ema urworlc Allan G. Bluman Higher Education Professor Emeritus, Community College of Allegheny County Me Graw Hill

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1 ema in urworlc Allan G. Bluman Professor Emeritus, Community College of Allegheny County Me Graw Hill Higher Education Boston Burr Ridge, IL Dubuque, IA Madison, Wl New York San Francisco St. Louis Bangkok Bogota Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto

2 Preface xiii Logic 73 Problem Solving 1 Mathematics in Our World: Rhind Papyrus 2 Introduction The Nature of Mathematical Reasoning Problem Solving Estimation 17 Sets 33 Estimation 19 Summary 28 Mathematics in Our World: Survey Blood Types 34 Introduction The Nature of Sets 35 Basic Concepts 35 Finite and Infinite Sets 38 Equal and Equivalent Sets Subsets and Set Operations 44 Subsets 44 Union and Intersection of Sets 47 Complement of a Set Venn Diagrams Using Sets to Solve Problems Infinite Sets 65 Summary Mathematics in Our World: "Alice in Logicland" 74 Introduction Statements Truth Tables 82 The Negation, Conjunction, and Disjunction 82 The Conditional and the Biconditional Statements 83 Truth Tables for Other Statements 85 Types of Statements 92 Tautologies and Self-Contradictions 92 Logically Equivalent Statements 93 Negations of Compound Statements 94 Variations of the Conditional Statement Arguments Euler Circles 106 Summary 113 Numeration Systems 117 Mathematics in Our World: Bar Codes 118 Introduction Early and Modern Numeration Systems 118 The Egyptian Numeration System 119 The Babylonian Numeration System 123 The Roman Numeration System 125 The Hindu-Arabic Numeration System 127 vii

3 viii Contents 4-2 Base Number Systems 131 Base Five System 131 Converting Base Five Numbers to Base Ten Numbers 132 Converting Base Ten Numbers to Base Five Numbers 133 Other Number Bases 135 Converting Base Ten Numbers to Other Base Numbers 138 Base Numbers and Computers Operations in Base Numbers 142 Summary 154 The Real Number System 159 Mathematics in Our World: Are You Sure It's Fat Free? 160 Introduction The Natural Numbers 160 Prime and Composite Numbers 160 Prime Factorizations 162 Greatest Common Factor 164 Least Common Multiple The Integers 170 Definition of Integers 170 Addition and Subtraction of Integers 171 Multiplication and Division of Integers 174 Order of Operations The Rational Numbers 181 Definition of Rational Numbers 181 Reducing Fractions 184 Changing Fractions to Higher Terms 184 Operations with Fractions 185 Density of Rational Numbers 187 Fractions and Decimals The Irrational Numbers 194 Definition of an Irrational Number 194 Operations with Irrational Numbers The Real Numbers 203 Definition of Real Numbers 203 Properties of Real Numbers Exponents and Scientific Notation 211 Properties of Exponents 211 Scientific Notation Arithmetic and Geometric Sequences Arithmetic Sequences 221 Geometric Sequences 225 Application of Sequences 228 Summary 232 Other Mathematical Systems Mathematics in Our World: Game or Mathematical System? 240 Introduction 240 Clock Arithmetic 240 Addition on the Twelve-Hour Clock 240 Multiplication on the Twelve-Hour Clock 241 Subtraction on the Twelve-Hour Clock 243 Properties of the Twelve-Hour Clock 244 Modular Systems 248 Operations for the Modular Systems 249 Properties of the Modular Systems 251 Application of Modular Systems 253 Mathematical Systems without Numbers 256 Operations for Systems without Numbers 256 Properties for Systems without Numbers 257 Summary 263 Topics in Algebra 267 Mathematics in Our World: Medicine and Children 268 Introduction Fundamental Concepts of Algebra 268 Basic Definitions 268 The Distributive Property 269 Combining Like Terms 270 Simplifying Algebraic Expressions 271 Evaluating Algebraic Expressions and Formulas Solving Linear Equations 278 Basic Definitions 278 Solving the Four Basic Types of Equations 278 Solving Equations in General 281

4 Contents ix Solving Equations Containing Fractions 283 Solving an Equation or Formula for a Specific Variable Applications of Linear Equations 289 Translating Verbal Statements into Symbols 289 Solving Word Problems Solving Linear Inequalities 295 Graphing Inequalities on the Number Line 295 Solving Inequalities 295 Applications of Inequalities Ratio, Proportion, and Variation 302 Ratios 302 Proportions 303 Variation Solving Quadratic Equations 313 The Standard Form of a Quadratic Equation 313 Multiplying Binomials 314 Factoring Trinomials When a = Factoring a Trinomial When a ^ Solving Quadratic Equations by Factoring 321 Solving Quadratic Equations Using the Quadratic Formula 322 Applications of Quadratic Equations 323 Summary 327 Additional Topics in Algebra 333 Mathematics in Our World: Stopping Distance of an Automobile 334 Introduction The Rectangular Coordinate System and the Line 334 The Line 338 Intercepts 339 Slope 341 Horizontal and Vertical Lines 344 Applications of Linear Equations Systems of Linear Equations 349 Solving a Linear System Graphically 350 Solving a Linear System by Substitution 354 Solving a System by Addition/Subtraction (Elimination) 356 Applications of Linear Systems Systems of Linear Inequalities 365 Systems of Linear Inequalities Linear Programming Functions 377 Relations 377 Functions 377 The Vertical Line Test 379 Graphing Linear Functions 380 Quadratic Functions 382 Graphing the Parabola 382 Applications of Quadratic Functions 386 Exponential Functions 387 Applications of Exponential Functions 390 Summary 395 Consumer Mathematics 401 Mathematics in Our World: How Much Can You Afford to Pay for a Home? 402 Introduction Percent 402 Percent Conversions 402 Three Types of Percent Problems 405 Applications of Percents Interest 412 Simple Interest 412 Finding the Principal, Rate, and Time 414 Compound Interest 416 Effective Rate 418 Annuities Installment Buying 425 Unpaid Balance Method 426 Average Daily Balance Method 426 Annual Percentage Rate 429 Rule of 78s Home Ownership 435 Finding the Monthly Payments and Total Interest 435 Computing an Amortization Schedule 437 Property Tax Markup and Markdown 442 Markup on Cost 442 Markup on Selling Price 444 Conversions 445 Markdown 446 Summary 451

5 Contents Geometry 457 Mathematics in Our World: The Force of the Wind 458 Introduction Points, Lines, Planes, and Angles 458 Points, Lines, and Planes 458 Pairs of Angles Triangles 468 Types of Triangles 469 The Pythagorean Theorem 471 Similar Triangles Polygons and Perimeter 479 Polygons 479 Perimeter Areas of Polygons and the Circle 485 Areas of Polygons 485 The Circle Surface Area and Volume Right Triangle Trigonometry 500 The Trigonometric Ratios 501 Angle of Elevation and Angle of Depression Networks 509 Summary 515 Probability and Counting Techniques 523 Mathematics in Our World: Would You Bet Your Life? 524 Introduction Basic Concepts of Probability 524 Sample Spaces 524 Classical Probability 525 Probability and Sets 528 Empirical Probability Tree Diagrams, Tables, and Sample Spaces 534 Tree Diagrams 535 Tables Odds and Expectation 544 Expectation The Addition Rules for Probability 551 Probability and Sets The Multiplication Rules and Conditional Probability 560 Conditional Probability 565 Probability and Sets The Fundamental Counting Rule and Permutations 570 Factorial Notation 572 Permutations Combinations Probability Using Permutations and Combinations 583 Summary 589 Statistics 595 Mathematics in Our World: Who Is a Typical First-Time Home Buyer? 596 Introduction The Nature of Statistics and Organizing Data 596 Populations and Samples 597 Sampling Methods 597 Descriptive and Inferential Statistics 598 Frequency Distributions 598 Stem and Leaf Plots Picturing Data 607 The Bar Graph and Pie Graph 608 The Histogram and Frequency Polygon 610 Time Series Graphs Measures of Average 620 The Mean 620 The Median 623 The Mode 625 Comparing the Mean, Median, and Mode 627 The Midrange Measures of Variation 633 Range 633 Variance and Standard Deviation 633 Interpretation of the Standard Deviation Measures of Position The Normal Distribution 643 Summary of the Properties of a Normal Distribution 644 The Standard Normal Distribution 644 Finding Areas Under the Standard Normal Distribution 645

6 Contents XI 12-7 Applications of the Normal Distribution Correlation and Regression Analysis 660 Scatter Plots 660 The Correlation Coefficienr 662 Regression 666 Correlation and Causation 669 Summary 672 Supplement: Misuses of Statistics The Pairwise Comparison Method and Approval Voting 709 The Pairwise Comparison Method 709 Arrow's Impossibility Theorem 712 Approval Voting 713 Tie Breaking 714 Summary 719 Voting Methods 689 Mathematics in Our World: Academy Awards 690 Introduction Preference Tables and the Plurality Method 691 Preference Tables 691 The Plurality Method The Borda Count Method and the Pluralitywith-Elimination Method 699 The Borda Count Method 699 The Plurality-with-Elimination Method 703 Appendix A Measurement 723 Appendix B Trigonometric Ratios 739 Appendix C Area Under the Standard Normal Distribution 741 Appendix D Significant Values for the Correlation Coefficient 743 Appendix E Using the TI-83 Plus Graphing Calculator 744 Appendix F Credits 760 Appendix G Answers 763 Index 1-1

Integrated Arithmetic and Basic Algebra

211 771 406 III T H I R D E D I T I O N Integrated Arithmetic and Basic Algebra Bill E. Jordan Seminole Community College William P. Palow Miami-Dade College Boston San Francisco New York London Toronto