek mathematics, it makes me wonder what were the main factors that made these people such great thinkers. Besides this idea of free thinking, what else were influencing these people that made them achieve this greatness. The Greeks made major developments in philosophy, politics, mathematics, physics, and many other fields. These ideas are commonly used today like the democracy in our government or Euclid's axioms in geometry.
Sources:

http://www.mathopenref.com/thales.html

http://www.mathopenref.com/pythagoras.html

http://www.mathopenref.com/euclid.html

What Is Math?

Mathematics is a generalized term that encompasses many different branches within. To name a few there is Geometry that deals with shapes, Calculus that deals with rates of change, Arithmetic that deals with counting and many more. To ask what is mathematics is like asking what is life. There is no definitive answer.

One of the significant discoveries within the history of mathematics would have to be the discovery of zero. The Mayans independently discovered the concept of zero without any communication from the outside world. Islamic civilizations discovered this concept as well and many other important mathematical findings.

Off of the top of my head and doing no research, the next important discovery would have to go to Issac Newton and Gottfried Leibniz for their work in the development of infinitesimal calculus.

Mathematical tools are also very important to consider. Starting with an early counting object called an abacus and progressing to the TI-84 calculator took not only a lot of time, but a lot of mathematics.

There are many more of these discoveries and yet there are still more to be found. To rank them in significance seems odd but chronologically might make more sense. Mathematics builds on itself. We first learn Arithmetic and then move on to basic Geometry and Algebra. We do not start with Calculus and work backwards.