{"product_id":"extending-the-frontiers-of-mathematics-isbn-9780470412220","title":"Extending the Frontiers of Mathematics","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. \u003ci\u003eExtending the Frontiers of Mathematics: Inquiries into proof and argumentation\u003c\/i\u003e 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. \u003cp\u003eIntroduction: A brief travel guide to the journey ahead xi\u003c\/p\u003e \u003cp\u003eTo the student: A great adventure xii\u003c\/p\u003e \u003cp\u003eTo the instructor: The mathematics to come xiv\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Puzzles and patterns: A precursor to proofs 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eSenators and snakes 2\u003c\/p\u003e \u003cp\u003eCheckerboards undercover 3\u003c\/p\u003e \u003cp\u003eFlipping cards and coins 6\u003c\/p\u003e \u003cp\u003eJuggling balls 7\u003c\/p\u003e \u003cp\u003eCounting on a good hand and a good grade 8\u003c\/p\u003e \u003cp\u003eStepping back 10\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Bringing theorems to justice: Exposing the truth through rigorous proof 11\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eMaking a statement 11\u003c\/p\u003e \u003cp\u003eThe truth, the whole truth, and other stuff besides the truth 13\u003c\/p\u003e \u003cp\u003eArguing the case through proof 16\u003c\/p\u003e \u003cp\u003eThe domino effect of bringing everyone down through guilt by induction 19\u003c\/p\u003e \u003cp\u003eStepping back 26\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Delving into the dependable digits: Counting on counting numbers 27\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eDivide and conquer 27\u003c\/p\u003e \u003cp\u003eSolving equations by simple division 28\u003c\/p\u003e \u003cp\u003eNow ready for prime time 31\u003c\/p\u003e \u003cp\u003eStepping back 32\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Going around in circles: The art of modular arithmetic 33\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eThe mod side of mathematics 33\u003c\/p\u003e \u003cp\u003eOld-school algebra gets an extreme makeover 35\u003c\/p\u003e \u003cp\u003eThe power of congruences 36\u003c\/p\u003e \u003cp\u003ePublic secret codes 37\u003c\/p\u003e \u003cp\u003eStepping back 39\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 The irrational side of numbers: A world of nonrepeating digits 41\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eDivisible and indivisible 41\u003c\/p\u003e \u003cp\u003eForgoing fractions 42\u003c\/p\u003e \u003cp\u003eIrrationalit-e 43\u003c\/p\u003e \u003cp\u003eStepping back 43\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Discovering how to function in mathematics: Moving beyond ordinary relations 45\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eGet ready, get set 45\u003c\/p\u003e \u003cp\u003eFun with functions 47\u003c\/p\u003e \u003cp\u003eMoving onto an intimate one-to-one relationship\u003c\/p\u003e \u003cp\u003ewith functions 48\u003c\/p\u003e \u003cp\u003eAn intuitively-believable-yet-challenging-to-verify correspondence 50\u003c\/p\u003e \u003cp\u003eStepping back 51\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Infinity: Understanding the unending 53\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eComparing cardinality 53\u003c\/p\u003e \u003cp\u003eCountable collections 54\u003c\/p\u003e \u003cp\u003eThe return of power sets 55\u003c\/p\u003e \u003cp\u003eA pair of paradoxes 56\u003c\/p\u003e \u003cp\u003eStepping back 56\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Recursively defined functions: The next generation 57\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eMoving ahead by looking back 57\u003c\/p\u003e \u003cp\u003eForgoing convergence—A formal look at power series 58\u003c\/p\u003e \u003cp\u003eGenerating closed formulas through generating functions 60\u003c\/p\u003e \u003cp\u003eStepping back 63\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Discrete thoughts of counting: Quantifying complicated quantities 65\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003ePigeons without a home 65\u003c\/p\u003e \u003cp\u003ePERMUTATION = UP TO A REMINT 66\u003c\/p\u003e \u003cp\u003eA combination that unlocks counting 67\u003c\/p\u003e \u003cp\u003eStepping back 69\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Quantifying uncertainty with probability: A likely story? 71\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eWhat are the chances? 71\u003c\/p\u003e \u003cp\u003eA coin toss? 73\u003c\/p\u003e \u003cp\u003eWhat’s the deal? 73\u003c\/p\u003e \u003cp\u003eStepping back 74\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 The subtle art of connecting the dots: Edging up to graphs 75\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eBridging the graph 75\u003c\/p\u003e \u003cp\u003eMaking the circuit 77\u003c\/p\u003e \u003cp\u003eBranching out to trees 79\u003c\/p\u003e \u003cp\u003eCompletely connected graphs 80\u003c\/p\u003e \u003cp\u003eStepping back 81\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Just plane graphs: Drawing without being cross 83\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eDrawing without crossing 83\u003c\/p\u003e \u003cp\u003eSeemingly similar graphs 85\u003c\/p\u003e \u003cp\u003eSensing symmetry 88\u003c\/p\u003e \u003cp\u003eStepping back 89\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Visible and invisible universes: Geometric vignettes 91\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eSizing up symmetry 91\u003c\/p\u003e \u003cp\u003eKeeping an artful eye on art galleries 95\u003c\/p\u003e \u003cp\u003eUp, up, and away—The fourth dimension 97\u003c\/p\u003e \u003cp\u003eStepping back 100\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 A synergy between geometry and numbers: Circles and Pythagorean triples 101\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eRightful triangles 101\u003c\/p\u003e \u003cp\u003eDetermining which triangles are allright 102\u003c\/p\u003e \u003cp\u003eA rational look at the circle 103\u003c\/p\u003e \u003cp\u003eStepping back 104\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 The mathematical mysteries within a sheet of paper: Unfolding pattern and structure 105\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eGetting into the fold 105\u003c\/p\u003e \u003cp\u003eThe story unfolds 106\u003c\/p\u003e \u003cp\u003eAdding arithmetic to generate a wild parity 107\u003c\/p\u003e \u003cp\u003eStepping back 108\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 Take it to the limit: An initial approach to analysis 109\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eGetting really close 109\u003c\/p\u003e \u003cp\u003eGrasping the cloudlike Cantor set 111\u003c\/p\u003e \u003cp\u003eTurning to ternary 112\u003c\/p\u003e \u003cp\u003eStepping back 114\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Uninterrupted thoughts of continuity: A jump-free journey 115\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eFunctions without lifting a finger 115\u003c\/p\u003e \u003cp\u003eValuing intermediate values 116\u003c\/p\u003e \u003cp\u003eA devilish staircase 117\u003c\/p\u003e \u003cp\u003eStepping back 119\u003c\/p\u003e \u003cp\u003e\u003cb\u003e18 An abstract world of algebra: Reconciling with your x 121\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eGrasping at groups 122\u003c\/p\u003e \u003cp\u003eA group mentality 123\u003c\/p\u003e \u003cp\u003eBuilding a group table 124\u003c\/p\u003e \u003cp\u003eStepping back 125\u003c\/p\u003e \u003cp\u003e\u003cb\u003e19 Cycles and curves: Algebraic structure in numbers and geometry 127\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eThe subculture of subgroups 127\u003c\/p\u003e \u003cp\u003eCyclic groups 128\u003c\/p\u003e \u003cp\u003eA circular group 128\u003c\/p\u003e \u003cp\u003eStepping back 131\u003c\/p\u003e \u003cp\u003e\u003cb\u003e20 Further frontiers 133\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix 1 Hints, remarks, and leading questions 139\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix 2 A proof primer: The proof is in the pudding 151\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Preproof prep 151\u003c\/p\u003e \u003cp\u003e2 Penning the proof 152\u003c\/p\u003e \u003cp\u003e3 Post proof 153\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix 3 Commentary for instructors 155\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eHow I used this material in my classroom 157\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix 4 A math personality questionnaire 161\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eAcknowledgments 163\u003c\/p\u003e \u003cp\u003eIndex 165\u003c\/p\u003e  \u003cp\u003e\u003cstrong\u003eDr. Edward Burger\u003c\/strong\u003e 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.\u003cbr\u003eHe 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\", \u003cem\u003eReader's Digest\u003c\/em\u003e's Annual Special Issue.? He has also received the \"Award of Excellence\", for 'educational mathematics videos that break new ground', from \u003cem\u003eTechnology \u0026amp; Learning\u003c\/em\u003e magazine.\u003cbr\u003eHis research interests include Algebraic Number Theory, Diophantine Analysis, \u003cem\u003ep-adic\u003c\/em\u003e Analysis, Geometry of Numbers, and?the Theory of Continued Fractions.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989189771493,"sku":"NP9780470412220","price":113.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9780470412220.jpg?v=1761783145","url":"https:\/\/k12savings.com\/products\/extending-the-frontiers-of-mathematics-isbn-9780470412220","provider":"K12savings","version":"1.0","type":"link"}