The Heart of Mathematics: An Invitation to Effective Thinking / Edition 3

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Overview

The Heart of Mathematics: An invitation to effective thinking --now in its second edition--succeeds at reaching non-math, non-science-oriented readers and encourages them to discover the mathematics inherent in the world around them. Infused throughout with the authors' humor and enthusiasm, The Heart of Mathematics introduces readers to the most important and interesting ideas in mathematics while inspiring them to actively engage in mathematical thinking.
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Product Details

  • ISBN-13: 9780470424766
  • Publisher: Wiley, John & Sons, Incorporated
  • Publication date: 11/9/2009
  • Series: Key Curriculum Press Series , #118
  • Edition description: Older Edition
  • Edition number: 3
  • Pages: 286
  • Sales rank: 496,490
  • Product dimensions: 8.14 (w) x 10.12 (h) x 1.55 (d)

Meet the Author

Edward B. Burger is professor of mathematics in the Department of Mathematics and Statistics at Williams College. He graduated from Connecticut College in 1985, where he earned B.A. Summa Cum Laude with Distinction in Mathematics, and received his Ph.D. in mathematics from the University of Texas at Austin. He did his postdoctoral work at the University of Waterloo in Canada. Dr. Burger has received numerous awards including:  the Award of Excellence, for "educational mathematics videos that break new ground from the Technology & Learning magazine, and the Distinguished Achievement Award, for Educational Video Technology from The Association of Educational Publishers.  He was honored as one of the "100 Best of America", Listed in Reader's Digest's Annual Special Issue as Best Math Teacher.  He also received the Residence Life Academic Teaching Award, University of Colorado at Boulder and the Robert W. Hamilton Book Award, for "The Heart of Mathematics".

Michael Starbird Michael Starbird is a University Distinguished Teaching Professor of Mathematics at The University of Texas at Austin. He has received more than a dozen teaching awards including the Mathematical Association of America’s 2007 national teaching award and several university-wide teaching awards based largely on his course in mathematics for liberal arts students. Starbird brings intriguing mathematics to general audiences through his classes, lectures, books, and video courses. In 1989, Starbird was UT’s Recreational Sports Super Racquets Champion.

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Table of Contents

But First, a Word from our Sponsors  ix
Welcome!  xi
Surfing the Book  xv
Chapter One: Fun and Games: An Introduction to Rigorous Thought  2
1.1  Silly Stories Each with a Moral  4
Conundrums that Evoke Techniques of Effective Thinking
1.2  Nudges  4
Leading Questions and Hints for Resolving the Stories
1.3  The Punch Lines  18
Solutions and Further Commentary
1.4  From Play to Power  27
Discovering Strategies of Thought for Life
Chapter Two: Number Contemplation  38
2.1  Counting  40
How the Pigeonhole Principle Leads to Precision Through Estimation
2.2  Numberical Patterns in Nature  49
Discovering the Beauty of the Fibonacci Numbers
2.3  Prime Cuts of Numbers  64
How the Primes Numbers are the Building Blocks of All Natural Numbers
2.4  Crazy Clocks and Checking Out Bars  82
Cyclical Clock Arithmetic and Bar Codes
2.5  Public Secret Codes adn How to Become a Spy  95
Encrypting Information Using Modula Arithmetic and Primes
2.6  The Irrational Side of Numbers  110
Are There Numbers Beyond Fractions?
2.7  Get Real  121
The Point of Decimals and Pinpointing Numbers on the Real Line
Chapter Three: Infinity  136
3.1 Beyond Numbers  138
What Does Infinity Mean?
3.2  Comparing the Infinite  145
Pairing Up Collections via a One-to-One Corerspondence
3.3  The Missing Member  162
Georg Cantor Answers: Are Some Infinities Larger Than Others?
Chapter Four: Geometric Gems  206
4.1 Pythagoras and His Hypotenuse  208
How a Puzzle Leads to the Proof of one of the Gems of Mathematics
4.2  A View of an Art Gallery 218
Using Computational Geometry to Place Security Cameras in Musuems
4.3  The Sexiest Rectangle  232
Finding Aesthetics in Life, Art, and Math Through the Golden Rectangle
4.4  Soothing Symmetry and Spinning Pinwheels  249
Can a Floor be Tiled without any Repeating Pattern?
4.5  The Platonic Solids Turn Amorous  269
Discovering the Symmetry and Interconnections Among the Platonic Solds
4.6  The Shape of Reality?  289
How Straight Lines Can Bend in Non-Euclidean Geometries
4.7  The Fouther Dimension  307
Can You See It?
Chapter Five: Contortions of Space  326
5.1  Rubger Sheet Geometry  328
Discovering the Topological Idea of Equivalence by Distortion
5.2  The Band That Wouldn't Stop Playing 46
Experimenting with the Mobius Band and Klein Bottle
5.3  Feeling Edgy?  359
Exploring Relationships Among Vertices, Edges, and Faces
5.4  Knots and Links  374
Untangling Ropes and Rings
5.5  Fixed Points, Hot Loops, and Rainy Days  389
How the Certainty of Fixed Points Implies Certain Weather Phenomena
Chapter Six: Chaos and Fractals  402
6.1  Images  404
Viewing a Gallery of Fractals
6.2  The Dynamics of Change  412
Can Change be Modeled by Repeated Applications of Simple Processes?
6.3  The Infinitely Detailed Beuty of Fractals  430
How to Create Workds of Ifinite Intricacy THrough Repeated PRocesses
6.4  The Mysterious Art of Imaginary Fractals  458
Creating Juila and Mandelbrot Sets by Stepping Out in the Complex Plane
6.5  Predetermined Chase  482
How Repeated SImple Processes Result in Utter Chaos
6.6  Between Dimnsions  503
Can the Dimensions of Fractals Fall Through the Cracks?
Chapter Seven:  Taming Uncertainty 514
7.1  Chance Surprises  516
Some Scenarioos Involving Chance that Confound our Intuition
7.2  Predicting the Future in an Uncertain World 523
How to MEasure Uncertainty Using the Idea of Probability
7.3  Random Thoughtss  541
Are Coincidences as Truly Amazing as They First Appear?
7.4  Down for the Count 554
Systematically Counting all Possible Outcomes
7.5  Collecting Data Rather than Dust  571
The Power and Pitfalls fo Statistics
7.6  What the Average American Has  585
Different Means of Describing Data
7.7  Parenting Peas, Twins, and Hypotheses  610
Making Inferences from Data
Chapter Eight: Deciding Wisely  628
Applications of Rigorous THinking
8.1  Great Expectations  630
Deciding How to Weigh the Unknown Future
8.2  Risk  645
Deciding Personal and PUblic Policy
8.3  Money matters 663
Deciding Between Faring Well and Welfare
8.4  Peril at the Polls 682
Deciding Who Actually Wins an Election
8.5
Cutting Cake for Greedy People  700
Deciding How to Slice Up Scarce Resources
Farewell 716
Acknowledgements: Second Edition 719
Acknowledgements: First Edition  722
Hints and Solutions 725
Index 747
Credits 758
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