Extending the Frontiers of Mathematics
Inquiries into Proof and Augmentation
Will be published approx. on 1. July 2007
Book
Paperback/Softback
192 pages
978-0-470-41222-0 (ISBN)
Description
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 readers to experience this progression of mathematical discovery. Burger's informal, humorous writing style and intriguing puzzles engage readers throughout the learning process. The innovative text offers a discovery-based approach that is ideal.
Reviews / Votes
From the reviews: "Extending the Frontiers of Mathematics: Inquiries into Proof and Argumentation a ] offers an attractive and very plausible approach. The level of sophistication seems just about right for first or second-year college students. a ] Although most students who might use this book would have studied calculus, the text is useful for a broader group because only one or two modules require any knowledge of calculus. This is a carefully conceived and well-thought-out approach to the a ~introduction to proofa (TM) course, and ita (TM)s worth a careful look." (William J. Satzer, MathDL, July, 2007) From the reviews:"Extending the Frontiers of Mathematics: Inquiries into Proof and Argumentation ??? offers an attractive and very plausible approach. The level of sophistication seems just about right for first or second-year college students. ??? Although most students who might use this book would have studied calculus, the text is useful for a broader group because only one or two modules require any knowledge of calculus. This is a carefully conceived and well-thought-out approach to the ???introduction to proof??? course, and it??'s worth a careful look." (William J. Satzer, MathDL, July, 2007)
More details
Person
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.
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.
Content
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