A Mathematics Sampler Topics for the Liberal Arts

by Berlinghoff, William P.; Grant, Kerry E.; Skrien, Dale

• ISBN13: 9780742502024
• ISBN10: 0742502023
• Edition: 5th
• Format: Hardcover
• Copyright: 2001-01-10
• Publisher: Rowman & Littlefield Publishers

Summary

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.

Table of Contents

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