ROBERT LAURENCE BABER is Professor Emeritus in the Department of Computing and Software at McMaster University, Canada. A Fellow of the BCS, The Chartered Institute for IT, he has published numerous journal articles in his areas of research interest, which include mathematical modeling and the conception, planning, and design of computer-based systems for technical and business applications.

• List of Tables xiiPreface xvPart A Introductory Overview1 Introduction 31.1 What Is Language? 41.2 What Is Mathematics? 51.3 Why Use Mathematics? 71.4 Mathematics and Its Language 81.5 The Role of Translating English to Mathematics in Applying Mathematics 91.6 The Language of Mathematics vs. Mathematics vs. Mathematical Models 111.7 Goals and Intended Readership 121.8 Structure of the Book 141.9 Guidelines for the Reader 152 Preview: Some Statements in English and the Language of Mathematics 192.1 An Ancient Problem: Planning the Digging of a Canal 202.2 The Wall Around the Ancient City of Uruk 212.3 A Numerical Thought Puzzle 232.4 A Nursery Rhyme 242.5 Making a Pot of Tea 252.6 Combining Data Files 262.7 Selecting a Telephone Tariff 272.8 Interest on Savings Accounts, Bonds, etc. 282.9 Sales and Value-Added Tax on Sales of Goods and Services 282.10 A Hand of Cards 302.11 Shear and Moment in a Beam 302.12 Forming Abbreviations of Names 322.13 The Energy in Earth’s Reflected Sunlight vs. That in Extracted Crude Oil 33Part B Mathematics and Its Language3 Elements of the Language of Mathematics 373.1 Values 373.2 Variables 383.3 Functions 393.4 Expressions 423.4.1 Standard Functional Notation 423.4.2 Infix Notation 433.4.3 Tree Notation 533.4.4 Prefix and Postfix Notation 583.4.5 Tabular Notation 613.4.6 Graphical Notation 623.4.7 Figures, Drawings, and Diagrams 653.4.8 Notation for Series and Quantification 663.4.9 Specialized Notational Forms for Certain Expressions 723.4.10 Advantages and Disadvantages of the Different Notational Forms 753.5 Evaluating Variables, Functions, and Expressions 783.5.1 Complete (Total) Evaluation 793.5.2 Partial Evaluation 803.5.3 Undefined Values of Functions and Expressions 823.6 Representations of Values vs. Names of Variables 854 Important Structures and Concepts in the Language of Mathematics 874.1 Common Structures of Values 874.1.1 Sets 884.1.2 Arrays (Indexed Variables), Subscripted Variables, and Matrices 924.1.3 Sequences 944.1.4 The Equivalence of Array Variables, Functions, Sequences, and Variables 964.1.5 Direct Correspondence of Other Mathematical Objects and Structures 964.1.6 Relations 994.1.7 Finite State Machines 1004.2 Infinity 1024.3 Iterative Definitions and Recursion 1034.4 Convergence, Limits, and Bounds 1054.5 Calculus 1114.6 Probability Theory 1174.6.1 Mathematical Model of a Probabilistic Process 1184.6.2 Mean, Median, Variance, and Deviation 1214.6.3 Independent Probabilistic Processes 1264.6.4 Dependent Probabilistic Processes and Conditional Probabilities 1284.7 Theorems 1304.8 Symbols and Notation 1315 Solving Problems Mathematically 1335.1 Manipulating Expressions 1345.2 Proving Theorems 1405.2.1 Techniques and Guidelines for Proving Theorems 1415.2.2 Notation for Proofs 1425.2.3 Lemmata and Examples of Proofs 1435.2.4 Additional Useful Identities 1485.3 Solving Equations and Other Boolean Expressions 1505.4 Solving Optimization Problems 153Part C English, the Language of Mathematics, and Translating Between Them6 Linguistic Characteristics of English and the Language of Mathematics 1596.1 Universe of Discourse 1596.2 Linguistic Elements in the Language of Mathematics and in English 1626.2.1 Verbs, Clauses, and Phrases 1646.2.2 Nouns and Pronouns 1666.2.3 Adjectives, Adverbs, and Prepositional Phrases 1666.2.4 Conjunctions 1666.2.5 Negation 1676.2.6 Parts of Speech and Naming Conventions for Functions and Variables 1696.3 Cause and Effect 1766.4 Word Order 1776.5 Grammatical Agreement 1806.6 Verbs: Tense, Mood, Voice, Action vs. State or Being, Stative 1846.7 Ambiguity 1876.8 Style 1896.9 Limitations and Extendability of the Language of Mathematics 1926.10 The Languages Used in Mathematical Text 1936.11 Evaluating Statements in English and Expressions in the Language of Mathematics 1956.12 Meanings of Boolean Expressions in an English Language Context 1956.13 Mathematical Models and Their Interpretation 1976.13.1 Dimensions of Numerical Variables 1996.13.2 An Example of a Mathematical Model and Its Interpretation 2007 Translating English to Mathematics 2097.1 General Considerations 2107.2 Sentences of the Form “… Is (a) …” (Singular Forms) 2157.3 Sentences of the Form “…s Are …s” (Plural Forms) 2187.4 Percent, Per …, and Other Low-Level Equivalences 2217.5 Modeling Time and Dynamic Processes in the Language of Mathematics 2237.5.1 Dynamic Processes in Continuous Time 2237.5.2 Dynamic Processes in Discrete Time Steps 2257.6 Questions in Translations from English to Mathematics 2347.7 Summary of Guidelines for Translating English to the Language of Mathematics 2367.8 Accuracy, Errors, and Discrepancies in Mathematical Models 2387.8.1 Errors Translating the Actual Problem into English 2397.8.2 Errors Translating the English Text into a Mathematical Model 2407.8.3 Errors Transforming the Mathematical Model into a Mathematical Solution 2417.8.4 Errors Translating the Mathematical Solution into an English Specification 2437.8.5 Errors Implementing the English Specification of the Solution 2438 Examples of Translating English to Mathematics 2458.1 Students with the Same Birthday 2468.2 Criterion for Searching an Array 2488.2.1 Search for Any Occurrence of a Value in an Array 2488.2.2 Search for the First Occurrence of a Value in an Array 2528.3 Specifying the Initial State of a Board Game 2568.3.1 Initialization of a Game Board: A Correct Solution 2568.3.2 Initialization of a Game Board: A Wrong “Solution” 2628.4 Price Discounts 2648.4.1 Flat Discounts 2648.4.2 Discount Rates Depending on Quantity 2658.4.3 Buy 2, Get 1 Free 2678.5 Model of a Very Small Economy 2688.6 A Logical Puzzle 2738.6.1 English Statement of the Puzzle 2738.6.2 Restatement of the Puzzle 2738.6.3 General Assumptions 2748.6.4 The Values, Variables, and Functions in the Mathematical Model 2758.6.5 The Interpretation of Values, Variables, Functions, and Sets 2778.6.6 The Mathematical Model 2788.6.7 Solving the Puzzle 2818.7 Covering a Modified Chess Board with Dominoes 2878.8 Validity of a Play in a Card Game 2898.8.1 The Rules of Play 2898.8.2 Translating the Rules of Play 2898.8.3 Identifying the Noun Phrases in the English Text 2908.8.4 Developing the Mathematical Model 2918.9 The Logical Paradox of the Barber of Seville 2948.9.1 English Statement of the Paradox 2958.9.2 Mathematical Model 2968.10 Controlling the Water Level in a Reservoir: Simple On/Off Control 2978.10.1 English Statement of the Requirements 2988.10.2 The Mathematical Variables and Their Interpretation 2988.10.3 The Mathematical Model 2988.10.4 Shortcomings of the Simple On/Off Control 2998.11 Controlling the Water Level in a Reservoir: Two-Level On/Off Control 2998.11.1 English Statement of the Requirements 2998.11.2 Interpretation 2998.11.3 The Mathematical Model 3008.12 Reliable Combinations of Less Reliable Components 3018.12.1 A Door Closure Sensor 3018.12.2 Increased Reliability with Additional Redundant Door Sensors 3028.12.3 The Complete Mathematical Model for the Redundant Door Sensing Systems 3068.13 Shopping Mall Door Controller 3098.13.1 Persons’ View of the Door 3098.13.2 Physical Devices Associated with the Door 3098.13.3 The Door Controller’s Inputs and Outputs 3108.13.4 Required Responses of the Door Controller 3108.13.5 Method of Operation of the Controller 3108.13.6 The Variables 3118.13.7 Interpretation of the Variables 3138.13.8 The Mathematical Model 3148.13.9 The Controller Function 3158.13.10 Constructing the Controller Function Table 3168.13.11 The Complete Controller Function Table 357Part D Conclusion9 Summary 3659.1 Transforming English to Mathematics: A Language—Not a Mathematical—Problem 3659.2 Advantages of the Language of Mathematics for Reasoning and Analyzing 3669.3 Comparison of Key Characteristics of English and the Language of Mathematics 3669.4 Translating from English to the Language of Mathematics: Interpretation 3689.5 Translating from English to the Language of Mathematics: Approach and Strategy 369Appendix A Representing Numbers 371Appendix B Symbols in the Language of Mathematics 376Appendix C Sets of Numbers 379Appendix D Special Structures in Mathematics 382Appendix E Mathematical Logic 385Appendix F Waves and the Wave Equation 389Appendix G Glossary: English to the Language of Mathematics 395Appendix H Programming Languages and the Language of Mathematics 398Appendix I Other Literature 400Index 407

“This text presents a new and original point of view on mathematics that will be useful for simplifying applications of mathematics to practical problems by translating English statements of a problem into the Language of Mathematics. The reviewer shares the author's opinion that \this book will improve and increase the reader's insight into mathematics and how to utilize it in practice.”  (Zentralblatt MATH, 2012)