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Extending the Frontiers of Mathematics: Inquiries into Proof and Augmentation

June 2008, ©2007
Extending the Frontiers of Mathematics: Inquiries into Proof and Augmentation (EHEP000280) cover image
Mathematicians do not know in advance if their assertions are true, so they need to verify assertions with rigorous proofs or produce counterexamples, then attempt to salvage the assertions by transforming them into theorems. Extending the Frontiers of Mathematics: Inquiries into proof and argumentation invites students to experience this progression of mathematical discovery.  Burger's informal, humorous writing style and intriguing puzzles engage students throughout the learning process.  The innovative text offers a discovery-based approach that is ideally suited for use in a proofs course, a discrete math course, or any bridge course to more abstract mathematics.
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Contents
Introduction: A brief travel guide to the journey ahead .    .    .    .    .    .    .    .    . xi
To the student: A great adventure . . . . . . . . . . xii
To the instructor: The mathematics to come . . . . . . xiv
1. Puzzles and patterns: A precursor to proofs .    .    .    .    .    .    .    .    .    .    .    .    .  1
Senators and snakes . . . . . . . . . . . . . . . 2
Checkerboards undercover . . . . . . . . . . . . . 3
Flipping cards and coins . . . . . . . . . . . . . 6
Juggling balls . . . . . . . . . . . . . . . . . 7
Counting on a good hand and a good grade . . . . . . . 8
Stepping back . . . . . . . . . . . . . . . . . 10
2. Bringing theorems to justice: Exposing the truth through rigorous proof . . 11
Making a statement . . . . . . . . . . . . . . . 11
The truth, the whole truth, and other stuff besides the truth . 13
Arguing the case through proof . . . . . . . . . . . 16
The domino effect of bringing everyone down through guilt by induction . . . . . . . . . . . 19
Stepping back . . . . . . . . . . . . . . . . . 26
3. Delving into the dependable digits: Counting on counting numbers .    .    .    . 27
Divide and conquer . . . . . . . . . . . . . . . 27
Solving equations by simple division . . . . . . . . . 28
Now ready for prime time . . . . . . . . . . . . . 31
Stepping back . . . . . . . . . . . . . . . . . 32
4. Going around in circles: The art of modular arithmetic .    .    .    .    .    .    .    .    . 33
The mod side of mathematics . . . . . . . . . . . 33
Old-school algebra gets an extreme makeover . . . . . . 35
The power of congruences . . . . . . . . . . . . 36
Public secret codes . . . . . . . . . . . . . . . 37
Stepping back . . . . . . . . . . . . . . . . . 39
5. The irrational side of numbers: A world of nonrepeating digits .    .    .    .    .    . 41
Divisible and indivisible . . . . . . . . . . . . . 41
Forgoing fractions . . . . . . . . . . . . . . . 42
Irrationalit-e . . . . . . . . . . . . . . . . . 43
Stepping back . . . . . . . . . . . . . . . . . 43
6. Discovering how to function in mathematics: Moving beyond ordinary relations .    .    .    .    .    .    .    .    .    .    .    .    .    .    .    . 45
Get ready, get set . . . . . . . . . . . . . . . . 45
Fun with functions . . . . . . . . . . . . . . . 47
Moving onto an intimate one-to-one relationship
with functions . . . . . . . . . . . . . . . 48
An intuitively-believable-yet-challenging-to-verify correspondence . . . . . . . . . . . . . . . 50
Stepping back . . . . . . . . . . . . . . . . . 51
7. Infinity: Understanding the unending .    .    .    .    .    .    .    .    .    .    .    .    .    .    . 53
Comparing cardinality . . . . . . . . . . . . . . 53
Countable collections . . . . . . . . . . . . . . 54
The return of power sets . . . . . . . . . . . . . 55
A pair of paradoxes . . . . . . . . . . . . . . . 56
Stepping back . . . . . . . . . . . . . . . . . 56
8. Recursively defined functions: The next generation .    .    .    .    .    .    .    .    .    . 57
Moving ahead by looking back . . . . . . . . . . . 57
Forgoing convergence—A formal look at power series . . . 58
Generating closed formulas through generating functions . 60
Stepping back . . . . . . . . . . . . . . . . . 63
9. Discrete thoughts of counting: Quantifying complicated quantities . . . . 65
Pigeons without a home . . . . . . . . . . . . . 65
PERMUTATION = UP TO A REMINT . . . . . . . . 66
A combination that unlocks counting . . . . . . . . 67
Stepping back . . . . . . . . . . . . . . . . . 69
10. Quantifying uncertainty with probability: A likely story? .    .    .    .    .    .    .    . 71
What are the chances? . . . . . . . . . . . . . . 71
A coin toss? . . . . . . . . . . . . . . . . . 73
What’s the deal? . . . . . . . . . . . . . . . . 73
Stepping back . . . . . . . . . . . . . . . . . 74
11. The subtle art of connecting the dots: Edging up to graphs . . . . . . . 75
Bridging the graph . . . . . . . . . . . . . . . 75
Making the circuit . . . . . . . . . . . . . . . 77
Branching out to trees . . . . . . . . . . . . . . 79
Completely connected graphs . . . . . . . . . . . 80
Stepping back . . . . . . . . . . . . . . . . . 81
12. Just plane graphs: Drawing without being cross .    .    .    .    .    .    .    . 83
Drawing without crossing . . . . . . . . . . . . . 83
Seemingly similar graphs . . . . . . . . . . . . . 85
Sensing symmetry . . . . . . . . . . . . . . . 88
Stepping back . . . . . . . . . . . . . . . . . 89
13. Visible and invisible universes: Geometric vignettes .    .    .    .    .    .    .    .    .    91
Sizing up symmetry . . . . . . . . . . . . . . . 91
Keeping an artful eye on art galleries . . . . . . . . . 95
Up, up, and away—The fourth dimension . . . . . . . 97
Stepping back . . . . . . . . . . . . . . . . 100
14. A synergy between geometry and numbers:
Circles and Pythagorean triples .    .    .    .    .    .    .    .    .    .    101
Rightful triangles . . . . . . . . . . . . . . . 101
Determining which triangles are allright . . . . . . . 102
A rational look at the circle . . . . . . . . . . . . 103
Stepping back . . . . . . . . . . . . . . . . 104
15. The mathematical mysteries within a sheet of paper: Unfolding pattern and structure .    .    .    .    .    .    .    .    .    .    .    .    .    .    .    .   105
Getting into the fold . . . . . . . . . . . . . . 105
The story unfolds . . . . . . . . . . . . . . . 106
Adding arithmetic to generate a wild parity . . . . . . 107
Stepping back . . . . . . . . . . . . . . . . 108
16. Take it to the limit: An initial approach to analysis . . . . . . . . . . 109
Getting really close . . . . . . . . . . . . . . 109
Grasping the cloudlike Cantor set . . . . . . . . . . 111
Turning to ternary . . . . . . . . . . . . . . . 112
Stepping back . . . . . . . . . . . . . . . . . 114
17. Uninterrupted thoughts of continuity: A jump-free journey .    .    .    .    .    115
Functions without lifting a finger . . . . . . . . . . 115
Valuing intermediate values . . . . . . . . . . . . 116
A devilish staircase . . . . . . . . . . . . . . . 117
Stepping back . . . . . . . . . . . . . . . . . 119
18. An abstract world of algebra: Reconciling with your x . . . . . . . 121
Grasping at groups . . . . . . . . . . . . . . . 122
A group mentality . . . . . . . . . . . . . . . 123
Building a group table . . . . . . . . . . . . . 124
Stepping back . . . . . . . . . . . . . . . . 125
19. Cycles and curves: Algebraic structure in numbers and geometry .    .    .    .   127
The subculture of subgroups . . . . . . . . . . . 127
Cyclic groups . . . . . . . . . . . . . . . . 128
A circular group . . . . . . . . . . . . . . . 128
Stepping back . . . . . . . . . . . . . . . . . 131
20. Further frontiers . . . . . . . . . . . . . . . . . . . . . . 133
Appendix 1.Hints, remarks, and leading questions .    .    .    .    .    139
Appendix 2.A proof primer: The proof is in the pudding . . . . . . . . . 151
1. Preproof prep . . . . . . . . . . . . . . . . . 151
2. Penning the proof . . . . . . . . . . . . . . . . 152
3. Post proof . . . . . . . . . . . . . . . . . . 153
Appendix 3.Commentary for instructors .    .    .    .    .    .    .    .    .    .    .    .    .    .  155
How I used this material in my classroom . . . . . . . . 157
Appendix 4.A math personality questionnaire .    .    .    .    .    .    .    .    .    .    .    .  161
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 163
Index .    .    .    .    .    .    .    .    165
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  • Offers puzzles and patterns as precursors to proofs
  • The “prove and extend or disprove and salvage” instruction is repeated, providing a consistent framework for approaching the problems
  • Investigative in nature, this discovery-based text leads but doesn’t tell
  • Brevity gives it flexibility; it can be used as either a core or supplementary text
  • Highly engaging writing style that is informal and humorous
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Instructors Resources
Wiley Instructor Companion Site
Instructor Resources Extending the Frontiers of M athematics: Inquiries into proof and argumentation
Instructor Resources to accompany Extending the Frontiers of Mathematics.
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Extending the Frontiers of Mathematics: Inquiries into Proof and Augmentation
ISBN : 978-0-470-53033-7
April 2009, ©2007
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Extending the Frontiers of Mathematics: Inquiries into Proof and Augmentation
ISBN : 978-0-470-41222-0
171 pages
June 2008, ©2007
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