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MAT Exam Expository Papers

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Perfect Numbers:, Diana French May 2010

Perfect Numbers:, Diana French

MAT Exam Expository Papers

While this topic of “Perfect Numbers” was completely new to me (at least to the degree at which it is discussed within this paper,) I found it very intriguing and believe there is still much information and mathematical discovery in it for me. There were many historical points of interest, and I found it difficult to whittle them down to a manageable size for the intent of this paper. Likewise, there were many facts and peculiarities I found interesting and certainly worthy of consideration. However, to maintain anything close to a reasonable length of discussion as outlined in the guidelines ...

Powers, Marci Ostmeyer Jul 2009

Powers, Marci Ostmeyer

MAT Exam Expository Papers

Powers, or exponents, give us a convenient, shorthand way to denote that a number is multiplied by itself a given number of times. The number being multiplied is called the base, and the number of times it appears in the product is the power.

Sicherman Dice, Scott Johnsen Jul 2009

Sicherman Dice, Scott Johnsen

MAT Exam Expository Papers

Many different types of games involve the use of numbered dice. The most common type of die used is the standard six-sided die with the numbers one through six used on its six sides. What if there was another way of numbering a set of dice (using only positive integers) that would create the same probability outcomes as those of a standard set of dice? The February 1978 issue of Scientific American reports that George Sicherman discovered such a numbering (Broline, 1979). Sicherman discovered that two cubic die numbered 1-2-2-3-3-4 and 1-3-4-5-6-8 have the same sum probabilities as do the ...

The Monty Hall Problem, Brian Johnson Jul 2009

The Monty Hall Problem, Brian Johnson

MAT Exam Expository Papers

In the game show "Let's Make a Deal", host Monty Hall would present a contestant with three doors. Behind one door is a car and behind the other two doors are goats. The contestant picks one door, after which Monty opens one of the other doors, showing a goat. Monty then offers the contestant the opportunity to keep the same door or switch to the other unopened door; the contestant will get to keep whatever is behind that door. Should the contestant switch?

Palindromes, Stephanie Fuehrer Jul 2009

Palindromes, Stephanie Fuehrer

MAT Exam Expository Papers

A palindrome is a word or phrase that reads the same forward and backward, such as the word “level” and the phrase “Madam, I’m Adam.” Numbers whose digits read the same forward and backward are also called palindromes, such as the numbers 22, 1234321, and 2002. Palindromic phrases can also occur in number form, for example the Universal day of Symmetry: 8:02 P.M. on February 20, 2002. If one were to look at the time on a twenty-four hour clock, it would read 20:02; the date can be read as the twentieth day of the second ...

Sudoku, Marlene Grayer Jul 2009

Sudoku, Marlene Grayer

MAT Exam Expository Papers

Many people have a magazine, a crossword puzzle, a newspaper, or a book on hand to fill idle time. I, however, prefer to tease my mind with a book of Sudoku puzzles (pronounced sue‐dah‐coup), trying to figure out what number goes where while resisting the temptation to peek in the back at the answer to make sure I am on the right track. For those of you who are unaware of what Sudoku is, it is a highly popular number game whose notoriety may even surpass the crossword or word search puzzles. It can be found in popular ...

Amicable Pairs, Lexi Wichelt Jul 2009

Amicable Pairs, Lexi Wichelt

MAT Exam Expository Papers

The ancient Greeks are often credited with making many new discoveries in the area of mathematics. Euclid, Aristotle, and Pythagoras are three such famous Greek mathematicians. One of their discoveries was the idea of an amicable pair. An Amicable pair is a pair of two whole numbers, each of which is the sum of the proper whole number divisors of the other.

Wythoff’S Game, Kimberly Hirschfeld-Cotton Jul 2008

Wythoff’S Game, Kimberly Hirschfeld-Cotton

MAT Exam Expository Papers

Wythoff’s Game, named after Willem Abraham Wythoff, is a well known game amongst number theorists. Dr. Wythoff, who received a Ph. D. in mathematics from the University of Amsterdam in 1898, described this game in these words: “The game is played by two persons. Two piles of counters are placed on a table, the number of each pile being arbitrary. The players play alternately and either take from one of the piles an arbitrary number of counters or from both piles an equal number. The player who takes up the last counter or counters, wins.” This game was previously ...

The Game Of Bridg-It, Sandy Dean Jul 2008

The Game Of Bridg-It, Sandy Dean

MAT Exam Expository Papers

On the surface, Bridg-It appears to be a simple game of connecting dots to form lines across the board. Playing Bridg-It is simple. Understanding and playing Bridg-It well is more complicated. To understand the theory and strategy behind Bridg-It, one must first understand certain elements of graph theory, such as disjoint spanning trees. Only then can one master the game of Bridg-It.

The Polar Coordinate System, Alisa Favinger Jul 2008

The Polar Coordinate System, Alisa Favinger

MAT Exam Expository Papers

Representing a position in a two-dimensional plane can be done several ways. It is taught early in Algebra how to represent a point in the Cartesian (or rectangular) plane. In this plane a point is represented by the coordinates (x, y), where x tells the horizontal distance from the origin and y the vertical distance. The polar coordinate system is an alternative to this rectangular system. In this system, instead of a point being represented by (x, y) coordinates, a point is represented by (r, θ) where r represents the length of a straight line from the point to the ...

De Bruijn Cycles, Val Adams Jul 2008

De Bruijn Cycles, Val Adams

MAT Exam Expository Papers

Problem 1. A robot is moving on a cyclic track. The track is marked at evenly spaced intervals with 0s and 1s, with a total of 8 marks. The robot can see the 3 marks closest to him. How should the 0s and 1s be put on the track so that the robot knows where on the track he is by just looking at the 3 closest marks?

Problem 2. The city of Konigsberg, Prussia is set on the Pregel River and includes two large islands, which are connected to each other and the mainland by seven bridges. Is it ...

Archimedean Solids, Anna Anderson Jul 2008

Archimedean Solids, Anna Anderson

MAT Exam Expository Papers

A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a threedimensional solid which consists of a collection ...

Mathematics And Evolution, Kacy Heiser Jul 2008

Mathematics And Evolution, Kacy Heiser

MAT Exam Expository Papers

What is game theory and how does it apply to Evolutionary Biology? Game theory was originally developed to help portray how individuals interact with each other and it was mainly used in the economics and political science fields. This area of mathematical study came about after World War II and was developed by John von Neumann and Oskar Morgenstern. It wasn’t until the 1970’s that George Price and John Maynard Smith began to apply game theory to evolution in trying to predict the behavior of animals in a species. There are two main evolutionary trends we will study ...

Lester’S Circle, Julie Hoaglund Jul 2008

Lester’S Circle, Julie Hoaglund

MAT Exam Expository Papers

Euclid is credited with most of the theorems in geometry textbooks today. Around 300 B.C., Euclid produced a thirteen volume publication called The Elements. These volumes include much information others had studied. Throughout history, many great mathematicians including Pasch, Hilbert, and Birkhoff have studied and tried to improve Euclidean geometry. Groups such as University of Chicago School Mathematics Project have made improvements on the Euclid’s axiomatic system.

The Kaprekar Routine, Emy Jones Jul 2008

The Kaprekar Routine, Emy Jones

MAT Exam Expository Papers

There is a story about a man named “Joe” whose wife sent him to the supermarket. Joe was never a very good listener – he tended to pick up on the major points of a conversation, but never seemed to get things in the right order. So when he arrived at the store, he headed straight for the garlic section (for he was sure that his wife had mentioned garlic). However, when he got there, he stared at the $10 bill in his hand. She had asked him to buy $4.95 worth of garlic, or was it $9.54, or ...

The Square Root Of I, Tiffany Lothrop Jan 2008

The Square Root Of I, Tiffany Lothrop

MAT Exam Expository Papers

While Girolamo Cardano was working on solving cubic and quadratic equations in 1539, he encountered some formulas that involved square roots of negative numbers. In 1545 Cardano published Ars Magna, where he presents the first recorded calculations that involve complex numbers. Then in 1572, Rafael Bombelli published the first three parts of his Algebra. He is known as the inventor of complex numbers, because he identifies some rules for working with them. Bombelli also shows how complex numbers are very important and useful. From Bombelli’s list of rules for adding, subtracting and multiplying the complex numbers, he was able ...

The Polygon Game, Kyla Hall Dec 2007

The Polygon Game, Kyla Hall

MAT Exam Expository Papers

The Polygon Game ‐ Take a regular, n‐sided polygon (i.e. a regular n‐gon) and the set of numbers, {1, 2, 3, …, (2n‐2), (2n‐1), 2n}. Place a dot at each vertex of the polygon and at the midpoint of each side of the polygon. Take the numbers and place one number beside each dot. A side sum is the sum of the number assigned to any midpoint plus the numbers assigned to the vertex on either side of the midpoint. A solution to the game is any polygon with numbers assigned to each dot for which all ...

Extending A Finite Sequence, Jessica Fricke Jul 2007

Extending A Finite Sequence, Jessica Fricke

MAT Exam Expository Papers

One of the most common mistakes in school mathematics is to list a few terms of a sequence and ask “What term comes next?” For example, a teacher may write: 1 2 4 … and ask what term comes next? The best answer is “almost anything could be the next term.” But people often find that answer unsatisfying. It helps, instead, to give an example. Perhaps, one might respond by saying: “The next number might be 8, but then again, it might be 7.” This answer might, of course, result in an angry teacher, so you need to be prepared with ...

Evaluating Polynomials, Thomas J. Harrington Jul 2007

Evaluating Polynomials, Thomas J. Harrington

MAT Exam Expository Papers

Computers use algorithms to evaluate polynomials. This paper will study the efficiency of various algorithms for evaluating polynomials. We do this by counting the number of basic operations needed; since multiplication takes much more time to perform on a computer, we will count only multiplications. This paper addresses the following: a) How many multiplications does it take to evaluate the one-variable polynomial, Σ= + + + + = n i i i n n a a x a x a x a x 0 2 0 1 2 ... when the operations are performed as indicated? (Remember that powers are repeated multiplications and must be counted ...

Ethnomathematics, Chad Larson Jul 2007

Ethnomathematics, Chad Larson

MAT Exam Expository Papers

When asked to think about a foreign country the first thing that comes to my mind is the language barrier and the customs that accompany that specific country. The culture of the citizens and how it differs from my culture are also things which peak my interest. Things which I view as “normal” may seem very odd to someone who lives thousands of miles away, and likewise, traditions that have been past down from generations of people from distant lands may seem peculiar to me. These customs and cultures of which I speak are also the things that make this ...

How To Graphically Interpret The Complex Roots Of A Quadratic Equation, Carmen Melliger Jul 2007

How To Graphically Interpret The Complex Roots Of A Quadratic Equation, Carmen Melliger

MAT Exam Expository Papers

As a secondary math teacher I have taught my students to find the roots of a quadratic equation in several ways. One of these ways is to graphically look at the quadratic and see were it crosses the x-axis. For example, the equation of y = x2 – x – 2, as shown in Figure 1, has roots at x = -1 and x = 2. These are the two places in which the sketched graph crosses the x-axis.

Experimentation With Two Formulas By Ramanujan, Daniel Schaben Jul 2007

Experimentation With Two Formulas By Ramanujan, Daniel Schaben

MAT Exam Expository Papers

Srinivasa Ramanujan was a brilliant mathematician, considered by George Hardy to be in the same class as Euler, Gauss, and Jacobi. His short life, marred by illness and tragic educational events, was unique in the history of mathematics. Mathematical discoveries are still being gleaned from his personal notebooks. Paper was a hard commodity to come by so his notebooks were a cluttered mix of pen over pencil mathematical hieroglyphics. The following highlights Ramanujan’s life in connection with Hardy, his work with ellipses, and his work with the partition function.

Master Of Arts In Teaching (Mat), Josh Severin Jul 2007

Master Of Arts In Teaching (Mat), Josh Severin

MAT Exam Expository Papers

The number zero is a very powerful tool in mathematics that has many different applications and rules. An interesting fact about the number zero is that according to our calendar (the Gregorian calendar), there is no “year zero” in our history. There is also no “zeroth” century as time is recorded from centuries B.C. to the 1st century A.D. However, certain calendars do have a year zero. In the astronomical year numbering system year zero is defined as year 1 BC. Buddhist and Hindu lunar calendars also have a year zero. In this paper I am going to ...

Hyperbolic Geometry, Christina L. Sheets Jul 2007

Hyperbolic Geometry, Christina L. Sheets

MAT Exam Expository Papers

In general, when one refers to geometry, he or she is referring to Euclidean geometry. Euclidean geometry is the geometry with which most people are familiar. It is the geometry taught in elementary and secondary school. Euclidean geometry can be attributed to the Greek mathematician Euclid of Alexandria. His work entitled The Elements was the first to systematically discuss geometry. Since approximately 600 B.C., mathematicians have used logical reasoning to deduce mathematical ideas, and Euclid was no exception. In his book, he started by assuming a small set of axioms and definitions, and was able to prove many other ...

The Volume Of A Platonic Solid, Cindy Steinkruger Jul 2007

The Volume Of A Platonic Solid, Cindy Steinkruger

MAT Exam Expository Papers

A regular tetrahedron and a regular octahedron are two of the five known Platonic Solids. These five “special” polyhedra look the same from any vertex, their faces are made of the same regular shape, and every edge is identical. The earliest known description of them as a group is found in Plato’s Timaeus, thus the name Platonic Solids. Plato theorized the classical elements were constructed from the regular solids. The tetrahedron was considered representative of fire, the hexahedron or cube represented earth, the octahedron stood for gas or air, the dodecahedron represented vacuum or ether (which is made up ...

Simple Statements, Large Numbers, Shana Streeks Jul 2007

Simple Statements, Large Numbers, Shana Streeks

MAT Exam Expository Papers

Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, as defined by Wikipedia (2007). Large numbers typically refer to large positive integers, or more generally, large positive real numbers, but may also be used in other contexts. Very large numbers often occur in fields such as mathematics, cosmology, and cryptography. Sometimes people refer to numbers as being “astronomically large”. However, it is easy to mathematically define numbers that are much larger than those even in astronomy. We are familiar with the large magnitudes, such as million or billion. In mathematics, we may know a ...

Pythagorean Triples, Diane Swartzlander Jul 2007

Pythagorean Triples, Diane Swartzlander

MAT Exam Expository Papers

Who was Pythagoras after which the Pythagorean Theorem is named? Pythagoras was born between 580-572 BC and died between 500-490 BC. Pythagoras and his students believed that everything was related to mathematics and that numbers were the ultimate reality. Very little is known about Pythagoras because none of his writings have survived. Many of his accomplishments may actually have been the work of his colleagues and students. Pythagoras established a secret cult called the Pythagoreans. His cult was open to both females and males and they lived a structured life consisting of religious teaching, common meals, exercise, reading and philosophical ...

Jean Baptiste Joseph Fourier, Gary Eisenhauer Jul 2007

Jean Baptiste Joseph Fourier, Gary Eisenhauer

MAT Exam Expository Papers

Jean Baptiste Joseph Fourier was born in Auxerre, France on March 21, 1768. He was the ninth of twelve children from his father’s second marriage. When he was nine, his mother died. The following year, his father, a tailor, also passed.

Order Of Operations And Rpn, Greg Vanderbeek Jul 2007

Order Of Operations And Rpn, Greg Vanderbeek

MAT Exam Expository Papers

There is not a wealth of information regarding the history of the notations and procedures associated with what is now called the “order of operations”. There is evidence that some agreed upon order existed from the beginning of mathematical study. The grammar used in the earliest mathematical writings, before mathematical notation existed, supports the notion of computational order (Peterson, 2000). It is clear that one person did not invent the rules but rather current practices have grown gradually over several centuries and are still evolving.

Distance, Rate, Time And Beyond, Janet Timoney Jul 2007

Distance, Rate, Time And Beyond, Janet Timoney

MAT Exam Expository Papers

In middle school mathematics, students learn to use the formula “distance equals rate times time,” usually expressed as d = r × t. Why not consider the formula distance = velocity × time? Does the term velocity mean something different than the term rate? We could also consider the variations of these formulas: distance ÷ time = rate, or distance ÷ rate = time. We can examine the definitions of these words and words which are very similar. After looking at the definitions of these words, maybe we will have a better understanding of how to use the formulas and of the meaning behind them.