Mathematical Excursions, 2nd Edition

Cover Page......Page 1
Title Page......Page 2
Copyright......Page 3
Contents......Page 4
CHAPTER 1 Problem Solving......Page 26
SECTION 1.1 Inductive and Deductive Reasoning......Page 27
Excursion The Game of Sprouts by John H. Conway......Page 35
SECTION 1.2 Problem Solving with Patterns......Page 40
Excursion Polygonal Numbers......Page 48
SECTION 1.3 Problem-Solving Strategies......Page 54
Excursion Routes on a Probability Demonstrator......Page 65
Summary......Page 71
Review Exercises......Page 72
Test......Page 75
CHAPTER 2 Sets......Page 77
SECTION 2.1 Basic Properties of Sets......Page 78
Excursion Fuzzy Sets......Page 83
SECTION 2.2 Complements, Subsets, and Venn Diagrams......Page 89
Excursion Subsets and Complements of Fuzzy Sets......Page 94
SECTION 2.3 Set Operations......Page 99
Excursion Union and Intersection of Fuzzy Sets......Page 106
SECTION 2.4 Applications of Sets......Page 111
Excursion Voting Systems......Page 117
SECTION 2.5 Infinite Sets......Page 122
Excursion Transfinite Arithmetic......Page 130
Summary......Page 134
Review Exercises......Page 135
Test......Page 137
CHAPTER 3 Logic......Page 138
SECTION 3.1 Logic Statements and Quantifiers......Page 139
Excursion Switching Networks......Page 146
SECTION 3.2 Truth Tables, Equivalent Statements, and Tautologies......Page 150
Excursion Switching Networks—Part II......Page 157
SECTION 3.3 The Conditional and the Biconditional......Page 161
Excursion Logic Gates......Page 167
SECTION 3.4 The Conditional and Related Statements......Page 170
Excursion Sheffer’s Stroke and the NAND Gate......Page 174
SECTION 3.5 Arguments......Page 177
Excursion Fallacies......Page 186
SECTION 3.5 Euler Diagrams......Page 189
Excursion Using Logic to Solve Puzzles......Page 194
Summary......Page 198
Review Exercises......Page 199
Test......Page 201
CHAPTER 4 Numeration Systems and Number Theory......Page 202
SECTION 4.1 Early Numeration Systems......Page 203
Excursion A Rosetta Tablet for the Traditional Chinese Numeration System......Page 208
SECTION 4.2 Place-Value Systems......Page 212
Excursion Subtraction via the Nines Complement and the End-Around Carry......Page 219
SECTION 4.3 Different Base Systems......Page 222
Excursion Information Retrieval via a Binary Search......Page 229
SECTION 4.4 Arithmetic in Different Bases......Page 233
Excursion Subtraction in Base Two via the Ones Complement and the End-Around Carry......Page 242
SECTION 4.5 Prime Numbers......Page 245
Excursion The Distribution of the Primes......Page 251
SECTION 4.6 Topics from Number Theory......Page 256
Excursion A Sum of the Divisors Formula......Page 263
Review Exercises......Page 267
Test......Page 269
CHAPTER 5 Applications of Equations......Page 270
SECTION 5.1 First-Degree Equations and Formulas......Page 271
Excursion Body Mass Index......Page 281
SECTION 5.2 Rate, Ratio, and Proportion......Page 288
Excursion Earned Run Average......Page 300
SECTION 5.3 Percent......Page 308
Excursion Federal Income Tax......Page 320
SECTION 5.4 Second-Degree Equations......Page 331
Excursion The Sum and Product of the Solutions of a Quadratic Equation......Page 338
Summary......Page 344
Review Exercises......Page 345
Test......Page 349
CHAPTER 6 Applications of Functions......Page 352
SECTION 6.1 Rectangular Coordinates and Functions......Page 353
Excursion Dilations of a Geometric Figure......Page 361
SECTION 6.2 Properties of Linear Functions......Page 365
Excursion Negative Velocity......Page 374
SECTION 6.3 Finding Linear Models......Page 378
Excursion A Linear Business Model......Page 385
SECTION 6.4 Quadratic Functions......Page 389
Excursion Reflective Properties of a Parabola......Page 395
SECTION 6.5 Exponential Functions......Page 402
Excursion Chess and Exponential Functions......Page 410
SECTION 6.6 Logarithmic Functions......Page 414
Excursion Benford’s Law......Page 424
Summary......Page 427
Review Exercises......Page 428
Test......Page 431
CHAPTER 7 Mathematical Systems......Page 432
SECTION 7.1 Modular Arithmetic......Page 433
Excursion Computing the Day of the Week......Page 440
SECTION 7.2 Applications of Modular Arithmetic......Page 444
Excursion Public Key Cryptography......Page 453
SECTION 7.3 Introduction to Group Theory......Page 456
Excursion Wallpaper Groups......Page 464
Summary......Page 470
Review Exercises......Page 471
Test......Page 473
CHAPTER 8 Geometry......Page 474
SECTION 8.1 Basic Concepts of Euclidean Geometry......Page 475
Excursion Preparing a Circle Graph......Page 487
SECTION 8.2 Perimeter and Area of Plane Figures......Page 494
Excursion Slicing Polygons into Triangles1......Page 510
SECTION 8.3 Properties of Triangles......Page 518
Excursion Topology......Page 526
SECTION 8.4 Volume and Surface Area......Page 532
Excursion Water Displacement......Page 541
SECTION 8.5 Introduction to Trigonometry......Page 547
Excursion Approximating the Value of Trigonometric Functions......Page 554
SECTION 8.6 Non-Euclidean Geometry......Page 558
Excursion Finding Geodesics......Page 567
SECTION 8.7 Fractals......Page 571
Excursion The Heighway Dragon Fractal......Page 581
Summary......Page 586
Review Exercises......Page 588
Test......Page 591
CHAPTER 9 The Mathematics of Graphs......Page 594
SECTION 9.1 Traveling Roads and Visiting Cities......Page 595
Excursion Pen-Tracing Puzzles......Page 606
SECTION 9.2 Efficient Routes......Page 614
Excursion Extending the Greedy Algorithm......Page 625
SECTION 9.3 Planarity and Euler’s Formula......Page 631
Excursion The Five Regular Convex Polyhedra......Page 638
SECTION 9.4 Map Coloring and Graphs......Page 643
Excursion Modeling Traffic Lights with Graphs......Page 651
Summary......Page 658
Review Exercises......Page 659
Test......Page 663
CHAPTER 10 The Mathematics of Finance......Page 666
SECTION 10.1 Simple Interest......Page 667
Excursion Day-of-the-Year Table......Page 674
SECTION 10.2 Compound Interest......Page 679
Excursion Consumer Price Index......Page 693
SECTION 10.3 Credit Cards and Consumer Loans......Page 699
Excursion Leasing versus Buying a Car......Page 710
SECTION 10.4 Stocks, Bonds, and Mutual Funds......Page 717
Excursion Treasury Bills......Page 723
SECTION 10.5 Home Ownership......Page 727
Excursion Home Ownership Issues......Page 735
Summary......Page 739
Review Exercises......Page 740
Test......Page 742
CHAPTER 11 Combinatorics and Probability......Page 744
SECTION 11.1 The Counting Principle......Page 745
Excursion Decision Trees......Page 751
SECTION 11.2 Permutations and Combinations......Page 755
Excursion Choosing Numbers in Keno......Page 764
SECTION 11.3 Probability and Odds......Page 768
Excursion The Value of Pi by Simulation......Page 777
SECTION 11.4 Addition and Complement Rules......Page 782
Excursion Keno Revisited......Page 788
SECTION 11.5 Conditional Probability......Page 792
Excursion Sharing Birthdays......Page 799
SECTION 11.6 Expectation......Page 803
Excursion Chuck-a-luck......Page 807
Summary......Page 812
Review Exercises......Page 813
Test......Page 816
CHAPTER 12 Statistics......Page 818
SECTION 12.1 Measures of Central Tendency......Page 819
Excursion Linear Interpolation and Animation......Page 827
SECTION 12.2 Measures of Dispersion......Page 832
Excursion A Geometric View of Variance and Standard Deviation1......Page 839
SECTION 12.3 Measures of Relative Position......Page 844
Excursion Stem-and-Leaf Diagrams......Page 850
SECTION 12.4 Normal Distributions......Page 856
Excursion Cut-Off Scores......Page 867
SECTION 12.5 Linear Regression and Correlation......Page 871
Excursion An Application of Linear Regression......Page 879
Summary......Page 886
Review Exercises......Page 888
Test......Page 892
CHAPTER 13 Apportionment and Voting......Page 894
SECTION 13.1 Introduction to Apportionment......Page 895
Excursion Apportioning the 1790 House of Representatives......Page 907
SECTION 13.2 Introduction to Voting......Page 915
Excursion Variations of the Borda Count Method......Page 928
SECTION 13.3 Weighted Voting Systems......Page 938
Excursion Blocking Coalitions and the Banzhaf Power Index......Page 946
Summary......Page 951
Review Exercises......Page 952
Test......Page 956
The Metric System of Measurement......Page 958
Solutions to Check Your Progress Problems......Page 966
Answers to Selected Exercises......Page 1020
Index......Page 1062
Web Appendix: Algebra Review......Page 1072