Students' contributions: The History of Infinity (10A Caleb Huang)

The concept of infinity has been an important component in the history and advancement of the knowledge in the areas regarding mathematics and philosophy since the beginning of the fourth century BC. The ancient Indian and Greek civilizations initially approached the concept of infinity from more of a conceptual philosophical aspect, rather than as a strict mathematical element. Ancient scholars, such as Aristotle, developed the foundations of the elements in infinity. Through the past several millenniums, the concept of infinity has been expanded and incorporated into the modern field of advanced infinitesimal calculus and the complex paradoxical philosophy in logic.

Caleb

Students' Contributions: Prisoner's Dilemma (by 9B Edward Shia)

Prisoner’s Dilemma

The game theory is applied mathematics used it strategic situations that range in many fields, such as: politics, economics, biology, philosophy, and international relations. The game theory basically explains that individual choice is affected by the choice of others. Prisoner’s Dilemma, or PD, provides a counterexample to the game theory. Prisoner’s Dilemma explains how individual choice is not affected by the choice of others.

Formulated by Merill Flood and Melvin Dresher, and later introduced with prison sentence payoffs by Albert W. Tucker, the prisoner’s dilemma is a problem in game theory that explains simply that individual choice is not based on the choice of others, regardless of what the others choose. In a typical PD scenario two people might not cooperate, even if cooperating gives the best penalties, because both people are angling for privileges. In a typical example of PD, two prisoners are caught by the police after a bank theft. They are put into separate cells, and obviously cannot communicate with each other. Since the police do not have enough evidence to convict them, the police cleverly offer the two prisoners the same deal. Each prisoner gets two choices, to confess or to remain silent. The police explain to both prisoners (in separate cells), that if both remain silent, their penalties would be shortened to a sentence of one year in jail. If both confess, both prisoners would have to serve a jail sentence of 5 years, but if either one of them confesses or the other remains silent, the prisoner that confessed can go free. The prisoner who remained silent will serve 10 years of jail sentence. The prisoner sentence payoffs are listed more clearly in the table below:

Prisoner A(Column) / B(Rows)	Confess	Remain Silent
Confess	5 years in jail for both prisoners	A: 10 years sentence, B: Freed
Remain Silent	A: Freed, B: 10 years sentence	1 year in jail for both prisoners

The prisoner’s dilemma explains the “self-profiting” choice in which a person would choose when presented with a two person conflict. Suppose prisoner A was given the two choices, being a rational thinker, prisoner A would think that if prisoner B confessed, it is better to confess so he would have to only serve 5 years in jail instead of 10 years, that is, if he didn’t confess. Similarly, prisoner A would think that if prisoner B did not confess, he could go free if he confessed. So overall, it is better to confess than to remain silent, still given that remaining silent would lessen the jail sentence for both prisoners. The same thinking applies on prisoner B. In this case, the prisoners are at a stage called the “dominant strategy equilibrium.” This means that both prisoners have a dominant strategy (to confess), and when dominant strategies are combined, this is the state called the “dominant strategy equilibrium.”

The prisoner’s dilemma successful presents the human instinct of “self-profiting” in practical real life situations. In one instance, 2 countries are given a choice, military expansion or reduction of weaponry. Both countries would profit if they choose military expansion, but if one country chooses to reduce weaponry, only one country would benefit (the one that choose military expansion). Regardless of what the other country chose, military expansion would be beneficial. So the countries would be more likely to choose to expand the military. In another practical example, countries are given to choice to reduce CO2 emissions, individual countries would benefit far greater to not reduce CO2 emissions than to other countries. Although it is beneficial to all countries if they cut the emissions all together, regardless of the choice the other countries chose, angling for profit, the countries are more likely to not alter the emission of CO2. Overall, the Prisoner’s Dilemma is both practical, and presents natural human responses in a realistic way when given two strategic choices, unaffected by the choice of others. 

Bibliography
1.    Wikipedia. Prisoner’s Dilemma. 5.16.2010. http://en.wikipedia.org/wiki/Prisoner's_dilemma
2.    Wikipedia. Game Theory. 5.16.2010. http://en.wikipedia.org/wiki/Game_theory
3.    Unknown. The Prisoner’s Dilemma. 5.16.2010. http://faculty.lebow.drexel.edu/McCainR//top/eco/game/dilemma.html

Students' Contributions: The Coastline Paradox (by 9B Angela Tien)

A coastline paradox is an observation based on intuition that landmasses/coastlines are not measured accurately. This proposition was first contemplated by L. F. Richardson (1881-1953). He thought that determining the length of a country’s coastline does not have a well defined length. In fact, he stated that the answer to the length of a coastline depends entirely upon the method used to measure it. Since landmasses can be shrunk down to any possible scale, there is no minimum scale. Any scale can be dwindled into a scale so incredibly small in measurements that it is almost impossible to measure. Therefore Richardson introduced a practical cogitation.

The order of the units being used to measure is the cogitation that Richardson has acknowledged. He affirmed that certain coastlines which are measured in miles, have measurements that ignored small variations less than a mile. Since the fjord is made up of many of the small variations, the scale of a fjord would be far from precise. When one use a ruler of a smaller length, the length of the actual border of the landmass increases. While, longer rulers diminish the more accurate length of the border, Richardson concluded that the smaller the units the border is measures in, the longer (precise) the border is.

Coastline paradoxes are also fractals. A fractal is an object or quantity that displays self-similarity (See textbook p. 590-591 --Mr. Ip). The curves of the coastline of a landmass would be similar to that of a Koch Snowflake. It is said that these curves have a dimension of either 1 or 2. Because the whole purpose of scales is to have the same measurement of the actual border as the scale, both would be a fractal since they’re similar.

It is impossible to actually measure the most precise length of any border because some even consider measuring atoms as well.

Students' Contributions: Rene Descartes and Analytic Geometry (by 9B Minnie Yuan)

Rene Descartes was both a famous philosopher and a great mathematician. He was born on March 31, 1596 in the southern part of France. In 1606, he entered the Jesuit College. In 1637, after he left the army, he had a quiet life and that’s when he started to have all his crazy ideas. He published a detailed report on how to use coordinates to locate points in space. That is known as analytic geometry, which is also better known as Cartesian geometry. Analytic geometry is used to find slopes, distances, midpoints, and stuff like that which you learned them in school. Despite his discovery in geometry, he also had a great contribution to philosophy. He is the person who said the famous quote: ‘‘I think, therefore I am. ” He had   written 3 important texts called Discourse on the Method of Rightly Conducting the Reason and Seeking Truth in the Sciences, Meditations on First Philosophy, and Principles of Philosophy.

I think Descartes is a really smart person. He is definitely a genius because he knows so many things in almost every area!  He is amazing to think about these weird formulas to calculate for distance and things like that. His contribution really made calculating math a lot easier. Without these formulas, we might still even be using a very complicated way to calculate math, or maybe, we won’t even be doing math! So in a way he is a great person, but in another way his discovery is kind of a mistake because he made us have to do math. Just kidding, I think he is a fabulous man. He is one of those people who contribute to the world, who makes people’s lives easier and better. He is a man that we should all respect and learn from.

Sources: 
http://www.answers.com/topic/ren-descartes
http://www.freeessays.cc/db/30/mdg2.shtml
http://www.renedescartes.com/Default.htm

Students' Contributions: Euclidean Geometry (by 9B Henry Chen)

Euclidean Geometry is the study of the properties of Euclidean space. Euclidean is the adjective of Euclid. Euclid is a mathematician in Greek, as known as Euclid the Alexandria. Often time people referred him "Father of Geometry". His book, Element, is one of the most significant and influential book in the history of mathematic. Euclidean Geometry is the principle in Element. Basically, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry. So, Euclidean geometry is made out of several basic postulates that has a dramatic influence on the later development of geometry. The property of Euclidean Geometry also reveals itself as a axiomatic system, which the statements are all true. Here are the postulates of Euclidean geometry in the book Element:

1.    To draw a straight line from any point to any point. - The statement means that a straight line can be drawn from on given point to another.2.    To produce [extend] a finite straight line continuously in a straight line. - The statement means that both ends on a single line can be extended endlessly.
3.    To describe a circle with any center and distance [radius]. - the statement means that a circle contains a center and all the points from the center with the same distance(radius) are joined to form a circle.
4.    That all right angles are equal to one another.
5.    The parallel postulate: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. - The statement means that the if two line is crossed by a transversal and the lines are parallel, then the same-side interior angles will be supplementary. If the angles are not supplementary and the sum is less than 180 degree, then the two lines will intercept at a certain point on the side of the angles.

Students' Contributions: The History of Pi and Its Approximation (by 9B Maggie Wang)

π is an irrational number whose value is the ratio of any circle’s circumference to its diameter. Its calculation continues to nowadays from the ancient times.

π was first discovered to begin agriculture and engineering skills since the permanent shelters were needed. In the late Greek period, Euclid (Right --Mr. Ip), a mathematician at the University of Alexandria in Egypt, provided many tools to future mathematicians to attack the π problems by publishing Elements. Archimedes (Lower right --Mr. Ip), who also studied at the University of Alexandria, approximated his value of π to about 22/7, which is still a common value today. After Archimedes was killed in the Roman conquest of Syracuse, the knowledge of π was demolished. It was in the decline of the Middle Ages that its knowledge was regained. Most of the historians believed that the Mayan value of π was more accurate than Europeans since Mayans were the top astronomers and had developed a very accurate calendar. During the Renaissance, π activity in Europe finally began to move again because of the increasing importance of mathematics for use in navigation and the creation of Arabic numerals. François Viète, a French lawyer, found the first formula for π based on an infinite sequence of algebraic operations. Viète also found the value of π to 9 decimal places. In the 1700's the invention of calculus by Sir Isaac Newton and Leibniz rapidly accelerated the calculation and theorization of π. In the late 1700's Lambert (Swiss) and Legendre (French) independently proved that π is irrational. Starting in 1949 with the ENIAC computer, digital systems have been calculating π to incredible accuracy. ENIAC was able to calculate 2,037 digits. Today, the approximation of pi is 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837297804995105973173281609631859502445945534690830264252230825334468503526193118817101000313783875288658753320838142061717766914730359825349042875546……

François Viète’s formula

The digits continue to 200 millions now. 

Sources:

http://edition.cnn.com/2010/TECH/03/12/pi.digits/index.html
http://library.thinkquest.org/C0110195/history/history.html

Students' Contributions: Archimedes and the Golden Crown (by 9B Tina Lin)

This is the 2nd entry in the "Students' Contributions" series.  You can read about interesting, unedited history/facts/problems of mathematics written by fellow students!  The writers may also share their thoughts on their topics! Please try to leave some comments (you don't need to log-in to leave a comment now!) to show some love!                         ----------Mr. Ip

 

Archimedes was a person like Leonardo Da Vinci, who was an expert in many different areas. He was a Greek mathematician, engineer, physicist, astronomer, and also an inventor. A biography about him had been written by his friend before, but it was lost. As a result, many details of his life are still unknown till today. Although the details of his life aren’t known by people today, almost everyone knows about his contributions to the “human knowledge”. The major subjects that Archimedes was famous for were mathematics and mechanical. He put a lot of effort into it, and came out with a lot of theories that no one ever thought of. In addition, he was sent to Alexandria by his father for his studies. After he went back to his birthplace again, the king kept him in his palace and sponsored him for his researches. Archimedes lived in this condition about ten years which helped him a lot with his discovery in different areas.

One of the famous stories about Archimedes was the “Golden Crown”. One day, King Hiero asked for a golden crown to be made for him. However, when the crown was finished, the king suspected if the person that made the crown had added any dishonest goldsmith into it. The king asked Archimedes to give him an answer according to his question without destroying the crown. 

This request gave Archimedes a really hard time. If he couldn’t cause any damage to the crown, it also meant that he wasn’t allowed to break the crown down and turn it into a regular shape, which was the simplest way for him to calculate. He was thinking about this question every moment in his daily life. Finally, one day while he was taking a bath, he found the answer of it. There was some water in his bathtub. When he put himself into it, the water came out of the bathtub. He then realized that the volume of the water that came out of the bathtub was equal to the volume of him. This was how Archimedes first came up with the idea on how to solve the king’s question.

However, his final accurate answer came out along with his theory of buoyancy. He compared the different volume of water that came out when he put the crown and the same amount of gold that the crown should have. Last he successfully proved that the person who made the crown was dishonest.

 

Archimedes almost gave his entire life to his country, because he not only discovered many theory in various areas that no one ever thought of, but he also came out with a lot of military inventions to help his country with the war. I really admire his knowledge. Unfortunately, he died because of the war, or I believe that he could have found out more unbelievable discoveries. Even before his death, his last words were still talking about circle. I guess that’s something I can never do in my entire life-putting so much effort on things just to look for an answer that he wanted.


Sources of the pictures:
http://celebrity.50g.com/htmlpage/photo.htm
http://en.wikipedia.org/wiki/Archimedes
http://sweetcandybubble.blogspot.com/2009/07/blog-post.html
http://www.wretch.cc/album/Lionchain

Notice----------------------------------->

Hey kids, if you don't know by now, there are individual class blogs that relay information to your specific classes! Check the right hand side under the title "sub-blogs" for those links! For example, click on G09 Geometry to go to the Geometry class blog.

Go vote in the poll in you class blog and leave some comments so I know you have visited!

Students' Contributions: The Golden Ratio (by 9B Lucy Chu)

Golden ratio is popular not only in mathematics but also in many other areas. It is used in mathematics, arts, and sciences. The ratio is said to be the most pleasing to the eyes.

“Two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to (=) the ratio of the larger quantity to the smaller one.” Which means,

Golden ratio is said to be a irrational number.

Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry. Euclid first provided the definition of the golden ratio: “A straight line that is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.”

During the Renaissance, many artists and architects had applied the golden ratio to their works. Leonardo Da Vinci painted the Golden Ratio. Also, Mona Lisa by Da Vinci has applied the golden ratio. In addition, the ancient Greek building, the Parthenon has many of its proportions approximate the golden ratio.

Statistics have shown that golden ratio can be applied to people’s different body parts. People who fit almost perfectly to the golden ratio most are said to be perfectly beautiful. Golden ratio can also be used in modern architecture. Our 2x3 Id photo and 3x5 index card, are all calculated to be golden ratio. Also Artists could use the ratio for drawing portraits. I think the golden ratio is very important in aesthetics. It is also very important how people are able to apply it for daily life. It is a wonder how people could discovered the ratio that is perfect for anything. The golden ratio is a perfect example for applying math on other areas.

Source:

http://www.geom.uiuc.edu/~demo5337/s97b/discover.htm

Be like the 12th graders!

Look at how they laugh in my Math class? And they HOLD on to their textbooks so dearly....

BE LIKE them! LOVE MATH and you will enjoy my class too!