Fermat’s Last Theorem

In learning about this subject and thinking about complex concepts, it was pleasant to see the process of discovery. Going beyond academic values it was hart warming to see how some of the most brilliant minds in mathematics struggle with questions.

Fermat was a French judge that enjoyed studying mathematics as a hobby. He was studying Arithmetica which is book of Greek text on mathematics. Habitually, Fermat made notes in the margins of the book about the relevant subject that he was considering. When he arrived at the Pythagorean thermo (a²+b²=c²) and in thinking about the whole numbers that could satisfy the equation (3²+4²=5², 5²+12²=13²…) he advanced to realize that the set of whole numbers that satisfied the solution were infinite. However, almost simultaneously, he seemed to know that we will never find an answer to this equation if we changed the exponents (a³+b³=c³, a⁴+b⁴=c⁴… aⁿ+bⁿ=cⁿ). This is to say so long as the exponent was 2 the answer set is infinite and if the exponent was other than 2 the answer set is empty no matter how far one looks for the answer. But evaluating this idea by trial and error is impossible because when we start with the first number we have infinite number of other numbers to yet check. And after checking the first number we are not one step closer to the end of the set of numbers that we should be examining. This limitless nature of possibilities required a mathematical proof based on other previously established theorems. Surprisingly Fermat notes that he knows such a proof but the margin of the book is too small for the elegant proof that he has imagined.
After Fermat’s passing others look at the notes that he had written in the margins and one by one establish an explanation for each scribble and at last we are left with the problem of variations on the Pythagorean thermo for which no one was able to find a proof. The story remains unresolved until the twentieth century. A young man named Andrew Wiles reads about this in a book from his local library in Cambridge England and tries to solve the problem. He studies math and all through his studies he never forgets about this unsolved problem. Latter in his thirties as a professor he is exposed to a paper by a pair of Japanese mathematicians: Taniyama and Shimura, concerning a conjecture (unproven fact) about a different subject. Professor Wiles invested seven years of complete concentration in proving the Japanese conjecture so that he could use it as a tool to prove Fermat’s theorem.
After Wiles announced that he had found the proof and during the peer review process the complicated procedure was not able to be reproduce and it took another year before the issue could be put to rest.
All Elliptic Curves are Modular => Taniyama and Shimura conjecture => Fermat’s last theorem
So finally, even though professor Wiles proved the Fermat’s theorem, we still do not know what solution Fermat actually had in mind. The techniques that Wiles used were not known during Fermat time.
Works Cited
BBC, Horizons. “Fermat’s Last Theorem”. 11/29/2013 <http://www.youtube.com/watch?v=7FnXgprKgSE>.
Wikipedia Fermat’s Last Theorem Version. 11/29/2013 <http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem>.

Mathematics in Music

Pythagoras, perhaps listening to the sounds of tradesmen at work, started thinking about sounds that were pleasing to hear. We can imagine him watching a blacksmith strike an iron rod to make a sword and noticing that different length blades produced a different sound. Logically he would conclude that the length of a string on an instrument could determine the kind of sound that the instrument produces. These different kinds of sound eventually became known as different notes.
As he went about observing the sound an instrument produced with different length strings, he observed that if the change in length could be represented by a whole number ratio the relating note of music was also pleasing to the ear and harmonious when combined with other notes. Ratios like 9:8, 5:4, 4:3, 3:2, 5:3, 15:8, and 2:1 that today we call Pythagorean Intervals, were at that time recognized for their quality and without the knowledge of sound waves and frequency attributed to gods.
Today we understand wavelengths and can measure their frequencies. If we use a spectrophotometer to measure the frequency of an A note on a violin we should get a reading of 440 hz. Given the 32.5 cm length of the violin string to get to the E note, using Pythagorean Intervals, we shorten the string to 21 and 2/3 cm (32.5*2/3). If we then measure the frequency of the E note, we should get a reading of 660 hz. That “given two notes their frequencies and string length is inversely proportional” is a mathematical representation of the fact that the string producing the E note would be vibrating twice as fast then when it was outputting an A note . Proper fractions and inverse relationships maybe some of the more sophisticated examples of math in music yet at the most basic level each performance starts with “a 1”, “a 2”, “a 1 2 3 4”.

Sports & Mathematics

2 December 2013
From baseball averages to calculating the velocity of a basketball to statistics needed for gambling in sports, math seems to be everywhere. However, math’s graph theory provides a visual scheduling system for teams and players long before any action has taken place on the field. Graph theory classifies a complete graph as one where different points on the graph (vertices) share only one connection. This is to say that if number of points on the graph are represented by n than the number of connections between the points can be predicted to be n-1. If each point is a player or a team, given four players there will be three matches for each player before every player had a chance to face every other player.
todo: put img here
Figure above shows the relationship of player 1 with the other players. Using graph theory, we can show this relationship for all the players with this graph:
todo: put img here
Such a graph maybe an overkill for small number of players but is valuable in more confusing circumstances. The number of connections at each point on the graph is also called the degree of the graph. The degree of the graph above is 3. To calculate the degree of the graph, subtracting one from the number of vertices is easy enough to remember, so I do not know why the formula n[(n-1)/2] is used to complicate the issue! But, this is a good reason to study graph theory in more detail.

George Boole

Boole stood out from the list of mathematicians because of his relations to the Boolean concepts in computer science.  His father, a shoemaker, was interested in science and taught him his first lesson in mathematics. This interest in science led to involvements with a social group in their English town of Lincoln called Lincoln Mechanics' Institution. The group promoted education and held discussions about science.  After a while, Boole’s father became the librarian for the group and provided the conditions to allow Boole to discover and learn foreign languages and mathematics.  Given access to books, combined with encouragements of his father Boole’s self-starting personality lead the way.    
Boole had to support his parents and siblings early on. For a few years, he did this by teaching at village schools and when he turned twenty, he opened his own school in Lincoln. Sources I looked at about him wan to point out that his work and responsibilities left him with little time to further his study but my personal feeling is that the daily grind with students and re hashing the basics probably contributed to insights that worked in his advantage.

Foundations of Boole’s interest in math were as a tool to solve mechanical problems in instrument making. This practical focus evolved into his many papers; one in particular one on differential equations which gained him the Gold Medal of The Royal Society of London. His talent in breaking down a problem into smaller parts and using algebraic formulas to move toward a solution is the basis for Boolean logic we use in computers today. This allowed for logical problems of sentences and words to be presented as algebraic problems, which are eventually solved mechanically. The general idea of classifying objects into sets and replacing the given set with a symbol somehow (my understanding of it is very limited) made 0 and 1 a special set of numbers called idempotent numbers, or numbers that do not changed when multiplied by themselves. In what he referred to as “The Rule of 0” in his book The Laws of Thought Boole states that an argument is valid if and only if after writing it as an equation and restricting the values for the symbols to only 0 or 1 we arrive at a valid equation in algebra. Boolean algebra correlates the operation of multiplication to the word “AND” and addition to the word “OR”.
With the industrial revolution came the use of electricity, a perfect fit for the Boolean logic system. Dividing an abstract problem into its elemental parts that can only be allocated values of 0 or 1 and evaluating the parts in order to validate the larger problem is time-consuming and cumbersome. But, the mechanical system of evaluation combined with speed of electrical circuits make Boolean logic viable. George Boole’s self-educated research method affords us this luxury.

Works Cited
Encyclopedia Britannica, George Boole. 11/29/2013 <http://www.britannica.com/EBchecked/topic/73612/George-Boole>.
Stanford Encyclopedia of Philosophy, George Boole. (2010). 11/29/2013 <http://plato.stanford.edu/entries/boole/>
Reville, William, University College, Cork. “The Greatness of George Boole” (1996). 11/29/2013 <http://understandingscience.ucc.ie/pages/sci_georgeboole.htm>.

Counting and Recording Without Writing

17 November 2013
I can count to 10 on my figures without taking my eye off the job. Perhaps I use the beads and the string to make a strand of 10 beads. This way every time I count from 1 to 10, utilizing the fingers of my hands, I move one of the beads along. This allows me to count up to hundred without losing count. After the counting process is completed, I would use two sticks to record the account. On one stick, I would carve a grove all the way around. One for every bead I had counted, therefore each grove would mark 10 sheep. On another stick, I would mark straight notches to record the counts on my fingers, the ones that were not transferred to the beads. The owner would take the two sticks as the record of his stock.

Vectors

I have always wanted to know about vectors. So far, I only knew that in the computer world if I was dealing with a vector-based image then I could manipulate he size if that image without it becoming pixelated. There is a lot to learn in this area. My basic understanding is that in a simplest form a vector-based line is different from an image of a line. An image of a line is hard coded information about a selection of pixels that form that line. However, a vector-based line is information about the direction and the length of that line. This is the reason that they –vector-based graphics- are always clear and not pixelated. Once we know the direction and the length of the line, we are free to draw it in way that suite our local environment. How this leads to functions, arrays, and what happens with shapes (collections of vectors) should be a fascinating study.

Early History of Numeration

Why did people first need numbers?

Probably the need for numbers is closely tied with people’s sense of tracking items and events in their lives. The number of nights since they last saw the moon in full. The number of livestock that were in pasture yesterday. Number of sunny days required to have a successful harvest. The answers to these questions as well as many others had direct effect on people’s lives. The symbols that are place holders for these answers eventually evolved to become what we currently call numbers and the method of evaluating them is numeration.

What were some of the ways people first kept numerical records?

The easiest method would have been replacement of real object or event with mare accessible substitute. Early human’s fingers were perhaps the most accessible and portable way of substitution. To compensate for the limitations of 10 fingers or 20 fingers and toes tally sticks and knotted strings were used. This led to further refinement and efficiency by grouping. A group could then be represented by a symbol pressed in clay or painted on a surface.

What were the number systems of ancient cultures like?

Once groping and symbolism combined to add efficiency to the system two general systems began to emerge. The literal systems such as the Egyptian and Chines systems used symbols that were noticeable in their daily lives. For instance Egyptians denoted one million with a figure of an astonished man. More abstract systems such as the Roman or Babylonians used more abstract symbols that resembles the alphabet.

Charles Lutwidge Dodgson

What other name did he use?

To come up with a pen name, Dodgson took his given name “Charles Lutwidge” and translated it to Latin: “Carolus Ludovicus”, reversed the first and last name: “Ludovicus Carolus”, and retranslated it into English: Lewis Carroll.

What are some of his published works under that name?

The Alice books are probably the best known of his publishing. His first book under his pen name was “Solitude” published in 1856. The Lewis Carroll society’s website has a long list of publications including “A Tangled Tale”, “Through the Looking Glass”, and “Rhyme? & Reason?”.

What was he known for under his real name?

He was a mathematician and lectured at Oxford England. He liked playing chess and designed many puzzle games. He spent 24 years perfecting his art of photography. He also served as Anglican deacon at Oxford.

What did he do for a living?

Lewis Carroll’s day job was teaching mathematics in college of Christ Church at Oxford University. He also published textbooks and academic papers on math and logic.

How he is specifically related to what we have studied?

Fifty eight of over two hundred items published by him were scholarly works on to mathematics and logic. He wrote nearly two dozen texts for students in arithmetic, algebra, plane geometry, trigonometry and analytical geometry. Many of his lectures and diaries were also published posthumously.
The Lewis Carroll society
http://lewiscarrollsociety.org.uk/pages/aboutcharlesdodgson/life.html
Modern Mechanicx
http://blog.modernmechanix.com/lewis-carroll-mathematician/
10 Wonderful facts about Lewis Carroll
http://www.youtube.com/watch?v=Rz8SiiTpODI
Alice In Wonderland - w/ historical commentary
http://www.youtube.com/watch?v=pxZ38Dq754Q&feature=share&list=PLWWN2YAAAlpfSOohWhn1SL_eklz3lPErj

Reasoning by Transitivity

The basic definition of reasoning by transitivity is easy to neglect. It is remarkable how regularly we use this concept of logic in our everyday life. The classic example a=b, b=c, therefore a=c is quick to say and easy to understand. But as I thought about the subject I realized that this concept is far more prevalent. I thought of various ways that we measure things in our daily lives and it became obvious. Looking at the ruler on my desk I realized that I do not have a tangible idea for what an inch is. But someone else, some trusted authority, does and has marked my ruler accordingly. So as I measure a line on a page all I know for sure is that the line I drew is equal to the distance between 0 and 1 on my ruler. An additional premise is that the distance between 0 and 1 on my ruler defines 1 inch. Therefore, by the power of Reasoning by Transitivity I can declare that the line I drew is one inch long. This means the validity of my declaration is completely related to the accuracy of my ruler’s manufacturer.

Evolution of Logic in Relation to Evolution of Numeration

Study of chapter four added perspective to the mecanical routine of dissecting word problems, applying memorized formulas, carrying out the underlying aritmatic, and discovering the answer to the prblem; namely practice of mathemathics. We regularly call math the language of science. The simpleast reason may be that if we can express an idea mathematically we have a bether path to suport or deny logical validity of such idea. This makes me wonder how cloesly did the early man, perhaps subconsciously, dabbled in logic as it started to form numbers and keep records.
An older, more learned member of clan may remember that clan members equal to the number of fingers on both hands went hunting and returned with food. Another day members equal to number of fingres on one hand went hunting and returned empty handed. Fairly shortly after realizing numbers.

What I liked and didn’t like about the set theory so far

The idea of two sets being equivalent is both helpful and confusing. I have to remind myself that equivalent sets are not necessarily equal sets. And this is the major source of confusion. Sets that have the same number of members are equivalent. This means that the determinant is the cardinality of sets that we are comparing, not their content. A funny example may be that the set of meals one might have in a day {breakfast, lunch, dinner} is equivalent but not equal to the set of arms on a clock {hour, minute, second}. I want to think that logical thinking requires that math (the language of science) provide a comparison system beyond equality. And the concept is probably not very complicated if I look at it from correct perspective. If I have a set of jet fighters and a set of fighter pilots comparing the individual members of each set with each other may be useless. But knowing that the two sets are equivalent allows me to say that I have a pilot for every jet, and that is a useful statement.

Uncommon symbols and their meaning

Last week’s lecture started discussion of logic and in designated “~” as the symbol for negation. I knew that in various programming languages “!” is used to refer to the opposite of a value, as in: if ! (foo.class = bar) then quz. So my mind wondered about different math symbols and I remembered an old question I had been meaning to look up. Namely, the difference between two stroke equal signs and three stork equal sign. A few google searches provided the answer. The difference between the two symbols only becomes evident when a variable is present in the statements involving the equal sign. The three stroke equal sign is used when the statement is valid regardless of what we replace the variable with. And the two stroke equal sign is designated for situations where only a limited set of values can replace the variable.
(x+2)^2≡x^2+4x+4 is valid for any replacement of x whereas (x+2)^2=16 is only valid if x is 2 or -6. So, the three stroke equal sign designates an identity, more on that later.