P1: TIX/XYZ P2: ABC JWBS073-FM JWBS073-Baber August 4, :47 Printer: Markono LIST OF TABLES PREFACE

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1 LIST OF TABLES PREFACE PART A INTRODUCTORY OVERVIEW xiii xv 1 Introduction What Is Language? What Is Mathematics? Why Use Mathematics? Mathematics and Its Language The Role of Translating English to Mathematics in Applying Mathematics The Language of Mathematics vs. Mathematics vs. Mathematical Models Goals and Intended Readership Structure of the Book Guidelines for the Reader 15 2 Preview: Some Statements in English and the Language of Mathematics An Ancient Problem: Planning the Digging of a Canal The Wall Around the Ancient City of Uruk A Numerical Thought Puzzle A Nursery Rhyme Making a Pot of Tea Combining Data Files Selecting a Telephone Tariff Interest on Savings Accounts, Bonds, etc Sales and Value-Added Tax on Sales of Goods and Services A Hand of Cards Shear and Moment in a Beam Forming Abbreviations of Names The Energy in Earth s Reflected Sunlight vs. That in Extracted Crude Oil 33 COPYRIGHTED MATERIAL vii

2 viii PART B MATHEMATICS AND ITS LANGUAGE 3 Elements of the Language of Mathematics Values Variables Functions Expressions Standard Functional Notation Infix Notation Tree Notation Prefix and Postfix Notation Tabular Notation Graphical Notation Figures, Drawings, and Diagrams Notation for Series and Quantification Specialized Notational Forms for Certain Expressions Advantages and Disadvantages of the Different Notational Forms Evaluating Variables, Functions, and Expressions Complete (Total) Evaluation Partial Evaluation Undefined Values of Functions and Expressions Representations of Values vs. Names of Variables 85 4 Important Structures and Concepts in the Language of Mathematics Common Structures of Values Sets Arrays (Indexed Variables), Subscripted Variables, and Matrices Sequences The Equivalence of Array Variables, Functions, Sequences, and Variables Direct Correspondence of Other Mathematical Objects and Structures Relations Finite State Machines Infinity Iterative Definitions and Recursion Convergence, Limits, and Bounds Calculus Probability Theory Mathematical Model of a Probabilistic Process Mean, Median, Variance, and Deviation 121

3 ix Independent Probabilistic Processes Dependent Probabilistic Processes and Conditional Probabilities Theorems Symbols and Notation Solving Problems Mathematically Manipulating Expressions Proving Theorems Techniques and Guidelines for Proving Theorems Notation for Proofs Lemmata and Examples of Proofs Additional Useful Identities Solving Equations and Other Boolean Expressions 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 Universe of Discourse Linguistic Elements in the Language of Mathematics and in English Verbs, Clauses, and Phrases Nouns and Pronouns Adjectives, Adverbs, and Prepositional Phrases Conjunctions Negation Parts of Speech and Naming Conventions for Functions and Variables Cause and Effect Word Order Grammatical Agreement Verbs: Tense, Mood, Voice, Action vs. State or Being, Stative Ambiguity Style Limitations and Extendability of the Language of Mathematics The Languages Used in Mathematical Text Evaluating Statements in English and Expressions in the Language of Mathematics 195

4 x 6.12 Meanings of Boolean Expressions in an English Language Context Mathematical Models and Their Interpretation Dimensions of Numerical Variables An Example of a Mathematical Model and Its Interpretation Translating English to Mathematics General Considerations Sentences of the Form Is (a) (Singular Forms) Sentences of the Form s Are s (Plural Forms) Percent, Per, and Other Low-Level Equivalences Modeling Time and Dynamic Processes in the Language of Mathematics Dynamic Processes in Continuous Time Dynamic Processes in Discrete Time Steps Questions in Translations from English to Mathematics Summary of Guidelines for Translating English to the Language of Mathematics Accuracy, Errors, and Discrepancies in Mathematical Models Errors Translating the Actual Problem into English Errors Translating the English Text into a Mathematical Model Errors Transforming the Mathematical Model into a Mathematical Solution Errors Translating the Mathematical Solution into an English Specification Errors Implementing the English Specification of the Solution Examples of Translating English to Mathematics Students with the Same Birthday Criterion for Searching an Array Search for Any Occurrence of a Value in an Array Search for the First Occurrence of a Value in an Array Specifying the Initial State of a Board Game Initialization of a Game Board: A Correct Solution Initialization of a Game Board: A Wrong Solution Price Discounts Flat Discounts Discount Rates Depending on Quantity 265

5 xi Buy 2, Get 1 Free Model of a Very Small Economy A Logical Puzzle English Statement of the Puzzle Restatement of the Puzzle General Assumptions The Values, Variables, and Functions in the Mathematical Model The Interpretation of Values, Variables, Functions, and Sets The Mathematical Model Solving the Puzzle Covering a Modified Chess Board with Dominoes Validity of a Play in a Card Game The Rules of Play Translating the Rules of Play Identifying the Noun Phrases in the English Text Developing the Mathematical Model The Logical Paradox of the Barber of Seville English Statement of the Paradox Mathematical Model Controlling the Water Level in a Reservoir: Simple On/Off Control English Statement of the Requirements The Mathematical Variables and Their Interpretation The Mathematical Model Shortcomings of the Simple On/Off Control Controlling the Water Level in a Reservoir: Two-Level On/Off Control English Statement of the Requirements Interpretation The Mathematical Model Reliable Combinations of Less Reliable Components A Door Closure Sensor Increased Reliability with Additional Redundant Door Sensors The Complete Mathematical Model for the Redundant Door Sensing Systems Shopping Mall Door Controller Persons View of the Door Physical Devices Associated with the Door The Door Controller s Inputs and Outputs Required Responses of the Door Controller Method of Operation of the Controller 310

6 xii The Variables Interpretation of the Variables The Mathematical Model The Controller Function Constructing the Controller Function Table The Complete Controller Function Table 357 PART D CONCLUSION 9 Summary Transforming English to Mathematics: A Language Not a Mathematical Problem Advantages of the Language of Mathematics for Reasoning and Analyzing Comparison of Key Characteristics of English and the Language of Mathematics Translating from English to the Language of Mathematics: Interpretation 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