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The Heart of Mathematics an Invitation to Effecti Ve Thinking 4E


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1. FUN AND GAMES An Introduction to Rigorous Thought 1.1 Silly Stories, Each with a Moral Conundrums that Evoke Techniques of Effective Thinking 1.2 Nudges Leading Questions and Hints for Resolving the Stories 1.3 The Punch Lines Solutions and Further Commentary 1.4 From Play to Power Discovering Strategies of Thought for Life 2. NUMBER CONTEMPLATION 2.1 Counting How the Pigeonhole Principle Leads to Precision Through Estimation 2.2 Numerical Patterns in Nature Discovering the Beauty of the Fibonacci Numbers 2.3 Prime Cuts of Numbers How the Prime Numbers Are the Building Blocks of All Natural Numbers 2.4 Crazy Clocks and Checking Out Bars Cyclical Clock Arithmetic and Bar Codes 2.5 Public Secret Codes and How to Become a Spy Encrypting Information Using Modular Arithmetic and Primes 2.6 The Irrational Side of Numbers Are There Numbers Beyond Fractions? 2.7 Get Real The Point of Decimals and Pinpointing Numbers on the Real Line 3. INFINITY 3.1 Beyond Numbers What Does In?nity Mean? 3.2 Comparing the Infinite Pairing Up Collections via a One-to-One Correspondence 3.3 The Missing Member Georg Cantor Answers: Are Some In?nities Larger than Others? 3.4 Travels Toward the Stratosphere of Infinities The Power Set and the Question of an In?nite Galaxy of In?nities 3.5 Straightening Up the Circle Exploring the In?nite Within Geometrical Objects 4. GEOMETRIC GEMS 4.1 Pythagoras and His Hypotenuse How a Puzzle Leads to the Proof of One of the Gems of Mathematics 4.2 A View of an Art Gallery Using Computational Geometry to Place Security Cameras in Museums 4.3 The Sexiest Rectangle Finding Aesthetics in Life, Art, and Math Through the Golden Rectangle 4.4 Soothing Symmetry and Spinning Pinwheels Can a Floor Be Tiled Without Any Repeating Pattern? 4.5 The Platonic Solids Turn Amorous Discovering the Symmetry and Interconnections Among the Platonic Solids 4.6 The Shape of Reality? How Straight Lines Can Bend in Non-Euclidean Geometries 4.7 The Fourth Dimension Can You See It? 5. CONTORTIONS OF SPACE 5.1 Rubber Sheet Geometry Discovering the Topological Idea of Equivalence by Distortion 5.2 The Band That Wouldn't Stop Playing Experimenting with the Moebius Band and Klein Bottle 5.3 Knots and Links Untangling Ropes and Rings 5.4 Fixed Points, Hot Loops and Rainy Days How the Certainty of Fixed Points Implies Certain Weather Phenomena 6. MODELING OUR WORLD THROUGH GRAPHS 6.1 Circuit Training From the Koenigsberg Bridge Puzzle to Graphs 6.2 Feeling Edgy? Exploring Relationships Among Vertices, Edges, and Faces 6.3 Plane Old Graphs Drawing in the Plane and Coloring Maps 6.4 Networking Using Graphical Models to Find the Shortest, Closest, and Cheapest 7. FRACTALS AND CHAOS 7.1 Images Viewing a Gallery of Fractals 7.2 The Infinitely Detailed Beauty of Fractals How to Create Works of In?nite Intricacy Through Repeated Processes 7.3 Between Dimensions Can the Dimensions of Fractals Fall Through the Cracks? 7.4 Mysterious Art of Imaginary Fractals Creating Julia and Mandelbrot Sets by Stepping Out in the Complex Plane 7.5 They Dynamics of Change Can Change Be Modeled by Repeated Applications of Simple Processes? 7.6 Predetermined Chaos How Repeated Simple Processes Result in Utter Chaos 8. TAMING UNCERTAINTY 8.1 Chance Surprises Some Scenarios Involving Chance That Confound Our Intuition 8.2 Predicting the Future in an Uncertain World How to Measure Uncertainty Using the Idea of Probability 8.3 Random Thoughts Are Coincidences as Truly Amazing as They First Appear? 8.4 Down for the Count Systematically Counting All Possible Outcomes 8.5 Drizzling, Defending, and Doctoring Probability in Our World and Our Lives 9. MEANING FROM DATA 9.1 Stumbling Through a Minefield of Data Inspiring Statistical Concepts Through Pitfalls 9.2 Getting Your Data to Shape Up Organizing, Describing, and Summarizing Data 9.3 Looking at Super Models Mathematically Described Distributions 9.4 Go Figure Making Inferences from Data 9.5 War, Sports, and Tigers Statistics Throughout Our Lives 10. DECIDING WISELY Applications of Rigorous Thinking 10.1 Great Expectations Deciding How to Weigh the Unknown Future 10.2 Risk Deciding Personal and Public Policy 10.3 Money Matters Deciding Between Faring Well and Welfare 10.4 Peril at the Polls Deciding Who Actually Wins an Election 10.5 Cutting Cake for Greedy People Deciding How to Slice Up Scarce Resources

About the Author

Dr. Edward Burger is a professor mathematics at Williams College in Williamstown, MA. He received his BA from Connecticut College and his PhD from University of Texas at Austin. He has received numerous awards including: the Nelson Bushnell Prize, for Scholarship and Teaching, Williams College, being listed among the top 100 best Math Teachers in the "100 Best of America", Reader's Digest's Annual Special Issue. He has also received the Award of Excellence, for "educational mathematics videos that break new ground", from Technology & Learning magazine. His research interests include Algebraic Number Theory, Diophantine Analysis, padic Analysis, Geometry of Numbers, and the Theory of Continued Fractions.

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