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Copyright, ii
Preface, ix
1. What is Meant by “An Advanced Perspective”?, 1
1. What is Meant by “An Advanced Perspective”?, 1
An Example of Concept Analysis: Parallelism, 1
An Example of Problem Analysis: Matching an Average, 5
An Example of Mathematical Connections: + and ·, 10
Part I. Algebra and Analysis with Connections to Geometry, 17
2. Real Numbers and Complex Numbers, 17
Unit 2.1. The Real Numbers, 18
2.1.1. Rational numbers and irrational numbers, 20
2.1.2. The number line and decimal representation of real numbers, 25
2.1.3. Periods of periodic decimals, 34
2.1.4. The distributions of various types of real numbers, 40
Unit 2.2. The Complex Numbers, 47
2.2.1. The complex numbers and the complex plane, 48
2.2.2. The geometry of complex number arithmetic, 54
Chapter Projects, 63
Bibliography, 66
3. Functions, 67
Unit 3.1. The Definitions, Historical Evolution, and Basic Machinery of Functions, 68
3.1.1. What is a function?, 68
3.1.2. Problem analysis: from equations to functions, 76
3.1.3. Some types of functions, 80
Unit 3.2. Properties of Real Functions, 86
3.2.1. Analyzing real functions, 87
3.2.2. Composition and inverse functions, 95
3.2.3. Monotone real functions, 101
3.2.4. Limit behavior of real functions, 105
Unit 3.3. Problems Involving Real Functions, 113
3.3.1. Fitting linear and exponential functions to data, 113
3.3.2. Fitting polynomial functions to data, 121
3.3.3. An extended analysis of the box problem, 125
Chapter Projects, 130
Bibliography, 133
4. Equations, 135
Unit 4.1. The Concept of Equation, 136
4.1.1. Equality, equivalence, and isomorphism, 136
4.1.2. Solving equations, 140
Unit 4.2. Algebraic Structures and Solving Equations, 146
4.2.1. Solving equations of the form a * x = b, 146
4.2.2. Solving equations of the form ax + b = cx + d, 149
4.2.3. Quadratic and other polynomial equations, 153
Unit 4.3. The Solving Process, 160
4.3.1. Generalized addition and multiplication properties of equality, 160
4.3.2. Applying the same function to both sides of an equation, 164
4.3.3. Solving inequalities, 167
4.3.4. Extended analysis: averages of speeds, 173
Chapter Projects, 178
Bibliography, 178
5. Integers and Polynomials, 179
Unit 5.1. Natural Numbers, Induction, and Recursion, 180
5.1.1. Recursion and proof by mathematical induction, 180
5.1.2. Mathematical induction, 186
5.1.3. More applications of mathematical induction, 193
5.1.4. An extended analysis of an induction situation, 197
Unit 5.2. Divisibility Properties of the Integers, 204
5.2.1. The Division Algorithm, 204
5.2.2. Divisibility of integers, 208
5.2.3. Solving linear Diophantine equations, 212
5.2.4. The Fundamental Theorem of Arithmetic, 218
5.2.5. Base representation of positive integers, 224
Unit 5.3. Divisibility Properties of Polynomials, 229
5.3.1. The Division Algorithm for polynomials, 229
5.3.2. The Euclidean Algorithm and prime factorization for polynomials, 236
Chapter Projects, 241
Bibliography, 243
6. Number System Structures, 245
Unit 6.1. The Systems of Modular Arithmetic, 246
6.1.1. Integer congruence, 246
6.1.2. Applications of integer congruence to calendars and cryptology, 252
6.1.3. The Chinese Remainder Theorem, 256
Unit 6.2. Number Fields, 261
6.2.1. Ordered fields, 262
6.2.2. Archimedean and complete ordered fields, 267
6.2.3. The structure of the complex number system, 271
Chapter Projects, 273
Bibliography, 274
Part II. Geometry with Connections to Algebra and Analysis, 275
7. Congruence, 275
Unit 7.1. Euclid and Congruence, 275
7.1.1. Euclid’s Elements, 275
7.1.2. Deduction and proof, 283
7.1.3. General properties of definitions, 290
7.1.4. Definitions of congruence from Euclid to modern times, 295
Unit 7.2. The Congruence Transformations, 302
7.2.1. Translations, 302
7.2.2. Rotations, 306
7.2.3. Reflections, 313
7.2.4. Glide reflections, 321
7.2.5. Are there other congruence transformations?, 324
7.2.6. Congruent graphs, 328
Unit 7.3. Symmetry, 333
7.3.1. Reflection symmetry, 333
7.3.2. Other congruence transformation symmetries, 339
Unit 7.4. Traditional Congruence Revisited, 344
7.4.1. Sufficient conditions for congruence, 344
7.4.2. Concept analysis: analyzing a geometric figure, 351
7.4.3. General theorems about congruence, 353
Chapter Projects, 357
Bibliography, 358
8. Distance and Similarity, 361
Unit 8.1. Distance, 362
8.1.1. What is distance?, 362
8.1.2. Minimum distance problems, 368
8.1.3. Extended analysis: locus problems, 375
8.1.4. Distance on the surface of a sphere, 379
Unit 8.2. Similar Figures, 383
8.2.1. When are two figures similar?, 384
8.2.2. Similarity of graphs, 391
8.2.3. Similar polygons, 396
8.2.4. Similar arcs, 400
8.2.5. When many theorems become one, 404
8.2.6. Types of similarity transformations, 408
Unit 8.3. Distances within Figures, 414
8.3.1. Geometric means, 415
8.3.2. Similarity and parallel lines, 419
Chapter Projects, 428
Bibliography, 429
9. Trigonometry, 431
Unit 9.1. Angle Measure and the Trigonometric Ratios, 432
9.1.1. Angle measure and arc length, 432
9.1.2. The trigonometric ratios, 434
9.1.3. Extended analysis: indirect measurement problems, 440
Unit 9.2. The Trigonometric Functions and Their Connections, 444
9.2.1. The trigonometric functions, 444
9.2.2. Modeling with trigonometric functions, 450
9.2.3. The historical and conceptual evolution of trigonometry, 456
Unit 9.3. Properties of the Sine and Cosine Functions, 459
9.3.1. Algebraic properties of the trigonometric functions, 460
9.3.2. Geometric properties of the sine and cosine functions, 463
9.3.3. Analytical properties of the sine and cosine functions, 469
Chapter Projects, 474
Bibliography, 476
10. Area and Volume, 477
Unit 10.1. Area, 478
10.1.1. What is area?, 478
10.1.2. Area formulas for triangles, 486
10.1.3. Extended analysis: the line through a given point minimizing area, 492
10.1.4. From polygons to regions bounded by curves, 500
10.1.5. The problem of quadrature, 506
10.1.6. Area as representing probability, 508
Unit 10.2. Volume, 511
10.2.1. What is volume?, 512
10.2.2. From cubes to polyhedra, 515
10.2.3. From polyhedra to spheres, 523
Unit 10.3. Relationships among Area, Volume, and Dimension, 528
10.3.1. Surface area, 528
10.3.2. The Isoperimetric Inequalities, 532
10.3.3. The Fundamental Theorem of Similarity, 536
10.3.4. Fractional dimension, 541
Chapter Projects, 544
Bibliography, 546
11. Axiomatics and Euclidean Geometry, 547
Unit 11.1. Constructing Euclidean Geometry, 548
11.1.1. Axioms for incidence, 548
11.1.2. Axioms for betweenness, 552
11.1.3. Congruence and the basic figures, 556
11.1.4. Geometry without the Parallel Postulate, 564
11.1.5. Euclid’s Fifth Postulate, 568
Unit 11.2. The Cartesian Model for Euclidean Geometry, 571
11.2.1. The Cartesian coordinate system, 571
11.2.2. Verifying the definition of Euclidean geometry: the relationship between a mathematical theory and its models, 576
Chapter Projects, 580
Bibliography, 582
Index, 583

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