Extending the Frontiers of Mathematics

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, p-adic Analysis, Geometry of Numbers, and?the Theory of Continued Fractions.

• Introduction: A brief travel guide to the journey ahead xiTo the student: A great adventure xiiTo the instructor: The mathematics to come xiv1 Puzzles and patterns: A precursor to proofs 1Senators and snakes 2Checkerboards undercover 3Flipping cards and coins 6Juggling balls 7Counting on a good hand and a good grade 8Stepping back 102 Bringing theorems to justice: Exposing the truth through rigorous proof 11Making a statement 11The truth, the whole truth, and other stuff besides the truth 13Arguing the case through proof 16The domino effect of bringing everyone down through guilt by induction 19Stepping back 263 Delving into the dependable digits: Counting on counting numbers 27Divide and conquer 27Solving equations by simple division 28Now ready for prime time 31Stepping back 324 Going around in circles: The art of modular arithmetic 33The mod side of mathematics 33Old-school algebra gets an extreme makeover 35The power of congruences 36Public secret codes 37Stepping back 395 The irrational side of numbers: A world of nonrepeating digits 41Divisible and indivisible 41Forgoing fractions 42Irrationalit-e 43Stepping back 436 Discovering how to function in mathematics: Moving beyond ordinary relations 45Get ready, get set 45Fun with functions 47Moving onto an intimate one-to-one relationshipwith functions 48An intuitively-believable-yet-challenging-to-verify correspondence 50Stepping back 517 Infinity: Understanding the unending 53Comparing cardinality 53Countable collections 54The return of power sets 55A pair of paradoxes 56Stepping back 568 Recursively defined functions: The next generation 57Moving ahead by looking back 57Forgoing convergence—A formal look at power series 58Generating closed formulas through generating functions 60Stepping back 639 Discrete thoughts of counting: Quantifying complicated quantities 65Pigeons without a home 65PERMUTATION = UP TO A REMINT 66A combination that unlocks counting 67Stepping back 6910 Quantifying uncertainty with probability: A likely story? 71What are the chances? 71A coin toss? 73What’s the deal? 73Stepping back 7411 The subtle art of connecting the dots: Edging up to graphs 75Bridging the graph 75Making the circuit 77Branching out to trees 79Completely connected graphs 80Stepping back 8112 Just plane graphs: Drawing without being cross 83Drawing without crossing 83Seemingly similar graphs 85Sensing symmetry 88Stepping back 8913 Visible and invisible universes: Geometric vignettes 91Sizing up symmetry 91Keeping an artful eye on art galleries 95Up, up, and away—The fourth dimension 97Stepping back 10014 A synergy between geometry and numbers: Circles and Pythagorean triples 101Rightful triangles 101Determining which triangles are allright 102A rational look at the circle 103Stepping back 10415 The mathematical mysteries within a sheet of paper: Unfolding pattern and structure 105Getting into the fold 105The story unfolds 106Adding arithmetic to generate a wild parity 107Stepping back 10816 Take it to the limit: An initial approach to analysis 109Getting really close 109Grasping the cloudlike Cantor set 111Turning to ternary 112Stepping back 11417 Uninterrupted thoughts of continuity: A jump-free journey 115Functions without lifting a finger 115Valuing intermediate values 116A devilish staircase 117Stepping back 11918 An abstract world of algebra: Reconciling with your x 121Grasping at groups 122A group mentality 123Building a group table 124Stepping back 12519 Cycles and curves: Algebraic structure in numbers and geometry 127The subculture of subgroups 127Cyclic groups 128A circular group 128Stepping back 13120 Further frontiers 133Appendix 1 Hints, remarks, and leading questions 139Appendix 2 A proof primer: The proof is in the pudding 1511 Preproof prep 1512 Penning the proof 1523 Post proof 153Appendix 3 Commentary for instructors 155How I used this material in my classroom 157Appendix 4 A math personality questionnaire 161Acknowledgments 163Index 165