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
