The Heart of Mathematics: An Invitation to Effective Thinking

Abstract

CHAPTER ONE: Fun and Games: An introduction to rigorous thought CHAPTER TWO: Number Contemplation Section 2.1. Counting [Pigeonhole principle]. Section 2.2. Numerical Patterns in Nature: [Fibonacci numbers]. Section 2.3. Prime cuts of numbers [Prime numbers]. Section 2.4. Crazy clocks and checking out bars [Modular arithmetic]. Section 2.5. Secret Codes and How to Become a Spy [RSA public key cryptography]. Section 2.6. The irrational side of numbers [Irrational numbers]. Section 2.7. Get real [The real number line]. CHAPTER THREE: Infinity Section 3.1. Beyond Numbers [An introduction to one-to-one correspondence]. Section 3.2. Comparing the Infinite [Examples of one-to-one correspondences]. Section 3.3. The Missing Member [Cantor's diagonalization proof that |N|<|R|]. Section 3.4. Travels Toward the Stratosphere of Infinities [Power set theorem]. Section 3.5. Straightening up the circle [Geometrical correspondences]. CHAPTER FOUR: Geometric Gems Section 4.1. Pythagoras and his hypotenuse [Blaskara's elegant proof]. Section 4.2. A view of an art gallery [A view-obstruction question from computational geometry]. Section 4.3. The sexiest rectangle [The Golden Rectangle]. Section 4.4. Soothing symmetry and spinning pinwheels [Aperiodic tilings]. Section 4.5. The Platonic Solids Turn Amorous [Symmetry and duality in the Platonic Solids]. Section 4.6. The shape of reality? [Non-Euclidean geometries]. Section 4.7. The Fourth Dimension [Geometry through analogy]. CHAPTER FIVE: Contortions of Space Section 5.1. Rubber sheet geometry [Topological equivalence by distortion]. Section 5.2. The Band That Wouldn't Stop Playing [Mobius Band and Klein Bottle]. Section 5.3. Circuit training. [The Euler circuit theorem]. Section 5.4. Feeling edgy? [The Euler characteristic]. Section 5.5. Knots and links [A little knot theory]. Section 5.6. Fixed Points, Hot Loops, and Rainy Days [The Brouwer Fixed Point Theorem]. CHAPTER SIX: Fractals and Chaos Section 6.1. Images [A gallery of fractals]. Section 6.2. The infinitely detailed beauty of fractals [Creating fractals through repeated processes]. Section 6.3. Between dimensions [Fractal dimension]. Section 6.4. The mysterious art of imaginary fractals [Julia and Mandelbrot Sets]. Section 6.5. The Dynamics of Change [Repeated applications of simple processes]. Section 6.6. Predetermined chaos [Deterministic chaos]. CHAPTER SEVEN: Taming Uncertainty Section 7.1. Chance surprises [Unexpected scenarios involving chance]. Section 7.2. Predicting the future in an uncertain world [Probability]. Section 7.3. Random thoughts [Coincidences]. Section 7.4. Down for the count [Systematic counting]. Section 7.5. Dizzling, Defending, and Doctoring [Probability of Precipitation, game theory, Bayesian probability] CHAPTER EIGHT: Meaning from Data Section 8.1. Stumbling Through a Minefield of Data [Pitfalls of statistics]. Section 8.2. Getting Your Data to Shape Up [Organizing, describing, and summarizing data] Section 8.3. Looking at Super Models [Mathematically described distributions] Section 8.4. Go Figure [Making inferences from data, hypothesis testing] Section 8.5. War, Sports, and Tigers [Cause and effect and correlation, Simpson's Paradox, famous applications of inference] CHAPTER NINE: Deciding Wisely Section 9.1. Great Expectations [Expected value] Section 9.2. Risk [Deciding personal and public safety] Section 9.3. Money Matters [Compound interest] Section 9.4. Peril at the polls [voting] Section 9.5. Cutting cake for greedy people [fair division]