EE+in+Mathematics

**Extended Essay In Mathematics**
Sample Past EEs (from IB - marked)

IB Guiding Notes and Assessment Criteria **

Introduction**  It is not easy for students in their first or second semester in Grade 11 to come up with ideas for an extended essay in Mathematics. On the other hand, there is a wide range of possible essay types and a lot of freedom as to how a good essay can develop. As in any subject, the essay Research Question in Mathematics needs to be well focused. Some general types of topic are:

     It is possible to bring in Mathematics from outside the IB Higher course, but it is perfectly alright to apply Higher Level techniques to model real world problems.
 * Applying Mathematics** modelling in Engineering, Sciences, Economics etc using Calculus, Discrete Mathematics, Probability, Statistics etc
 * Proving Theorems** for example in Number Theory, Discrete Mathematics, Probability, Geometry etc.
 * Origin and Development of a Branch of Mathematics** with emphasis on the Mathematics and not the History
 * Links between Different Branches of Mathematics** common structures exist in seemingly unrelated areas
 * Interaction between Mathematics and Technology** either use of technology to solve a mathematical problem or investigating mathematics arising from the existence of a technology.

If you have a possible essay topic, please check it with your teacher. Your topic needs to be checked to make sure it is suitable for a Mathematics essay and that it is feasible given the time and your current knowledge.

**Some Books which might give Extended Essay Ideas**
 David Acheson 1089 and All That - A Journey into Mathematics Amir D Aczel The Mystery of the Aleph: Mathematics, the Kabbalah and the Search for Infinity Peter Beckmann A History of Pi Albert Beiler Recreations in the Theory of Numbers Elwyn R Berlekamp John H Conway Winning Ways For Your Mathematical Plays Volume 1 & Richard K Guy Ken Binmore Fun and Games: A Text on Game Theory John H Conway The Book of Numbers & Richard K Guy John H Conway On Numbers and Games John Derbyshire Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics Keith Devlin Mathematics: The New Golden Age Heinrich Dorrie One Hundred Great Problems of Elementary Mathematics: Their History and Solution William Dunham The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems and Personalities William Dunham Euler: The Master of Us All Rob Eastaway Why Do Buses Come in Threes?: The Hidden Mathematics of Everyday Life & Jeremy Wyndham Rob Eastaway How Long is a Piece of String?: More Hidden Mathematics in Everyday Life & Jeremy Wyndham G Gandolfo Economic Dynamics Liang-shin Hahn Complex Numbers & Geometry Peter M Higgins Mathematics for the Imagination Peter M Higgins Mathematics for the Curious Robert Kaplan The Art of the Infinite: Our Lost Language of Numbers Robert Kaplan The Nothing That Is: A Natural History of Zero Eli Maor To Infinity and Beyond: Cultural History of the Infinite Eli Maor E: Story of a Number Eli Maor Trigonometric Delights Barry Mazur Imagining Numbers: (Particularly the Square Root of Minus Fifteen Paul J Nahin An Imaginary Tale: The Story of the Square Root of Minus One Paul J Nahin When Least is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible Edward Packel The Mathematics of Games and Gambling Theoni Pappas The Joy of Mathematics: Discovering Mathematics All Around You Samual H Preston Demography: Measuring and Modelling Population Processes Karl Sabbagh Dr. Riemann's Zeros Marcus du Sautoy The Music of the Primes: Why an Unsolved Problem in Mathematics Matters Bruce Schechter My Brain is Open: The Mathematical Journeys of Paul Erdos Charles Seife Zero: The Biography of a Dangerous Idea Simon Singh Fermat's Last Theorem Simon Singh The Code Book: The Secret History of Codes and Code-breaking Alexei Sossinsky Knots: Mathematics with a Twist Ian Stewart Does God Play Dice?: The New Mathematics of Chaos Ian Stewart From Here to Infinity Ian Stewart The Magical Maze: Seeing the World Through Mathematical Eyes Robin Wilson Four Colours Suffice: How the Map Problem was Solved

**Some Past Essay Topics**
 1999-2001

Modelling Populations Using Leslie Matrices  The Solution of Cubic and Quartic Equations  Modelling Wind Farm Efficiencies Using Difference Equations  <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Fractals and their Dimensions <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">

2000-2002

<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Modelling Population Growth Using Differential Equations <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Linear Codes for Error Detection and Correction <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">

2001-2003 <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">An Analysis of Winning Strategies in the Game of Prim <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Using Complex Numbers to Analyse A.C. Circuits <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Pre-computer Calculation Methods Applied to Spherical Geometry <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">

2002-2004 <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Trans-shipment Models <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Shapes of Ropes and Beams <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Modelling the Day/Night Boundary of the Geochron <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">The Double Pendulum and Chaos <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">

2003-2005 <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">A Comparison of Fibonacci Sequences and Random Fibonacci Sequences <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">A Statistical Study of the Random Walk Hypothesis in Stock Market Indices <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">

<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Spherical Geometry and Trigonometry <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Numerical Methods for Finding Real and Complex Solutions of Polynomial Equations <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Fitting Curves to Data <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Using Calculus to Prove Kepler’s Laws of Planetary Motion <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">A Comparison of Proof Methods in Geometry <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Iterated Mappings and Chaos <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Circles and Points Associated with Triangles <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Map Projections and their Properties <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">The Mathematics of Perspective <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Map Colouring <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Probability ‘Paradoxes’ <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Convergence of Iterated Powers <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Convergence of the Newton-Raphson Method for Solution of Equations <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Quaternions A Computer Investigation of Queues Prime Numbers and Cryptography <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Farey Fractions and their applications <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">The Chinese Remainder Theorem and Applications <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Representation of Integers as the Sums of Squares of Integers <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> <span style="font-family: 'Arial','sans-serif'; font-size: 11pt; mso-fareast-font-family: 'Times New Roman'; text-decoration: none; text-underline: none;">
 * Other Possible Topics** - there may be several different essays associated with the topics below. Each essay would have a more focused title. There are lots of other possible topics

===<span style="font-family: 'Arial','sans-serif'; font-size: 11pt; mso-fareast-font-family: 'Times New Roman'; text-decoration: none; text-underline: none;">A Short Synopsis of Some Past Mathematics Essay Topics <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> === =<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman';">Modelling Populations Using Leslie Matrices = <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LesMat; msobookmark: LesMat;">How does knowledge of the fertility and survival rates for different age groups in a population enable one to predict the long-term numbers in each age group? This situation gives rise to a particular type of <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LesMat; msobookmark: LesMat; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">matrix <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LesMat; msobookmark: LesMat;"> (the Leslie matrix) and is intimately linked with <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LesMat; msobookmark: LesMat; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">eigenvalues and eigenvectors <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LesMat; msobookmark: LesMat;"> and <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LesMat; msobookmark: LesMat; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">difference equations <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LesMat; msobookmark: LesMat;">. The behaviour of the population numbers depends on the dominant (biggest in magnitude) eigenvalue and eigenvector of the Leslie matrix. Various theorems can be proved about this situation.

<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: CubQua; msobookmark: CubQua;">The quadratic formula for solving a general quadratic equation has been known for thousands of years. It was not until the 15th and 16th centuries that progress was made in solving cubic equations (i.e. equations containing an x3 term) and quartic equations (i.e. equations containing an x4 term). In this essay the methods are explained and illustrated with examples. <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> Wind farms contain arrays of wind turbines and, depending on the wind direction, these can interfere with each other as one wind turbine reduces the wind reaching its neighbours. Simple one and two dimensional arrays are studied using <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">difference equation <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> models.
 * <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">The Solution of Cubic and Quartic Equations **
 * Modelling Wind Farm Efficiencies Using Difference Equations**

=<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman';">Fractals and their Dimensions = <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">Is it possible for a geometric shape to have an infinite measure in one dimension and zero measure in the dimension above? It is and such shapes can meaningfully be thought of as having fractional dimensions in between the two integer values. Fractional dimensional shapes are called fractals and this essay studies examples such as the Cantor set, Minkowski square, Koch snowflake curve and Menger sponge to find their fractal dimensions.

<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Differential equations <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> are used to study the growth of population numbers under a variety of simplifying assumptions. Single populations are considered and the interaction of predator and prey species is modelled. Both analytical and numerical methods are used to solve the equations.
 * Modelling Population Growth Using Differential Equations**

<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">Modern digital communication and storage methods usually rely on streams of 0’s and 1’s. It is possible that some of these are corrupted and changed so it is important to be able to detect if an error has been made and if possible correct it. In this essay, general principles of error detection and correction are discussed Linear codes involving <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">matrices <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> and <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">finite arithmetic <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> are studied in detail.
 * <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">Linear Codes for Error Detection and Correction **

**<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">An Analysis of Winning Strategies in the Game of Prim ** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> Prim is a two person game related to NIM. In this essay, general principles for analysing two person games are discussed and applied to Prim. Winning positions are identified and strategies to force a win from those positions are studied.

**<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">Using Complex Numbers to Analyse A.C. Circuits ** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> A.C. circuits contain A.C. generators and components such as resistors, capacitors and inductances. <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Differential Equations <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> derived from Kirchhof's Laws can be used to solve for the currents in the circuit. If only the steady state currents are required, <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">complex numbers <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> in the form of complex impedances can be used to solve for the currents. This essay shows how differential equations arise in circuits, shows how to solve some of them and establishes the link with complex numbers.

**<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">Pre-computer Calculation Methods Applied to Spherical Geometry ** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> How did previous generations of Scientists and Mathematicians perform complex calculations without the aid of computers or mechanical calculating devices? Apart from amazing feats of longhand multiplication and division, they used aids such as trigonometric identities (the prosthaphaeretic method) and logarithms. This essays shows how these work when applied to the problem of calculating distances and angles on the surface of a sphere such as the earth.

**<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">Trans-shipment Models ** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> How should a manufacturing company store its products in depots and arrange for the delivery of the products to customers in the most cost efficient way? This is a complicated Linear Programming problem which may be tackled using the Simplex Method. The object is to identify the amount of goods to be delivered to storage depots and customers in order to minimise cost.

**<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">Shapes of Ropes and Beams ** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> What is the shape of a hanging rope or chain? Is this the same shape as the load bearing cables of a suspension bridge? What shapes do beams take up under different loading conditions? This essay shows how to model these situations and obtain solutions using calculus.

**<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">Modelling the Day/Night Boundary of the Geochron ** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> The Geochron shows the day/night boundary over a map projection of the earth. The shape of the boundary varies with the time of year and is at its simplest at the equinoxes where everywhere has equal day and night. This essay seeks to model the day/night boundary on the surface of the earth using geometry and then looks at the final map projection stage.

**<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">The Double Pendulum and Chaos ** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> A simple pendulum describes periodic motion with a fixed period. This can be modelled using a <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">differential equation <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> and for small oscillations, an approximate equation can be solved to describe the Simple Harmonic Motion. The double pendulum (one suspended from another pendulum) can be modelled with two differential equations and must be solved approximately on a computer. What types of motion are possible for the double pendulum? This essay solves the equations on a spreadsheet to investigate whether chaotic motion can result.

**<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">A Comparison of Fibonacci Sequences and Random Fibonacci Sequences ** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> The Fibonacci sequence starts 1, 1, 2, 3, 5,....where each term after the second term is the sum of the two preceding values. What happens if instead of adding the preceding two terms, we decide at random whether to add or subtract the preceding two terms? Will the resulting sequences behave in the same way as the original Fibonacci sequence or will there be major differences?

**<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">A Statistical Study of the Random Walk Hypothesis in Stock Market Indices ** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> A random walk results when the next movement has no memory of the earlier stages of the walk. It has been proposed that stock prices and stock market indices may follow a random walk, especially if there is sufficient interest in the stock or index to believe that all possible factors are already included in the current stock price. In this essay, various statistical tests of the random walk hypothesis are applied to data on individual stocks and stock indices.

**<span style="font-family: 'Arial','sans-serif'; font-size: 11pt;">A Short Synopsis of Other Possible Mathematics Essay Topics ** <span style="font-family: 'Arial','sans-serif'; font-size: 11pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">

<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: SpherGeom; msobookmark: SpherGeom;">The ancient Greeks knew how to find shortest distances and angles on the surface of a sphere using trigonometry. This became useful knowledge for long distance navigation of the approximately spherical earth. In the nineteenth century, spherical geometry was seen as an example of a Non-Euclidean Geometry where there were no parallel lines. An essay on this topic could focus on practical applications of spherical trigonometry or it might concentrate on a comparison of theorems between Euclidean Geometry and spherical geometry.
 * Sperical Geometry and Trigonomerty**

<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">**Equations** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Polynomial equations <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> of degrees 2,3 or 4 (quadratic, cubic and quartic) can be solved analytically using formulae in terms of the coefficients (e.g. the quadratic formulae). In practical applications, it is often necessary to know the roots of higher degree polynomial equations. An essay could study some of the methods for finding approximate numerical solutions to these equations, including roots which are <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">complex numbers <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">.

Given the //x// and //y// coordinates of a set of points, how do we find the equation of a curve which goes through all the points so that we may estimate //y// for any //x// or estimate //x// for any //y//? This is the interpolation problem. One possibility is to join neighbouring points by a straight line, giving piecewise linear interpolation. We might try to fit all the points using a polynomial curve or we might use splines. These are polynomial curves fitting some neighbouring points and which join up nicely with polynomial curves fitting other points. Another possible area of investigation is ‘best’ approximation of the points by a simple curve such as a straight line or parabola. We have to define what we mean by ‘best’ here.
 * Fitting Curves to Data**

One of the milestones of renaissance Mathematics was the use by Newton of the newly invented calculus together with Newton’s Universal Law of Gravitation to prove Kepler’s three laws of planetary motion. In this essay, the student would aim to derive the laws using modern notation and techniques. Other related topics are Rutherford's experiment with alpha particles and gold foil and trajectories of space probes around Jupiter.
 * Using Calculus to Prove Kepler’s Laws of Planetary Motion**

The ancient Greeks proved a lot of their theorems in geometry using congruent and similar shapes and ratios. The development of coordinate geometry by Descartes lead to the use of algebra to show geometric results. <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Complex numbers <span style="font-family: 'Arial','sans-serif'; font-size: 9pt;"> also provided a tool for studying plane geometry and the introduction of vectors in the 19th century gave yet another method for proving geometrical theorems. An essay in this topic would take a variety of theorems and illustrate the proof approaches by different methods. <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman';">
 * A Comparison of Proof Methods in Geometry**
 * Iterated Mappings and Chaos**

=<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; font-weight: normal; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman';">In the last twenty to thirty years a lot of progress has been made in understanding the intricate behaviour of sequences defined recursively i.e. //an// where //an+1// is defined in terms of //an// and earlier elements of the sequence. The sequence may ‘settle down’ to a fixed number, may cycle around a set of numbers or may behave chaotically. In this essay, the student could perform computer experiments with some maps and compare the results with the reported known general properties of such maps. = <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> **<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">Circles and Points Associated with Triangles ** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">You will be aware of the circumcentre and circumcircle for a triangle, the former being the point of intersection of the perpendicular bisectors of the sides of the triangle and the latter being the circle through the vertices. These are among many interesting points and circles associated with a triangle. It is envisaged that an essay on this topic would aim to prove one or more of the major results showing the relationship between some such points and circles for a triangle.

**<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">Map Projections and their Properties ** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">The surface of the earth can be approximated by a sphere. How does one represent this on a flat surface by a map? Many such projections exist and they distort the true surface in different ways. Some preserve directions, a very useful property for navigation. Some preserve area. This essay would study the mathematics behind the map projections.

**<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">The Mathematics of Perspective ** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">One of the great advances in art in Europe in the renaissance period was the development of perspective as a means of giving the illusion of depth in a two-dimensional painting. Unsurprisingly mathematics, and in particular geometry is heavily involved in this. This essay would look at the mathematical ideas behind perspective drawing leading to projective geometry.

=<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman';">Map Colouring <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; font-weight: normal; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman';"> = =<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; font-weight: normal; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman';">How many colours does it take to colour a map on a flat surface or a sphere so that adjacent countries are given different colours? What about a map on a torus (a doughnut shape)? The answer is four in the former case and seven in the latter. This essay will look at the mathematical way to look at such a problem through graph theory. The proofs of the results above are out of the scope of an essay but it is possible to study the idea of graph colouring and prove some lesser colouring results about maps. = =<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman';">Probability ‘Paradoxes’ = <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">How many people is it necessary to have in a room so that the probability of at least two having the same birthday is more than 0.5? The answer is surprisingly small. How is the probability that a needle dropped on a grid of parallel lines crosses a line related to <span style="font-family: Symbol; font-size: 9pt; mso-bookmark: LinCodes; msoasciifontfamily: Arial; msobidifontfamily: Arial; msobookmark: LinCodes; msochartype: symbol; msohansifontfamily: Arial; msosymbolfontfamily: Symbol;">p <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">? Given n letters and n envelopes, why, if the envelopes are filled at random, is the probability that every letter is in the wrong envelope related to the mathematical constant e? There are many surprising results in the area of probability and a lot of interesting mathematics arises in studying such problems. <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msospacerun: yes;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> **<span style="font-family: 'Arial','sans-serif'; font-size: 9pt;">Convergence of Iterated Powers ** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">For what real values of //x// does x^(x^(x^(x....) converge i.e. make sense? This problem was considered by Euler. The problem may be investigated experimentally on a computer and the analytical solution of Euler (which involves the mathematical constant e) may be studied and explained. What happens if //x// is allowed to be a complex number? More computer investigations are possible. This problem has been completely solved in relatively recent times. =<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman';">Convergence of the Newton-Raphson Method for Solution of Equations = <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">Most equations of the form //f//(//x//) = 0 cannot be solved analytically in the way that quadratic equations can, but there are approximate numerical methods to obtain solutions as accurately as we want. One of these is the Newton-Raphson method which produces a sequence of numbers that hopefully converge (i.e. settle down) to a solution of the equation. How fast does the method converge? What effect does the starting guess have? What if <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">complex numbers <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> are allowed? There is ample scope here for an analytical and computer experimental study. =<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman';">Quaternions = <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">When <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">complex numbers <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> were invented, it turned out that they were useful to study geometry in two dimensions. Mathematicians started to look for other new types of ‘number’ that could be used to study three-dimensional Geometry. In the 19th century, the Irish mathematician Hamilton discovered a generalisation of <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">complex numbers <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">, called quaternions, which could be useful in the study of four-dimensional geometry. There are close links between the arithmetic of quaternions and vectors in three dimensions. Quaternions also give examples of finite and infinite groups (studied in the Sets, Relations, Groups option in IB Higher). This essay would involve the explanation of what quaternions are, their properties and some applications. =<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman';">A Computer Investigation of Queues = <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">Queuing is familiar in many aspects of life e.g. in Post Offices or banks; on the telephone; accessing computer networks etc. There is clearly randomness involved in both the arrivals to the queue and the service time for people in the queue and it is no surprise that probabilities are involved in the study of queues. In this essay, the student would create computer simulations on a spreadsheet of some simple queues and investigate the behaviour of the queue in terms of waiting time in the queue, number of customers in the queue etc. =<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman';">Prime Numbers and Cryptography = <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">Secret communication has been important in relations between states for thousands of years. In modern times, electronic communication and commerce has led to the need for secure means of communication for everyone. In this essay the student would study how prime numbers can be used to develop a public key cryptosystem. Applications to key distribution for conventional ciphers, authentication (how can we be sure that a message really comes from the person claimed) and digital signatures could also be studied. =<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman';">Farey Fractions and their applications = <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">A Farey sequence of order n contains all the fractions in their lowest terms with denominators at most n arranged in order. These fractions have interesting properties for approximating real numbers by fractions, solving certain equations with integers and error free computing.

Is it possible to find a whole number which leaves remainder 1 when divided by 2, remainder 2 when divided by 3, remainder 4 when divided by 5 and remainder 6 when divided by 7? The Chinese Remainder Theorem(CRT) shows that such a number exists and how to find it. The CRT can be used to solve certain equations with integers. It can also be used to give a representation of numbers which enables fast computation.
 * The Chinese Remainder Theorem and Applications**

<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman';">**Representation of Integers as the Sums of Squares of Integers** <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;">Which positive integers can be written as the sum of two squares of integers? E.g. 25 is 9+16. What about three squares? Can every integer be written as the sum of four squares? How many ways can an integer be written as the sum of two, three or four squares?

**<span style="font-family: 'Arial','sans-serif'; font-size: 11pt;">A Few More Titles ** <span style="font-family: 'Arial','sans-serif'; font-size: 11pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> Gödel's Incompleteness Theorems Acceleration Techniques for the Summation of Infinite Series The Mathematical Analysis of the Newton Raphson Method Maclaurin Series using Integration Kepler's Problem: A Mathematical Perspective Patterns arising in the Fibonacci Sequence Using Game Theory to Identify conditions for Cooperation in a Cournot Oligopoly The Diffie Hellman and RSA Cryptographic Methods: An Examination How Stable is the Behaviour of a Duffing's Equation on a Double Well Oscillator? An Investigation into the Construction of a Natural Number System using Set Theory The Isomorphic Relation between the Beltrami-Klein and Poincaré Models in Hyperbolic Geometry. The Stern-Brocot Tree, Viswanath's Constant and the Tension between Theoretical and Applied Mathematics Central Solitaire: Theory and Solutions Learning using Linear Regression and Running: An Investigation of Linear Regression and Non-Linear Regression Moving Complexity: Application of Dynamical Systems to Differential Calculus On Division by Zero A Study of Penrose Tesselations Application of Quaternions to Rubik's Cube Continued Fractions and Their Applications Methods of Determining the Eigenvalues of a Matrix Elements of Projective Geometry in Architectural Drawing The Law of Quadratic Reciprocity Developing a Mathematical Model to Improve Learning Efficiency Regular Continued Fractions with Positive Elements and Ladder Electrical Networks Torus Knots and Composition of Mathematical Knots Continuous Probability as an Estimator of Pi: Buffon's Needle Experiment Fixed Point Bifurcations in the Lorenz System Time Series Modelling of Monthly Road Accidents Simple Deterministic Model for Cumulative Number of SARS Cases An Investigation into Finding the Volume of Egg Shaped Ovals The Efficiency and Accuracy of Various Pi Calculating Formulae Pi is Irrational A Mathematical Model for Number of Doctors and Population Growth The Application of Linear Programming in Maximising the Profit of a Duck Keeping Farm Formulating Mathematical Expressions for Car Parking to aid Human Judgements The Distribution of Prime Numbers Solutions to Linear and Quadratic Diophantine Equations Using Continued Fractions Far Reaching Vertices: On the Domination Numbers of n-Cube Graphs Using Vector Methods to Prove Geometric Theorems Mathematical Foundation for the Formation of Sierpinski's Triangle Non-Standard Analysis in the Development of the Calculus with Applications in Mechanics Using an Exponential and Logistic Function to Model and Project Population Growth Applications of the Gamma Function Securing the Internet with the RSA Cryptosystem The Five Colour Theorem Zermelo-Fraenkel Theory of Sets and the Axiom of Choice Using Linear Programming to Determine the Optimum Profit Obtainable from a Tennis Tournament =<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes; msofareastfontfamily: 'Times New Roman';">Glossary of Terms = <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> Complex Number a number of the form //a// + //ib// where //a// and //b// are ordinary real numbers and //i// is a new number with the property that i2 = -1. These are studied in the IB Higher core syllabus.

<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> a relationship between terms in a sequence of numbers and earlier terms in the sequence. An example is the Fibonacci sequence recurrence relation //fn+2 = fn+1 + fn//.

<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> a relationship between variables involving derivatives e.g dy/dx = - 2xy. Some simple differential equations are studied in the IB Higher core syllabus and more are studied in the Series and Differential Equations option

<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: Matrix;">Eigenvalues are special numbers associated with a square matrix. To each eigenvalue there is a special vector direction called an eigenvector.

<span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: Matrix;"> if n is an integer, modulo n arithmetic works with the finite set of numbers 0, 1, 2,…., n-1. Arithmetic is carried out as normal except that only the remainder on dividing an answer by n is kept. E.g. if n=2, 1+1=0 or if n=5, 2*3=1. This is studied in the Discrete Mathematics option.

Graph Colouring In a graph, vertices are assigned colours so that any two vertices joined by an edge have different colours.

Graph Theory The study of structures, called graphs, containing vertices and vertices connected by edges. Graphs are useful for studying many different types of situation in pure and applied maths. There is some Graph Theory in the Discrete Mathematics option in IB Higher.

Matrix <span style="font-family: 'Arial','sans-serif'; font-size: 9pt; mso-bookmark: LinCodes; msobookmark: LinCodes;"> a rectangular array of numbers. Matrices can be added, subtracted and multiplied. This is a straightforward topic studied in the IB Higher core syllabus.

Polynomial Equation an equation involving sums of terms containing a number (a coefficient) times a power of //x//. The highest power of //x// is called the degree of the polynomial equation e.g. 3x5 + 4x3 – 2x2 – 7x + 4 is a polynomial equation of degree 5.