Volume V, Issue Two
XEVĆŽX
June 2013
The Mathematics Magazine of Ramaz
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Volume V, Issue Two
June 2013
Inside This Issue Profiles: Kenneth Appel Dies (5) Christian Golbach’s Conjecture (9)
Math & the Past: A History of Math Education (3) The Rise of Fractal Art (4) A History of Pi (11)
Math & Problem Solving: Math: Discovered or Invented? (5) Game Theory (6) Public Polling (7)
Mathematical Fun: Crossword (7) Backgammon (12)
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History of Math Education Jacob Berman `16 Math education has changed dramatically over the years. It started with simple counting and today our knowledge of math has improved more dramatically than ever before. Archaeologists have found hieroglyphic Egyptian numbers dating back to 3,000
education started up again but advances took place mainly in the sciences. In the 18th century the most influential mathematician was Leonhard Euler who made many contributions to topography and calculus, and even created his own number! But in the 20th century math education really took off. This was also a time in which new electronics and the Internet were created which made it much easier for people to learn about math. Calculators are an example of this, invented in the 1960s.
B.C.E. Egyptian calendars and simple equations written on papyrus have also been found, but it was not until 518 B.C.E. that dramatic changes in math education occurred.
Nowadays in America and in many modern countries, education is better than it has ever been before. With education mandatory in these countries, the level of math that the average American knows now is better than 99% of people who have lived before us. With the rapid development of new technology, math education can only keep getting better and better.
In 518 B.C.E., Pythagoras opened a school in Croton, Italy. Hundreds of followers came to learn with Pythagoras and they were sworn to secrecy. They developed many theories, most famously the Pythagorean Theorem. It was then that the name “mathematic” was even coined. It was from there on that math education spread more quickly throughout the world. Chinese mathematicians developed a place value system, while the Hindu-Arabic numeral system is still in use today, and the number “zero” was independently developed by the Babylonians, Mayans, and Indians. The book Elements, written by Euclid in 300 B.C.E., was so influential that Euclid is known as the “father of geometry” because of it. Soon after, trigonometry and 360 degree angles were discovered. Then in the middle ages advances in math slowed greatly. Around the time of the Renaissance math 3
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The Rise of Fractal Art Tess Solomon `16 40,000 years ago, prehistoric people left their only written legacy on the walls of caves, depicting animals and human hunters. Ancient Egyptian art was a combination of animal-like gods, people, and often, the Nile River, their main source of life. Classical Greek statues attempted to recreate the human body in stone with perfect proportions. The Renaissance emphasized realism, and artists attempted to put exact replicas of the world on paper. For as long as it has existed, art has attempted to imitate nature through devices artists learned. However, it wasn’t until 1982 that this desire to recreate the natural world could be understood with near-mathematical certainty. It was in 1982 that Benoit Mandelbrot published his hugely influential book, The Fractal Geometry of Nature (an expanded and revised version of its 1975 and 1977 predecessors.) The novel illustrated the importance of fractals in nature.
close—and seeing exactly the same pattern on a smaller scale. Imagine getting closer until you are two feet away from the sand—and the same pattern (or a very similar pattern) confronts you, this time in terms of centimeters inland and towards the ocean. Mandelbrot proved that fractals are an effective way of understanding much of nature: the shapes of clouds, the lines on a leaf, and the shape of trees, among many other examples. With this new understanding of patterns within nature, and with the relatively new advent of computer technology and programming, artists such as Scott Draves, Carlos Ginzburg, and William Latham used the mathematics of fractal geometry to express the phenomena of nature in a new way. This new art
But what is a fractal? Fractals are self-similar patters
form was not dictated by gentleness of hand or brushstroke, but by degrees in computer science and mathematics, and the ability to understand the complicated mathematics behind fractals (for example, School of Mathematics at Georgia Institute of Technology requires two years of calculus as a prerequisite.) that recur at varying dimensions of observation. They are figures with an infinite amount of detail: they do not become simpler when magnified, but remain as complex as they were without magnification. For example, consider a coastline. From an airplane, it is clearly not smooth but has huge undulations inland and towards the ocean, measured in kilometers. Imagine getting twice as
“Geometry is concerned with making our special intuitions objective,” Michael Barnsley writes in his book, Fractals Everywhere. Not only did Mandelbrot give us the mathematical understanding of a new form of artistic expression – but by objectifying our intuitions about nature into mathematical formulae, more complicated than we could ever imagine, he expanded our perception of the world around us from the largest to the smallest.
Life is complex: it has both real and imaginary components.
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Kenneth Appel Dies Sarah Ascherman `16 Kenneth I. Appel, the mathematician who proved that “four colors suffice,” died on April 19, at the age of 80. Appel was born in Brooklyn in 1932 and earned his Ph. D in math at the University of Michigan in 1959. With the help of an I.B.M computer, Appel and his colleague Wolfgang Haken helped to solve a long existing problem of colors on a map in 1976. Appel proved that only four colors were needed to make a world map or any map. After 1,200 hours, or 50 days, and 10 billion calculations made by the I.B.M computer, Appel and Haken’s proof deduced that four colors would indeed be adequate to make a map of the world’s
countries in which no two adjacent countries were the same color. In 1977, Appel and Haken’s work was published in the Illinois Journal of Mathematics. However, many mathematicians were not convinced with this proof that a computer helped formulate, and that they could not physically see, which consequently led to a debate about what a mathematical proof should involve. However, other experts have found this work truly inspirational; in fact, mathematician Kevin Short said that this proof “has spawned whole fields of study.”
Math: Discovered or Invented? Brandon Cohen `14 For centuries people have debated whether mathematics is discoverable, or simply invented by the minds of great mathematicians. While some rules such as PEMDAS are invented because an order needs to be established, the acts of multiplying, adding, subtracting, and dividing were actually discovered. Those operations were used in ancient times before an idea of “math.” Putting two things together is adding whether you know what adding is or not. But while those operations are discovered, i (√-1) was invented because mathematicians needed something that represented an unreal number. No number multiplied by itself can be negative so i is like the Liar’s Paradox in that it is impossible to determine what it actually means. The Liar’s paradox is very intriguing in that the truth of the statement can never be determined. It states that: if "this is a lie" is true, then the sentence is false, which would mean that it is actually true, which would mean that it is false,
and so on and so forth. Conundrums such as these were invented because they never actually exist in reality. Number theory is all discovered. While mathematicians give certain numbers names, their properties are not invented but rather discovered. When a new prime or Mersenne prime is found, the media describes it as having been discovered not invented. While Euclidean Geometry relies on certain postulates, it was also discovered. Pythagorean’s Theorem is something that was always true and simply wasn’t acknowledged until Pythagorean came along. Operations such as logarithms and matrices are just ways of formatting a specific verbal equation. Thus, one can conclude that almost all things in math are discovered because the concepts existed beforehand, they just didn’t have symbols that represented them; nonetheless, mathematics remains as a mix of discovery and invention. 5
Mathematics is made of 50 percent formulas, 50 percent proofs, and 50 percent imagination.
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Game Theory Jessica Gruenstein `14 In the article “Game Theory: Jane Austen Had It First,” (New York Times, April 22, 2013) Jennifer Schuessler summarizes a book written by Professor Michael Chwe about Jane Austen and the game theory. Chwe made the connection between the classic author and John von Neumann’s idea of social warfare when he was watching the movie “Clueless,” a comedy based off Austen’s famous novel “Emma.” John von Neumann’s “Theory of Games and Economic Behavior” (published in 1944) imagined social interaction as a series of moves and countermoves aimed at maximizing “payoff.” This book lays down the foundation for the modern Game Theory, a kind of study of strategic decision making. Chwe argues that though Neumann’s book was indeed groundbreaking, Austen had covered the “philosophical groundwork” for the Game Theory in her novels over a century earlier. One aspect of the Game Theory into which Austen delves is that of “Cluelessness,” when someone
considered to be a social inferior manipulates someone of a higher social standing, while their social differences make the “higher” party unsuspecting of manipulation. Chwe offers an example of Cluelessness in Austen’s Pride and Prejudice. In the novel, Lady Catherine de Bourgh demands that Elizabeth Bennet promise not to marry Mr. Darcy. When Elizabeth refuses to do so, Lady de
Bourgh tells this to Mr. Darcy, meaning to call attention to Elizabeth’s insolence. Lady de Bourgh does not know, however, that she is being manipulated by Ms. Bennet to let Mr. Darcy know that she is still interested in marrying him. Lady de Bourgh is completely unsuspecting of Elizabeth because she is of a lower social standing than Lady de Bourgh. One might apply this same thinking to military warfare. Calling an enemy by a name that implies weakness might impede one side’s ability to strategize clearly. This is an example of Cluelessness because thinking of an enemy as “animals” or “weaklings” might cause one side to underestimate or anticipate incorrectly the power of that party, completely changing the outcome of a battle. This book has received both acclaim and criticism. Some English scholars believe that Chwe’s book was rather unoriginal. Austen scholars “will not be surprised at all to see the depths of her grasp of strategic thinking and the way she anticipated a 20th-century field of inquiry,” said Laura J. Rosenthal, a specialist in 18th-century British literature at the University of Maryland. Others might say that Chwe’s idea, though perhaps already thought of, was well-presented and well-argued, and that it cogently blends the two worlds of psychology and literature. 6
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Volume V, Issue One
December 2012
Math Crossword Ben Kaplan `16
Public Polling Alex Weinberg `14
Democracy, by its nature, is fraught with uncertainty. Because the will of the people is so fickle, it is often difficult to predict the outcome of elections. That unpredictability has led many to try and discern the results of an election or a referendum prior to the election itself using scientific methods. The first recorded example of opinion polling was a
local straw poll conducted by The Harrisburg Pennsylvanian in 1824. The Pennsylvanian poll called the election for Andrew Jackson winning the presidency over John Quincy Adams 335-169. The success of the poll at predicting the winner of the state, and the popular vote in the country made opinion polling a popular phenomena during the 19th century, although polling remained local and 7
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Volume V, Issue Two un-scientific. Straw polls, however, are not the most accurate way to check the pulse of a people. This is evident in the disparity between the predictions of straw polls and scientific polls in the 1936 presidential election. The Literary Digest polled their readership, 2.3 million voters, on whether they would vote for Franklin Roosevelt or Alf Landon in the upcoming presidential election. The Digest called the election for Landon whereas George Gallup, who only polled 50,000 voters, predicted Roosevelt’s victory using a more demographically representative pool of respondents.
June 2013 representing a demographic group or wording the questions in such a way that skews the results towards one answer. Another common issue is nonresponse bias. If Democrats are 30% of the
From then on out, scientific polling grew much more popular. However scientific opinion polling is not exact as seen through population in total but make up only 10% of those who choose to pick up the phone then the poll will understandably misrepresent the population. Another common error is failing to call cell phones for responses. Around 30% of the the United States population is only reachable through a cellphone, and as a result polling firms that only call landlines often are biased due to the lack of cellphone-only owners included within the sample.
the failure of all polling firms in the UK General Election of 1992. All the polling firms predicted that the Labour Party would sweep in power but when the results came in, the Conservative Party had won a devastating electoral victory. Some problems with scientific polling include the propensity for sampling error. In order to accurately predict the opinions of the whole population, the firms need to have an representative sample of the total population. Common reasons for error include over or under
Correction for biases like these is part of the reason that poll aggregators like Nate Silver, Drew Linzer, and Josh Putnam are able to so accurately predict election results. For example, in the 2012 election Nate Silver correctly predicted the winner in all 50 states, including the nine swing states. Rasmussen, a standard polling firm, only correctly predicted three out of nine swing state races. Ultimately polling, like all statistics, can never be 100% accurate, even aggregate polling like Nate Silver’s; however the capacity for error is taken into account when publishing results. The value in polling comes not with snapshot views of the public’s opinion but with long-term trends in the thinking of the representative sample and with the nation as a whole. 8
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Golbach’s Conjecture Ben Kaplan `16 271 years ago a problem arose, and one month ago it was partially solved. That problem was the Golbach Conjecture, one of many theories regarding prime numbers. Christian Goldbach, a Prussian mathematician, stated in 1742 that every number greater then two is equaled to the sum of three prime numbers. A prime number is a number, which only has two factors – itself, and the number one. There are some possible traces of prime numbers dating back to ancient Egypt, but the earliest record specifically relating to the study of prime numbers comes from ancient Greece. The record is from Euclid’s book Euclid’s Elements. Euclid lived around the year 300 B.C.E. The study of prime numbers arose again in the 18th century by Goldbach and Euler. Ever since Goldbach made his original conjecture, his conjecture has been split up into two parts: the weak and strong. The weak conjecture says that all odd numbers greater then five are equal to the sum of three prime numbers. The strong conjecture states that all even numbers greater than
four are equal to the sum of two prime numbers. The greatest mathematicians have obsessed with these conjectures for centuries. These mathematicians include G. H. Hardy, who helped form a crucial ingredient for public cryptography, J. E. Littlewood, who helped Hardy develop the circle method, I .M. Vinogradov, who developed the circle method separately, and Terence Tao. Terence Tao came close to solving the conjecture last year when he proved that any odd integer is the sum of at most five prime numbers. But in early May this year Harald Andres Helfgott, from Paris, seems to have proven the weak theory. He uses the circle method in his proof. Even though it has not been formally published yet, top mathematicians believe his findings are true and have endorsed his proof.
A Perfect Bracket? Ben Rabinowitz `16
Yes, a perfect bracket can be made; however, it takes a lot of luck. As everyone knows, one of the rarest things to see in sports is a perfectly filled out March Madness bracket. People have been close however; no one has had a perfect article yet. One of the biggest factors that make it so difficult to make a perfect bracket is the fact that there are so many upsets that are difficult to predict. Also, if there is an upset in an early round, it affects later round scores as well, because you didn’t even have that team
advancing to that game. So, while it might seem easier than it actually is to fill out a perfect bracket of the 64 team field (excluding the First Four for all purposes, considering a 16-seed has never beaten a 1-seed in all of history, though it is looking like it could happen pretty soon, but those odds are for a different article) even if you fill out a 100 brackets it is still nearly impossible to have a perfect bracket.
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What is the difference between a Psychotic, a Neurotic and a mathematician? A Psychotic believes that 2+2=5. A Neurotic knows that 2+2=4, but it kills him. A mathematician simply changes the base.
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So, what are the actual odds of filling out a perfect bracket in March? Well, there is a simple way to go about this: take the number 2 (number of teams per game) and put it to the 63rd power (number of games in the tournament). That gives you approximately
who picked two 15-seeds to beat two 2-seeds last year, and FGCU to make it to the sweet sixteen this year? Probably, no one. But, you never know. And luckily for all of us, usually, to win the pool, a perfect bracket isn’t needed!
9,200,000,000,000,000,000 (9.2 quintillion!) So, the odds of filling out a perfect bracket are 1 to a very, very, very large number. But, this is not actually the odds that people can look at because it doesn’t take into account number one seeds beating sixteen seeds and other obvious wins for some games. So, according to DePaul math professor Jay Bergen, the actual odds are more like 1 to 128 billion. Sorry to say to all people filling out brackets in the future, but using those odds, only one bracket per 400 years will be perfect So, can anyone pick a perfect bracket? Technically speaking, yes, with a lot of luck involved. I mean
Exponential Growth Gabe Silverman `16
An exponential function is one represented by the equation y = ex. This graph is upward sloping, and increases at a much faster rate as x increases, because x is an exponent. For example, in our equation, if x = 2, then y = e2, or approximately 7.4, but if x = 3, then y = 20, and if x = 4, then y = 54.6, and so on. An exponential function will have a domain of all x, but a range of y>0, because a number to a degree can never equal a negative value. Exponential functions are often used to graph populations, because populations will grow in this manner if left alone with unlimited resources and no limiting factors. However, once the environment can’t hold a larger population, the
exponential growth levels off into what is known as the carrying capacity. The graph of exponential growth followed by the leveling off of a population is called a logistic growth curve, and is shaped like an “S”. For thousands of years, the human population grew very slowly. This was due to several limiting factors such as scarce food supplies, primitive medicinal technology, and disease caused by lack of hygiene. However, starting with the Industrial Revolution in the 18th century, the human population began growing exponentially. This was because advancements in agriculture, industry, trade, technology, medicine and
Q: How do you tell that you are in the hands of the Mathematical Mafia? A: They make you an offer that you can't understand.
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sanitation dramatically reduced the death rate, while the birth rate still remained high. Current estimates say that there could be up to 10 billion people by 2050. If this exponential growth continues uninterrupted, we could run out of space and resources to support the massive world population, let alone the damage down to the environment. However, most demographers, scientists who study the growth and density of populations, say that all is not lost. There is a concept called the demographic transition, which refers to the stages of human population growth. The hypothesis is that as
countries begin to modernize, households will have fewer children, and the birth rate will begin to fall until it eventually meets the death rate, as shown in the graph below. This concept can already be seen in developed countries like the United States, Japan, and much of Europe, whose population growth has slowed dramatically in the past 30 years. As more countries move toward this stage in the demographic transition, the population growth will level off into a logistic curve, so in effect, we are probably not doomed.
A History of π Max Teplitz `16 Pi, or in greek, π, is an irrational number that is used in everyday mathematics. Pi is approximately 3.14, (22/7, 355/113 and so on…) and also represents the number of times a circle’s diameter goes into its circumference. Roughly 4000 years ago, ancient Babylonians would find the area of a circle by squaring the radius and then multiplying it by 3. Here, pi is equal to roughly 3. Also, some other mathematics from the time indicate that Pi could also have been 3.125, which is closer to today’s definition. An ancient Egyptian mathematician named the Rhind Papyrus used pi at around 3.1605. However, the first accurate calculations known today are by Archimedes a Greek mathematician, and Zu Chongzhi a Chinese mathematician. The two different men used very different approaches to finding the number, both genius.
sided polygon, and did incredibly lengthy calculations, which get that the ratio of a circumference to a diameter is roughly 355/113, which he most likely achieved by calculating to the 9th decimal point, and hundreds of square roots! The greek letter π was originally introduced around 300 years ago, and the idea to use it has obviously stuck. Today, π is used for many purposes, including measuring the abilities of computers, it is used in radian measure, it can be used to measure AC voltage, and measures many types of waves. Pi is a great discovery and has many uses; many institutions have relied on Pi to develop as much as they have!
Archimedes used previous work from Pythagoras, known as the Pythagorean Theroem (a2 + b2 = c2) to find two different polygons where one was inscribed inside of the circle, and one where the circle was circumscribed inside of it. Through this method, Archimedes was not able to directly get to pi, but he did realize that pi was between 3 1/7 and 3 10/71. Zu Chongzhi’s method was slightly different, but we can’t be 100 percent sure because his work has been lost. He also probably would not have been able to access and use the work of Archimedes and Pythagoras, so he had to do it without those methods. Instead, he started with an inscribed 24,576 11
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A Good Mathematical Game of Backgammon Max Teplitz `16 Backgammon, often referred to as “Shesh-Besh” or “Tolleh”, is a two-person board game, in which players roll dice to move all their pieces onto one side of the board, called the “house”. The dice indicate the number of places a player may move their piece; one player’s pieces must be on one space, and if the opposite player has only one piece on that space it is taken off and they must roll back in. Many believe that the dice take up all the skill in the game, and that winning is random, however, there is much more skill involved in the game; the question that remains is how much. First off, there are odds involved with the dice. In backgammon, normally there are two moves, but when doubles are rolled (two of the same number) the number of moves increases to four. The most commonly rolled number is seven, whether it be 3 and 4, 2 and 5, or 6 and 1. After 7, 6 and 8 are the second most frequent, then 5 and 9, then 4 and 10, then 3 and 11, and finally 2 and 12 are the least common. Only half of all possible rolls can be doubles. The probability of rolling doubles is approximately 16 percent, the same as the chances
of rolling a seven.
Once the dice are out of the equation, the only thing left as a factor in the outcome is the player. The player is not only fighting against the dice rolls, but he is also fighting his opponent. The player starts out with his pieces four spaces away from his opponent, in the first half, and five spaces away in the second half. Once the players roll for first move, the players need either doubles, or rolls where the numbers on each die are separated by the number of spaces their pieces are away from each other, otherwise they will have open pieces, which can be taken off. The odds of the aforementioned move happening are definitely in the minority and so players must make do with what they have. Obviously as the game progresses, the players will have more opportunities to move their pieces together, and change their pieces’ positions from the original set up. Backgammon is a very complicated game that requires focus and skill. The majority of the skill is left to the player, to decide what his opponent will do. If he knows the odds of putting his opponent in a disadvantageous position, he will likely do better. What makes a good backgammon player is not what they roll, but what they do once they roll.
Faculty Advisor:
Contributors:
Dr. Koplon
Sarah Ascherman `16 Jacob Berman `16 Jessica Gruenstein `14 Ben Kaplan `16 Ben Rabinowitz `16 Gabe Silverman `16 Tess Solomon `16 Max Teplitz `16 Alex Weinberg `14
Editors: Brandon Cohen `14 Dan Korff-Korn `14 Layla Malamut `14
Theorem: There are two groups of people in the world; those who believe 12 that the world can be divided into two groups of people, and those who don't.