Extending the Frontiers of Mathematics Inquiries into Proof and Augmentation

ISBN-10: 0470412224

ISBN-13: 9780470412220

Edition: 2007

Authors: Edward B. Burger

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Book details

List price: $124.95
Copyright year: 2007
Publisher: John Wiley & Sons, Incorporated
Publication date: 6/10/2008
Binding: Paperback
Pages: 192
Size: 6.00" wide x 9.00" long x 0.50" tall
Weight: 0.682
Language: English

Introduction: A brief travel guide to the journey ahead
To the student: A great adventure
To the instructor: The mathematics to come
Puzzles and patterns: A precursor to proofs
Senators and snakes
Checkerboards undercover
Flipping cards and coins
Juggling balls
Counting on a good hand and a good grade
Stepping back
Bringing theorems to justice: Exposing the truth through rigorous proof
Making a statement
The truth, the whole truth, and other stuff besides the truth
Arguing the case through proof
The domino effect of bringing everyone down through guilt by induction
Stepping back
Delving into the dependable digits: Counting on counting numbers
Divide and conquer
Solving equations by simple division
Now ready for prime time
Stepping back
Going around in circles: The art of modular arithmetic
The mod side of mathematics
Old-school algebra gets an extreme makeover
The power of congruences
Public secret codes
Stepping back
The irrational side of numbers: A world of nonrepeating digits
Divisible and indivisible
Forgoing fractions
Irrationalit-e
Stepping back
Discovering how to function in mathematics: Moving beyond ordinary relations
Get ready, get set
Fun with functions
Moving onto an intimate one-to-one relationship with functions
An intuitively-believable-yet-challenging-to-verify correspondence
Stepping back
Infinity: Understanding the unending
Comparing cardinality
Countable collections
The return of power sets
A pair of paradoxes
Stepping back
Recursively defined functions: The next generation
Moving ahead by looking back
Forgoing convergence-A formal look at power series
Generating closed formulas through generating functions
Stepping back
Discrete thoughts of counting: Quantifying complicated quantities
Pigeons without a home
PERMUTATION = UP TO A REMINT
A combination that unlocks counting
Stepping back
Quantifying uncertainty with probability: A likely story?
What are the chances?
A coin toss?
What's the deal?
Stepping back
The subtle art of connecting the dots: Edging up to graphs
Bridging the graph
Making the circuit
Branching out to trees
Completely connected graphs
Stepping back
Just plane graphs: Drawing without being cross
Drawing without crossing
Seemingly similar graphs
Sensing symmetry
Stepping back
Visible and invisible universes: Geometric vignettes
Sizing up symmetry
Keeping an artful eye on art galleries
Up, up, and away-The fourth dimension
Stepping back
A synergy between geometry and numbers: Circles and Pythagorean triples
Rightful triangles
Determining which triangles are allright
A rational look at the circle
Stepping back
The mathematical mysteries within a sheet of paper: Unfolding pattern and structure
Getting into the fold
The story unfolds
Adding arithmetic to generate a wild parity
Stepping back
Take it to the limit: An initial approach to analysis
Getting really close
Grasping the cloudlike Cantor set
Turning to ternary
Stepping back
Uninterrupted thoughts of continuity: A jump-free journey
Functions without lifting a finger
Valuing intermediate values
A devilish staircase
Stepping back
An abstract world of algebra: Reconciling with your X
Grasping at groups
A group mentality
Building a group table
Stepping back
Cycles and curves: Algebraic structure in numbers and geometry
The subculture of subgroups
Cyclic groups
A circular group
Stepping back
Further frontiers
Hints, remarks, and leading questions
A proof primer: The proof is in the pudding
Preproof prep
Penning the proof
Post proof
Commentary for instructors
How I used this material in my classroom
A math personality questionnaire
Acknowledgments
Index
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