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Extending the Frontiers of Mathematics: Inquiries into Proof and Augmentation



Extending the Frontiers of Mathematics: Inquiries into Proof and Augmentation

Edward B. Burger

ISBN: 978-0-470-53033-7 July 2009 192 Pages

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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.

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


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

Offers puzzles and patterns as precursors to proofs