Mathematical Ideas, Miller C.D., Heeren V.E., Hornsby J., 2015

Mathematical Ideas, Miller C.D., Heeren V.E., Hornsby J., 2015.
   
   After twelve editions and over four decades, Mathematical Ideas continues to be one of the most popular textbooks in liberal arts mathematics education. We are proud to present the thirteenth edition of a text that offers non-physical science students a practical coverage that connects mathematics to the world around them. It is a flexible text that has evolved alongside changing trends but remains steadfast to its original objectives.




Pitfalls of Inductive Reasoning.
There are pitfalls associated with inductive reasoning. A classic example involves the maximum number of regions formed when chords are constructed in a circle. When two points on a circle are joined with a line segment, a chord is formed.

Locate a single point on a circle. Because no chords are formed, a single interior region is formed. See Figure 2(a) on the next page. Locate two points and draw a chord. Two interior regions are formed, as shown in Figure 2(b). Continue this pattern. Locate three points, and draw all possible chords. Four interior regions are formed, as shown in Figure 2(c). Four points yield 8 regions and five points yield 16 regions. See Figures 2(d) and 2(e).

The results of the preceding observations are summarized in Table 1. The pattern formed in the column headed “Number of Regions” is the same one we saw in Example 2(c), where we predicted that the next number would be 64. It seems here that for each additional point on the circle, the number of regions doubles.

Contents.
PREFACE.
ACKNOWLEDGMENTS.
ABOUT THE AUTHORS.
1 The Art of Problem Solving.
1.1. SOLVING PROBLEMS BY INDUCTIVE REASONING.
1.2. AN APPLICATION OF INDUCTIVE REASONING: NUMBER PATTERNS.
1.3. STRATEGIES FOR PROBLEM SOLVING.
1.4. NUMERACY IN TODAY'S WORLD.
2 The Basic Concepts of Set Theory.
2.1. SYMBOLS AND TERMINOLOGY 48.
2.2. VENN DIAGRAMS AND SUBSETS.
2.3. SET OPERATIONS.
2.4. SURVEYS AND CARDINAL NUMBERS.
3 Introduction to Logic.
3.1 STATEMENTS AND QUANTIFIERS.
3.2. TRUTH TABLES AND EQUIVALENT STATEMENTS.
3.3. THE CONDITIONAL AND CIRCUITS.
3.4. THE CONDITIONAL AND RELATED STATEMENTS.
3.5. ANALYZING ARGUMENTS WITH EULER DIAGRAMS.
3.6. ANALYZING ARGUMENTS WITH TRUTH TABLES.
4 Numeration Systems.
4.1. HISTORICAL NUMERATION SYSTEMS.
4.2. MORE HISTORICAL NUMERATION SYSTEMS.
4.3. ARITHMETIC IN THE HINDU-ARABIC SYSTEM.
4.4. CONVERSION BETWEEN NUMBER BASES.
5 Number Theory.
5.1. PRIME AND COMPOSITE NUMBERS.
5.2. LARGE PRIME NUMBERS.
5.3. SELECTED TOPICS FROM NUMBER THEORY.
5.4. GREATEST COMMON FACTOR AND LEAST COMMON MULTIPLE.
5.5. THE FIBONACCI SEQUENCE AND THE GOLDEN RATIO.
5.6. MAGIC SQUARES (ONLINE).
6 The Real Numbers and Their Representations.
6.1. REAL NUMBERS, ORDER, AND ABSOLUTE VALUE.
6.2. OPERATIONS, PROPERTIES, AND APPLICATIONS OF REAL NUMBERS.
6.3. RATIONAL NUMBERS AND DECIMAL REPRESENTATION.
6.4. IRRATIONAL NUMBERS AND DECIMAL REPRESENTATION.
6.5. APPLICATIONS OF DECIMALS AND PERCENTS.
7 The Basic Concepts of Algebra.
7.1. LINEAR EQUATIONS.
7.2. APPLICATIONS OF LINEAR EQUATIONS.
7.3. RATIO, PROPORTION, AND VARIATION.
7.4. LINEAR INEQUALITIES.
7.5. PROPERTIES OF EXPONENTS AND SCIENTIFIC NOTATION.
7.6. POLYNOMIALS AND FACTORING.
7.7. QUADRATIC EQUATIONS AND APPLICATIONS.
8 Graphs, Functions, and Systems of Equations and Inequalities.
8.1 THE RECTANGULAR COORDINATE SYSTEM AND CIRCLES.
8.2. LINES, SLOPE, AND AVERAGE RATE OF CHANGE.
8.3. EQUATIONS OF LINES.
8.4. LINEAR FUNCTIONS, GRAPHS, AND MODELS.
8.5. QUADRATIC FUNCTIONS, GRAPHS, AND MODELS.
8.6. EXPONENTIAL AND LOGARITHMIC FUNCTIONS, GRAPHS, AND MODELS.
8.7. SYSTEMS OF LINEAR EQUATIONS.
8.8. APPLICATIONS OF LINEAR SYSTEMS.
8.9. LINEAR INEQUALITIES, SYSTEMS, AND LINEAR PROGRAMMING.
9 Geometry.
9.1. POINTS, LINES, PLANES, AND ANGLES.
9.2. CURVES, POLYGONS, CIRCLES, AND GEOMETRIC CONSTRUCTIONS.
9.3. THE GEOMETRY OF TRIANGLES: CONGRUENCE, SIMILARITY, AND THE PYTHAGOREAN THEOREM.
9.4. PERIMETER, AREA, AND CIRCUMFERENCE.
9.5. VOLUME AND SURFACE AREA.
9.6. TRANSFORMATIONAL GEOMETRY.
9.7. NON-EUCLIDEAN GEOMETRY AND TOPOLOGY.
9.8. CHAOS AND FRACTAL GEOMETRY.
10 Counting Methods.
10.1. COUNTING BY SYSTEMATIC LISTING.
10.2. USING THE FUNDAMENTAL COUNTING PRINCIPLE.
10.3. USING PERMUTATIONS AND COMBINATIONS.
10.4. USING PASCAL'S TRIANGLE.
10.5. COUNTING PROBLEMS INVOLVING "NOT" AND "OR".
11 Probability.
11.1. BASIC CONCEPTS.
11.2. EVENTS INVOLVING "NOT" AND "OR".
11.3. CONDITIONAL PROBABILITY AND EVENTS INVOLVING "AND".
11.4. BINOMIAL PROBABILITY.
11.5. EXPECTED VALUE AND SIMULATION.
12 Statistics.
12.1. VISUAL DISPLAYS OF DATA.
12.2. MEASURES OF CENTRAL TENDENCY.
12.3. MEASURES OF DISPERSION.
12.4. MEASURES OF POSITION.
12.5. THE NORMAL DISTRIBUTION.
13 Personal Financial Management.
13.1. THE TIME VALUE OF MONEY.
13.2. CONSUMER CREDIT.
13.3. TRUTH IN LENDING.
13.4. THE COSTS AND ADVANTAGES OF HOME OWNERSHIP.
13.5. FINANCIAL INVESTMENTS.
14 Graph Theory.
14.1. BASIC CONCEPTS.
14.2. EULER CIRCUITS AND ROUTE PLANNING.
14.3. HAMILTON CIRCUITS AND ALGORITHMS.
14.4. TREES AND MINIMUM SPANNING TREES.
15 Voting and Apportionment.
15.1. THE POSSIBILITIES OF VOTING.
15.2. THE IMPOSSIBILITIES OF VOTING.
15.3. THE POSSIBILITIES OF APPORTIONMENT.
15.4. THE IMPOSSIBILITIES OF APPORTIONMENT.
ANSWERS TO SELECTED EXERCISES A-1.
CREDITS C-1.
INDEX OF APPLICATIONS 1-1.
INDEX I-7.



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