LIST OF TABLES PREFACE PART A INTRODUCTORY OVERVIEW xiii xv 1 Introduction 3 1.1 What Is Language? 4 1.2 What Is Mathematics? 5 1.3 Why Use Mathematics? 7 1.4 Mathematics and Its Language 8 1.5 The Role of Translating English to Mathematics in Applying Mathematics 9 1.6 The Language of Mathematics vs. Mathematics vs. Mathematical Models 11 1.7 Goals and Intended Readership 12 1.8 Structure of the Book 14 1.9 Guidelines for the Reader 15 2 Preview: Some Statements in English and the Language of Mathematics 19 2.1 An Ancient Problem: Planning the Digging of a Canal 20 2.2 The Wall Around the Ancient City of Uruk 21 2.3 A Numerical Thought Puzzle 23 2.4 A Nursery Rhyme 24 2.5 Making a Pot of Tea 25 2.6 Combining Data Files 26 2.7 Selecting a Telephone Tariff 27 2.8 Interest on Savings Accounts, Bonds, etc. 28 2.9 Sales and Value-Added Tax on Sales of Goods and Services 28 2.10 A Hand of Cards 30 2.11 Shear and Moment in a Beam 30 2.12 Forming Abbreviations of Names 32 2.13 The Energy in Earth s Reflected Sunlight vs. That in Extracted Crude Oil 33 COPYRIGHTED MATERIAL vii
viii PART B MATHEMATICS AND ITS LANGUAGE 3 Elements of the Language of Mathematics 37 3.1 Values 37 3.2 Variables 38 3.3 Functions 39 3.4 Expressions 42 3.4.1 Standard Functional Notation 42 3.4.2 Infix Notation 43 3.4.3 Tree Notation 53 3.4.4 Prefix and Postfix Notation 58 3.4.5 Tabular Notation 61 3.4.6 Graphical Notation 62 3.4.7 Figures, Drawings, and Diagrams 65 3.4.8 Notation for Series and Quantification 66 3.4.9 Specialized Notational Forms for Certain Expressions 72 3.4.10 Advantages and Disadvantages of the Different Notational Forms 75 3.5 Evaluating Variables, Functions, and Expressions 78 3.5.1 Complete (Total) Evaluation 79 3.5.2 Partial Evaluation 80 3.5.3 Undefined Values of Functions and Expressions 82 3.6 Representations of Values vs. Names of Variables 85 4 Important Structures and Concepts in the Language of Mathematics 87 4.1 Common Structures of Values 87 4.1.1 Sets 88 4.1.2 Arrays (Indexed Variables), Subscripted Variables, and Matrices 92 4.1.3 Sequences 94 4.1.4 The Equivalence of Array Variables, Functions, Sequences, and Variables 96 4.1.5 Direct Correspondence of Other Mathematical Objects and Structures 96 4.1.6 Relations 99 4.1.7 Finite State Machines 100 4.2 Infinity 102 4.3 Iterative Definitions and Recursion 103 4.4 Convergence, Limits, and Bounds 105 4.5 Calculus 111 4.6 Probability Theory 117 4.6.1 Mathematical Model of a Probabilistic Process 118 4.6.2 Mean, Median, Variance, and Deviation 121
ix 4.6.3 Independent Probabilistic Processes 126 4.6.4 Dependent Probabilistic Processes and Conditional Probabilities 128 4.7 Theorems 130 4.8 Symbols and Notation 131 5 Solving Problems Mathematically 133 5.1 Manipulating Expressions 134 5.2 Proving Theorems 140 5.2.1 Techniques and Guidelines for Proving Theorems 141 5.2.2 Notation for Proofs 142 5.2.3 Lemmata and Examples of Proofs 143 5.2.4 Additional Useful Identities 148 5.3 Solving Equations and Other Boolean Expressions 150 5.4 Solving Optimization Problems 153 PART C ENGLISH, THE LANGUAGE OF MATHEMATICS, AND TRANSLATING BETWEEN THEM 6 Linguistic Characteristics of English and the Language of Mathematics 159 6.1 Universe of Discourse 159 6.2 Linguistic Elements in the Language of Mathematics and in English 162 6.2.1 Verbs, Clauses, and Phrases 164 6.2.2 Nouns and Pronouns 166 6.2.3 Adjectives, Adverbs, and Prepositional Phrases 166 6.2.4 Conjunctions 166 6.2.5 Negation 167 6.2.6 Parts of Speech and Naming Conventions for Functions and Variables 169 6.3 Cause and Effect 176 6.4 Word Order 177 6.5 Grammatical Agreement 180 6.6 Verbs: Tense, Mood, Voice, Action vs. State or Being, Stative 184 6.7 Ambiguity 187 6.8 Style 189 6.9 Limitations and Extendability of the Language of Mathematics 192 6.10 The Languages Used in Mathematical Text 193 6.11 Evaluating Statements in English and Expressions in the Language of Mathematics 195
x 6.12 Meanings of Boolean Expressions in an English Language Context 195 6.13 Mathematical Models and Their Interpretation 197 6.13.1 Dimensions of Numerical Variables 199 6.13.2 An Example of a Mathematical Model and Its Interpretation 200 7 Translating English to Mathematics 209 7.1 General Considerations 210 7.2 Sentences of the Form Is (a) (Singular Forms) 215 7.3 Sentences of the Form s Are s (Plural Forms) 218 7.4 Percent, Per, and Other Low-Level Equivalences 221 7.5 Modeling Time and Dynamic Processes in the Language of Mathematics 223 7.5.1 Dynamic Processes in Continuous Time 223 7.5.2 Dynamic Processes in Discrete Time Steps 225 7.6 Questions in Translations from English to Mathematics 234 7.7 Summary of Guidelines for Translating English to the Language of Mathematics 236 7.8 Accuracy, Errors, and Discrepancies in Mathematical Models 238 7.8.1 Errors Translating the Actual Problem into English 239 7.8.2 Errors Translating the English Text into a Mathematical Model 240 7.8.3 Errors Transforming the Mathematical Model into a Mathematical Solution 241 7.8.4 Errors Translating the Mathematical Solution into an English Specification 243 7.8.5 Errors Implementing the English Specification of the Solution 243 8 Examples of Translating English to Mathematics 245 8.1 Students with the Same Birthday 246 8.2 Criterion for Searching an Array 248 8.2.1 Search for Any Occurrence of a Value in an Array 248 8.2.2 Search for the First Occurrence of a Value in an Array 252 8.3 Specifying the Initial State of a Board Game 256 8.3.1 Initialization of a Game Board: A Correct Solution 256 8.3.2 Initialization of a Game Board: A Wrong Solution 262 8.4 Price Discounts 264 8.4.1 Flat Discounts 264 8.4.2 Discount Rates Depending on Quantity 265
xi 8.4.3 Buy 2, Get 1 Free 267 8.5 Model of a Very Small Economy 268 8.6 A Logical Puzzle 273 8.6.1 English Statement of the Puzzle 273 8.6.2 Restatement of the Puzzle 273 8.6.3 General Assumptions 274 8.6.4 The Values, Variables, and Functions in the Mathematical Model 275 8.6.5 The Interpretation of Values, Variables, Functions, and Sets 277 8.6.6 The Mathematical Model 278 8.6.7 Solving the Puzzle 281 8.7 Covering a Modified Chess Board with Dominoes 287 8.8 Validity of a Play in a Card Game 289 8.8.1 The Rules of Play 289 8.8.2 Translating the Rules of Play 289 8.8.3 Identifying the Noun Phrases in the English Text 290 8.8.4 Developing the Mathematical Model 291 8.9 The Logical Paradox of the Barber of Seville 294 8.9.1 English Statement of the Paradox 295 8.9.2 Mathematical Model 296 8.10 Controlling the Water Level in a Reservoir: Simple On/Off Control 297 8.10.1 English Statement of the Requirements 298 8.10.2 The Mathematical Variables and Their Interpretation 298 8.10.3 The Mathematical Model 298 8.10.4 Shortcomings of the Simple On/Off Control 299 8.11 Controlling the Water Level in a Reservoir: Two-Level On/Off Control 299 8.11.1 English Statement of the Requirements 299 8.11.2 Interpretation 299 8.11.3 The Mathematical Model 300 8.12 Reliable Combinations of Less Reliable Components 301 8.12.1 A Door Closure Sensor 301 8.12.2 Increased Reliability with Additional Redundant Door Sensors 302 8.12.3 The Complete Mathematical Model for the Redundant Door Sensing Systems 306 8.13 Shopping Mall Door Controller 309 8.13.1 Persons View of the Door 309 8.13.2 Physical Devices Associated with the Door 309 8.13.3 The Door Controller s Inputs and Outputs 310 8.13.4 Required Responses of the Door Controller 310 8.13.5 Method of Operation of the Controller 310
xii 8.13.6 The Variables 311 8.13.7 Interpretation of the Variables 313 8.13.8 The Mathematical Model 314 8.13.9 The Controller Function 315 8.13.10 Constructing the Controller Function Table 316 8.13.11 The Complete Controller Function Table 357 PART D CONCLUSION 9 Summary 365 9.1 Transforming English to Mathematics: A Language Not a Mathematical Problem 365 9.2 Advantages of the Language of Mathematics for Reasoning and Analyzing 366 9.3 Comparison of Key Characteristics of English and the Language of Mathematics 366 9.4 Translating from English to the Language of Mathematics: Interpretation 368 9.5 Translating from English to the Language of Mathematics: Approach and Strategy 369 Appendix A Representing Numbers 371 Appendix B Symbols in the Language of Mathematics 376 Appendix C Sets of Numbers 379 Appendix D Special Structures in Mathematics 382 Appendix E Mathematical Logic 385 Appendix F Waves and the Wave Equation 389 Appendix G Glossary: English to the Language of Mathematics 395 Appendix H Programming Languages and the Language of Mathematics 398 Appendix I Other Literature 400 INDEX 407