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Tables of Contents for A Mathematics Sampler
Chapter/Section Title
Page #
Page Count
Preface
xiii
To the Student
xxi
Problems and Solutions
1
18
What Is Mathematics?
1
4
Problem Solving
5
5
It All Adds Up
10
5
The Mathematical Way of Thinking
15
4
Topics for Papers
17
1
For Further Reading
17
2
Mathematics of Patterns: Number Theory
19
48
What Is Number Theory?
19
7
Divisibility
26
3
Counting Divisors
29
13
Summing Divisors
42
4
Proper Divisors
46
3
Even Perfect Numbers
49
6
Mersenne Primes
55
7
LINK: Number Theory and Cryptography
62
5
Topics for Papers
64
1
For Further Reading
65
2
Mathematics of Axiom Systems: Geometries
67
62
What Is Geometry?
67
4
Euclidean Geometry
71
11
Euclid and Parallel Lines
82
9
Axiom Systems and Models
91
11
Consistency and Independence
102
7
Non-Euclidean Geometries
109
9
Axiomatic Geometry and the Real World
118
6
LINK: Axiom Systems and Society
124
5
Topics for Papers
127
1
For Further Reading
128
1
Mathematics of Chance: Probability and Statistics
129
104
The Gamblers
129
4
The Language of Sets
133
9
What Is Probability?
142
8
Counting Processes
150
10
LINK: Counting and the Genetic Code
160
3
Some Basic Rules of Probability
163
7
Conditional Probability
170
10
LINK: Probability and Marketing
180
4
What Is Statistics?
184
9
Central Tendency and Spread
193
12
Distributions
205
12
Generalization and Prediction
217
10
LINK: Statistics in the Psychology of Learning
227
6
Topics for Papers
231
1
For Further Reading
231
2
Mathematics of Infinity: Cantor's Theory of Sets
233
51
What Is Set Theory?
233
5
Infinite Sets
238
6
The Size of N
244
5
Rational and Irrational Numbers
249
6
A Different Size
255
7
Cardinal Numbers
262
4
Cantor's Theorem
266
4
The Continuum Hypothesis
270
3
The Foundations of Mathematics
273
5
LINK: Set Theory and Metaphysics
278
6
Topics for Papers
281
2
For Further Reading
283
1
Mathematics of Symmetry: Finite Groups
284
57
What Is Group Theory?
284
6
Operations
290
6
Some Properties of Operations
296
4
The Definition of a Group
300
4
Some Basic Properties of Groups
304
7
Subgroups
311
4
Lagrange's Theorem
315
7
Lagrange's Theorem Proved [Optional]
322
7
Groups of Symmetries
329
5
LINK: Groups in Music and in Chemistry
334
7
Topics for Papers
337
2
For Further Reading
339
2
Mathematics of Space and Time: Four-Dimensional Geometry
341
66
What Is Four-Dimensional Geometry?
341
2
One-Dimensional Space
343
5
Two-Dimensional Space
348
10
Three-Dimensional Space
358
11
Four-Dimensional Space
369
8
Cross Sections
377
9
Cylinders and Cones [Optional]
386
11
LINK: 4-Space in Fiction and in Art
397
10
Topics for Papers
404
2
For Further Reading
406
1
Mathematics of Connection: Graph Theory
407
48
What Is Graph Theory?
407
3
Some Basic Terms
410
7
Edge Paths
417
8
Vertex Paths
425
7
Crossing Curves
432
6
Euler's Formula
438
7
Looking Back
445
2
LINK: Diagraphs and Project Management
447
8
Topics for Papers
453
1
For Further Reading
454
1
Mathematics of Machines: Computer Algorithms
455
What Is a Computer?
455
The Traveling Salesman Problem
465
The Speed of a Computer
470
Algorithms and Sorting
473
Comparing Algorithms
480
Complexity Analysis
488
NP-Completeness
498
Implications of NP-Completeness
507
LINK: Algorithms, Abstraction, and Strategic Planning
513
Topics for Papers
520
For Further Reading
523
APPENDICES
A Basic Logic
525
A.1 Statements and Their Negations
525
A.2 Conjunctions and Disjunctions
531
A.3 Conditionals and Deduction
536
Topics for Papers
544
For Further Reading
544
B A Brief History of Mathematics
545
B.1 Preliminary Thoughts
545
B.2 From the Beginning to 600 B.C.
546
B.3 600 B.C. to A.D. 400
551
B.4 400 to 1400
558
B.5 The Fifteenth and Sixteenth Centuries
562
B.6 The Seventeenth Century
564
B.7 The Eighteenth Century
569
B.8 The Nineteenth Century
573
B.9 The Twentieth Century
580
Topics for Papers
588
For Further Reading
589
c Literacy in the Language of Mathematics
591
James O. Bullock
Answers to Most Odd-numbered Exercises
A1
Index
A31