Language of Mathematics : Utilizing Math in Practice.
Title:
Language of Mathematics : Utilizing Math in Practice.
Author:
Baber, Robert L.
ISBN:
9781118061718
Personal Author:
Edition:
1st ed.
Physical Description:
1 online resource (438 pages)
Contents:
THE LANGUAGE OF MATHEMATICS -- CONTENTS -- LIST OF TABLES -- PREFACE -- PART A INTRODUCTORY OVERVIEW -- 1 Introduction -- 1.1 What Is Language? -- 1.2 What Is Mathematics? -- 1.3 Why Use Mathematics? -- 1.4 Mathematics and Its Language -- 1.5 The Role of Translating English to Mathematics in Applying Mathematics -- 1.6 The Language of Mathematics vs. Mathematics vs. Mathematical Models -- 1.7 Goals and Intended Readership -- 1.8 Structure of the Book -- 1.9 Guidelines for the Reader -- 2 Preview: Some Statements in English and the Language of Mathematics -- 2.1 An Ancient Problem: Planning the Digging of a Canal -- 2.2 The Wall Around the Ancient City of Uruk -- 2.3 A Numerical Thought Puzzle -- 2.4 A Nursery Rhyme -- 2.5 Making a Pot of Tea -- 2.6 Combining Data Files -- 2.7 Selecting a Telephone Tariff -- 2.8 Interest on Savings Accounts, Bonds, etc. -- 2.9 Sales and Value-Added Tax on Sales of Goods and Services -- 2.10 A Hand of Cards -- 2.11 Shear and Moment in a Beam -- 2.12 Forming Abbreviations of Names -- 2.13 The Energy in Earth's Reflected Sunlight vs. That in Extracted Crude Oil -- PART B MATHEMATICS AND ITS LANGUAGE -- 3 Elements of the Language of Mathematics -- 3.1 Values -- 3.2 Variables -- 3.3 Functions -- 3.4 Expressions -- 3.4.1 Standard Functional Notation -- 3.4.2 Infix Notation -- 3.4.3 Tree Notation -- 3.4.4 Prefix and Postfix Notation -- 3.4.5 Tabular Notation -- 3.4.6 Graphical Notation -- 3.4.7 Figures, Drawings, and Diagrams -- 3.4.8 Notation for Series and Quantification -- 3.4.9 Specialized Notational Forms for Certain Expressions -- 3.4.10 Advantages and Disadvantages of the Different Notational Forms -- 3.5 Evaluating Variables, Functions, and Expressions -- 3.5.1 Complete (Total) Evaluation -- 3.5.2 Partial Evaluation -- 3.5.3 Undefined Values of Functions and Expressions.
3.6 Representations of Values vs. Names of Variables -- 4 Important Structures and Concepts in the Language of Mathematics -- 4.1 Common Structures of Values -- 4.1.1 Sets -- 4.1.2 Arrays (Indexed Variables), Subscripted Variables, and Matrices -- 4.1.3 Sequences -- 4.1.4 The Equivalence of Array Variables, Functions, Sequences, and Variables -- 4.1.5 Direct Correspondence of Other Mathematical Objects and Structures -- 4.1.6 Relations -- 4.1.7 Finite State Machines -- 4.2 Infinity -- 4.3 Iterative Definitions and Recursion -- 4.4 Convergence, Limits, and Bounds -- 4.5 Calculus -- 4.6 Probability Theory -- 4.6.1 Mathematical Model of a Probabilistic Process -- 4.6.2 Mean, Median, Variance, and Deviation -- 4.6.3 Independent Probabilistic Processes -- 4.6.4 Dependent Probabilistic Processes and Conditional Probabilities -- 4.7 Theorems -- 4.8 Symbols and Notation -- 5 Solving Problems Mathematically -- 5.1 Manipulating Expressions -- 5.2 Proving Theorems -- 5.2.1 Techniques and Guidelines for Proving Theorems -- 5.2.2 Notation for Proofs -- 5.2.3 Lemmata and Examples of Proofs -- 5.2.4 Additional Useful Identities -- 5.3 Solving Equations and Other Boolean Expressions -- 5.4 Solving Optimization Problems -- PART C ENGLISH, THE LANGUAGE OF MATHEMATICS, AND TRANSLATING BETWEEN THEM -- 6 Linguistic Characteristics of English and the Language of Mathematics -- 6.1 Universe of Discourse -- 6.2 Linguistic Elements in the Language of Mathematics and in English -- 6.2.1 Verbs, Clauses, and Phrases -- 6.2.2 Nouns and Pronouns -- 6.2.3 Adjectives, Adverbs, and Prepositional Phrases -- 6.2.4 Conjunctions -- 6.2.5 Negation -- 6.2.6 Parts of Speech and Naming Conventions for Functions and Variables -- 6.3 Cause and Effect -- 6.4 Word Order -- 6.5 Grammatical Agreement -- 6.6 Verbs: Tense, Mood, Voice, Action vs. State or Being, Stative -- 6.7 Ambiguity.
6.8 Style -- 6.9 Limitations and Extendability of the Language of Mathematics -- 6.10 The Languages Used in Mathematical Text -- 6.11 Evaluating Statements in English and Expressions in the Language of Mathematics -- 6.12 Meanings of Boolean Expressions in an English Language Context -- 6.13 Mathematical Models and Their Interpretation -- 6.13.1 Dimensions of Numerical Variables -- 6.13.2 An Example of a Mathematical Model and Its Interpretation -- 7 Translating English to Mathematics -- 7.1 General Considerations -- 7.2 Sentences of the Form "… Is (a) …" (Singular Forms) -- 7.3 Sentences of the Form "…s Are…s" (Plural Forms) -- 7.4 Percent, Per …, and Other Low-Level Equivalences -- 7.5 Modeling Time and Dynamic Processes in the Language of Mathematics -- 7.5.1 Dynamic Processes in Continuous Time -- 7.5.2 Dynamic Processes in Discrete Time Steps -- 7.6 Questions in Translations from English to Mathematics -- 7.7 Summary of Guidelines for Translating English to the Language of Mathematics -- 7.8 Accuracy, Errors, and Discrepancies in Mathematical Models -- 7.8.1 Errors Translating the Actual Problem into English -- 7.8.2 Errors Translating the English Text into a Mathematical Model -- 7.8.3 Errors Transforming the Mathematical Model into a Mathematical Solution -- 7.8.4 Errors Translating the Mathematical Solution into an English Specification -- 7.8.5 Errors Implementing the English Specification of the Solution -- 8 Examples of Translating English to Mathematics -- 8.1 Students with the Same Birthday -- 8.2 Criterion for Searching an Array -- 8.2.1 Search for Any Occurrence of a Value in an Array -- 8.2.2 Search for the First Occurrence of a Value in an Array -- 8.3 Specifying the Initial State of a Board Game -- 8.3.1 Initialization of a Game Board: A Correct Solution -- 8.3.2 Initialization of a Game Board: A Wrong "Solution".
8.4 Price Discounts -- 8.4.1 Flat Discounts -- 8.4.2 Discount Rates Depending on Quantity -- 8.4.3 Buy 2, Get 1 Free -- 8.5 Model of a Very Small Economy -- 8.6 A Logical Puzzle -- 8.6.1 English Statement of the Puzzle -- 8.6.2 Restatement of the Puzzle -- 8.6.3 General Assumptions -- 8.6.4 The Values, Variables, and Functions in the Mathematical Model -- 8.6.5 The Interpretation of Values, Variables, Functions, and Sets -- 8.6.6 The Mathematical Model -- 8.6.7 Solving the Puzzle -- 8.7 Covering a Modified Chess Board with Dominoes -- 8.8 Validity of a Play in a Card Game -- 8.8.1 The Rules of Play -- 8.8.2 Translating the Rules of Play -- 8.8.3 Identifying the Noun Phrases in the English Text -- 8.8.4 Developing the Mathematical Model -- 8.9 The Logical Paradox of the Barber of Seville -- 8.9.1 English Statement of the Paradox -- 8.9.2 Mathematical Model -- 8.10 Controlling the Water Level in a Reservoir: Simple On/Off Control -- 8.10.1 English Statement of the Requirements -- 8.10.2 The Mathematical Variables and Their Interpretation -- 8.10.3 The Mathematical Model -- 8.10.4 Shortcomings of the Simple On/Off Control -- 8.11 Controlling the Water Level in a Reservoir: Two-Level On/Off Control -- 8.11.1 English Statement of the Requirements -- 8.11.2 Interpretation -- 8.11.3 The Mathematical Model -- 8.12 Reliable Combinations of Less Reliable Components -- 8.12.1 A Door Closure Sensor -- 8.12.2 Increased Reliability with Additional Redundant Door Sensors -- 8.12.3 The Complete Mathematical Model for the Redundant Door Sensing Systems -- 8.13 Shopping Mall Door Controller -- 8.13.1 Persons' View of the Door -- 8.13.2 Physical Devices Associated with the Door -- 8.13.3 The Door Controller's Inputs and Outputs -- 8.13.4 Required Responses of the Door Controller -- 8.13.5 Method of Operation of the Controller -- 8.13.6 The Variables.
8.13.7 Interpretation of the Variables -- 8.13.8 The Mathematical Model -- 8.13.9 The Controller Function -- 8.13.10 Constructing the Controller Function Table -- 8.13.11 The Complete Controller Function Table -- PART D CONCLUSION -- 9 Summary -- 9.1 Transforming English to Mathematics: A Language-Not a Mathematical-Problem -- 9.2 Advantages of the Language of Mathematics for Reasoning and Analyzing -- 9.3 Comparison of Key Characteristics of English and the Language of Mathematics -- 9.4 Translating from English to the Language of Mathematics: Interpretation -- 9.5 Translating from English to the Language of Mathematics: Approach and Strategy -- Appendix A Representing Numbers -- Appendix B Symbols in the Language of Mathematics -- Appendix C Sets of Numbers -- Appendix D Special Structures in Mathematics -- Appendix E Mathematical Logic -- Appendix F Waves and theWave Equation -- Appendix G Glossary: English to the Language of Mathematics -- Appendix H Programming Languages and the Language of Mathematics -- Appendix I Other Literature -- INDEX.
Abstract:
A new and unique way of understanding the translation of concepts and natural language into mathematical expressions Transforming a body of text into corresponding mathematical expressions and models is traditionally viewed and taught as a mathematical problem; it is also a task that most find difficult. The Language of Mathematics: Utilizing Math in Practice reveals a new way to view this process-not as a mathematical problem, but as a translation, or language, problem. By presenting the language of mathematics explicitly and systematically, this book helps readers to learn mathematics¿and improve their ability to apply mathematics more efficiently and effectively to practical problems in their own work. Using parts of speech to identify variables and functions in a mathematical model is a new approach, as is the insight that examining aspects of grammar is highly useful when formulating a corresponding mathematical model. This book identifies the basic elements of the language of mathematics, such as values, variables, and functions, while presenting the grammatical rules for combining them into expressions and other structures. The author describes and defines different notational forms for expressions, and also identifies the relationships between parts of speech and other grammatical elements in English and components of expressions in the language of mathematics. Extensive examples are used throughout that cover a wide range of real-world problems and feature diagrams and tables to facilitate understanding. The Language of Mathematics is a thought-provoking book of interest for readers who would like to learn more about the linguistic nature and aspects of mathematical notation. The book also serves as a valuable supplement for engineers, technicians, managers, and consultants who would like to improve their ability to apply
mathematics effectively, systematically, and efficiently to practical problems.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
Genre:
Electronic Access:
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