A Mathematics Sampler: Topics for the Liberal Arts

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Rowman & Littlefield, Jan 1, 2001 - Mathematics - 602 pages
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This work presents mathematics as both science and art, focusing on the historical role of mathematics in our culture. It uses selected topics from modern mathematics - including computers, perfect numbers and four-dimensional geometry - to exemplify its distinctive features.
  

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Contents

PROBLEMS AND SOLUTIONS 11 WHAT IS MATHEMATICS?
1
12 PROBLEM SOLVING
5
13 IT ALL ADDS UP
10
14 THE MATHEMATICAL WAY OF THINKING
15
FOR FURTHER READING
17
MATHEMATICS OF PATTERNS NUMBER THEORY 21 WHAT IS NUMBER THEORY?
19
22 DIVISIBILITY
26
23 COUNTING DIVISORS
29
67 LANGRANGES THEOREM
315
68 LAGRANGES THEOREM PROVED OPTIONAL
322
69 GROUPS OF SYMMETRIES
329
GROUPS IN MUSIC AND IN CHEMISTRY
334
TOPICS FOR PAPERS
337
FOR FURTHER READING
339
MATHEMATICS OF SPACE AND TIME FOURDIMENSIONAL GEOMETRY 71 WHAT IS FOURDIMENSIONAL GEOMETRY?
341
72 ONEDIMENSIONAL SPACE
343

24 SUMMING DIVISORS
42
25 PROPER DIVISORS
46
26 EVEN PERFECT NUMBERS
49
27 MERSENNE PRIMES
55
NUMBER THEORY AND CRYPTOGRAPHY
62
TOPICS FOR PAPERS
64
FOR FURTHER READING
65
MATHEMATICS OF AXIOM SYSTEMS GEOMETRIES 31 WHAT IS GEOMETRY?
67
32 EUCLIDEAN GEOMETRY
71
33 EUCLID AND PARALLEL LINES
82
34 AXIOM SYSTEMS AND MODELS
91
35 CONSISTENCY AND INDEPENDENCE
102
36 NONEUCLIDEAN GEOMETRIES
109
37 AXIOMATIC GEOMETRY AND THE REAL WORLD
118
AXIOM SYSTEMS AND SOCIETY
124
TOPICS FOR PAPERS
127
FOR FURTHER READING
128
MATHEMATICS OF CHANCE PROBABILITY AND STATISTICS 41 THE GAMBLERS
129
42 THE LANGUAGE OF SETS
133
43 WHAT IS PROBABILITY?
142
44 COUNTING PROCESSES
150
COUNTING AND THE GENETIC CODE
160
46 SOME BASIC RULES OF PROBABILITY
163
47 CONDITIONAL PROBABILITY
170
PROBABILITY AND MARKETING
180
49 WHAT IS STATISTICS?
184
410 CENTRAL TENDENCY AND SPREAD
193
411 DISTRIBUTIONS
205
412 GENERALIZATION AND PREDICTION
217
STATISTICS IN THE PSYCHOLOGY OF LEARNING
227
FOR FURTHER READING
231
MATHEMATICS OF INFINITY CANTORS THEORY OF SETS 51 WHAT IS SET THEORY?
233
52 INFINITE SETS
238
53 THE SIZE OF N
244
54 RATIONAL AND IRRATIONAL NUMBERS
249
55 A DIFFERENT SIZE
255
56 CARDINAL NUMBERS
262
57 CANTORS THEOREM
266
58 THE CONTINUUM HYPOTHESIS
270
59 THE FOUNDATIONS OF MATHEMATICS
273
SET THEORY AND METAPHYSICS
278
TOPICS FOR PAPERS
281
FOR FURTHER READING
283
MATHEMATICS OF SYMMETRY FINITE GROUPS 61 WHAT IS GROUP THEORY?
284
62 OPERATIONS
290
63 SOME PROPERTIES OF OPERATIONS
296
64 THE DEFINITION OF A GROUP
300
65 SOME BASIC PROPERTIES OF GROUPS
304
66 SUBGROUPS
311
73 TWODIMENSIONAL SPACE
348
74 THREEDIMENSIONAL SPACE
358
75 FOURDIMENSIONAL SPACE
369
76 CROSS SECTIONS
377
77 CYLINDERS AND CONES OPTIONAL
386
4SPACE IN FICTION AND IN ART
397
TOPICS FOR PAPERS
404
FOR FURTHER READING
406
MATHEMATICS OF CONNECTION GRAPH THEORY 81 WHAT IS GRAPH THEORY?
407
82 SOME BASIC TERMS
410
83 EDGE PATHS
417
84 VERTEX PATHS
425
85 CROSSING CURVES
432
86 EULERS FORMULA
438
87 LOOKING BACK
445
DIGRAPHS AND PROJECT MANAGEMENT
447
TOPICS FOR PAPERS
453
FOR FURTHER READING
454
MATHEMATICS OF MACHINES COMPUTER ALGORITHMS 91 WHAT IS A COMPUTER?
455
92 THE TRAVELING SALESMAN PROBLEM
465
93 THE SPEED OF A COMPUTER
470
94 ALGORITHMS AND SORTING
473
95 COMPARING ALGORITHMS
480
96 COMPLEXITY ANALYSIS
488
97 NPCOMPLETENESS
498
98 IMPLICATIONS OF NPCOMPLETENESS
507
ALGORITHMS ABSTRACTION AND STRATEGIC PLANNING
513
TOPICS FOR PAPERS
520
FOR FURTHER READING
523
BASIC LOGIC A1 STATEMENTS AND THEIR NEGATIONS
525
A2 CONJUNCTIONS AND DISJUNCTIONS
531
A3 CONDITIONALS AND DEDUCTION
536
TOPICS FOR PAPERS
544
A BRIEF HISTORY OF MATHEMATICS B1 PRELIMINARY THOUGHTS
545
B2 FROM THE BEGINNING TO 600 BC
546
B3 600 BC to AD 400
551
B4 400 to 1400
558
B5 THE FIFTEENTH AND SIXTEENTH CENTURIES
562
B6 THE SEVENTEENTH CENTURY
564
B7 THE EIGHTEENTH CENTURY
569
B8 THE NINETEENTH CENTURY
573
B9 THE TWENTIETH CENTURY
580
TOPICS FOR PAPERS
588
FOR FURTHER READING
589
LITERACY IN THE LANGUAGE OF MATHEMATICS INTRODUCTION
591
Answers to Most Oddnumbered Exercises
A-1
Index
A-31
Copyright

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About the author (2001)

William P. Berlinghoff is visiting professor of mathematics at Colby College. Kerry E. Grant is professor of mathematics at Southern Connecticut State University. Dale Skrien is professor of computer science at Colby College

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