The Heart of Mathematics: An Invitation to Effective Thinking (LL)

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.

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

2.1  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

3.1  Beyond Numbers

What Does Infinity 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 Infinities Larger than Others?

3.4  Travels Toward the Stratosphere of Infinities

The Power Set and the Question of an Infinite Galaxy of Infinities

3.5 Straightening Up the Circle

Exploring the Infinite Within Geometrical Objects

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?

5.1  Rubber Sheet Geometry

Discovering the Topological Idea of Equivalence by Distortion

5.2  The Band That Wouldnt Stop Playing

Experimenting with the Möbius 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

6.1  Circuit Training

From the Königsberg 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

7.1  Images

Viewing a Gallery of Fractals

7.2 The Infinitely Detailed Beauty of Fractals

How to Create Works of Infinite 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

8.1  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

9.1  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

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