A Mathematics Sampler: Topics for the Liberal ArtsRowman & Littlefield, 2001 - 602 Seiten Now in its fifth edition, A Mathematics Sampler 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 the distinctive features of mathematics as an intellectual endeavor, a problem-solving tool, and a way of thinking about the rapidly changing world in which we live. A Mathematics Sampler also includes unique LINK sections throughout the book, each of which connects mathematical concepts with areas of interest throughout the humanities. The original course on which this text is based was cited as an innovative approach to liberal arts mathematics in Lynne Cheney's report, "50 HOURS: A Core Curriculum for College Students", published by the National Endowment for the Humanities. |
Inhalt
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 |
A-31 | |
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A Mathematics Sampler: Topics for the Liberal Arts William P. Berlinghoff,Kerry E. Grant,Dale Skrien Eingeschränkte Leseprobe - 2001 |
Häufige Begriffe und Wortgruppen
1-1 correspondence 4-space algebra algorithm axiom system bers called cardinal number century chapter cities construct contains coordinate cosets cross section crossing curve cube data set decimal defined definition described digits divisors edge path equal Euclid's Euclidean Euclidean geometry exactly example false Figure formula geometry given graph theory Hamilton graph hypersphere idea identity element integers Lagrange's Theorem length line segment logical mathematicians mathematics mean ment Mersenne primes natural numbers non-Euclidean geometries notation NP-complete number of elements odd numbers operation ordered pair outcomes parallel Parallel Postulate pattern perfect numbers plane possible Postulate prime factorization probability problem proof Proposition prove radius rational numbers represent sample space Selection Sort set-builder notation shortest tour solution solve square statement steps subgroup subset Table taxicab tesseract Theorem tion Traveling Salesman Problem triangle true vertex path vertices WRITING EXERCISES