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E-raamat: Language of Mathematics: Utilizing Math in Practice

(McMaster University, Canada)
  • Formaat: PDF+DRM
  • Ilmumisaeg: 13-Oct-2011
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781118061718
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  • Formaat: PDF+DRM
  • Ilmumisaeg: 13-Oct-2011
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781118061718
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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 processnot 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.

Arvustused

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)

List of Tables
xiii
Preface xv
PART A Introductory Overview
1 Introduction
3(16)
1.1 What Is Language?
4(1)
1.2 What Is Mathematics?
5(2)
1.3 Why Use Mathematics?
7(1)
1.4 Mathematics and Its Language
8(1)
1.5 The Role of Translating English to Mathematics in Applying Mathematics
9(2)
1.6 The Language of Mathematics vs. Mathematics vs. Mathematical Models
11(1)
1.7 Goals and Intended Readership
12(2)
1.8 Structure of the Book
14(1)
1.9 Guidelines for the Reader
15(4)
2 Preview: Some Statements in English and the Language of Mathematics
19(18)
2.1 An Ancient Problem: Planning the Digging of a Canal
20(1)
2.2 The Wall Around the Ancient City of Uruk
21(2)
2.3 A Numerical Thought Puzzle
23(1)
2.4 A Nursery Rhyme
24(1)
2.5 Making a Pot of Tea
25(1)
2.6 Combining Data Files
26(1)
2.7 Selecting a Telephone Tariff
27(1)
2.8 Interest on Savings Accounts, Bonds, etc.
28(1)
2.9 Sales and Value-Added Tax on Sales of Goods and Services
28(2)
2.10 A Hand of Cards
30(1)
2.11 Shear and Moment in a Beam
30(2)
2.12 Forming Abbreviations of Names
32(1)
2.13 The Energy in Earth's Reflected Sunlight vs. That in Extracted Crude Oil
33(4)
PART B Mathematics and Its Language
3 Elements of the Language of Mathematics
37(50)
3.1 Values
37(1)
3.2 Variables
38(1)
3.3 Functions
39(3)
3.4 Expressions
42(36)
3.4.1 Standard Functional Notation
42(1)
3.4.2 Infix Notation
43(10)
3.4.3 Tree Notation
53(5)
3.4.4 Prefix and Postfix Notation
58(3)
3.4.5 Tabular Notation
61(1)
3.4.6 Graphical Notation
62(3)
3.4.7 Figures, Drawings, and Diagrams
65(1)
3.4.8 Notation for Series and Quantification
66(6)
3.4.9 Specialized Notational Forms for Certain Expressions
72(3)
3.4.10 Advantages and Disadvantages of the Different Notational Forms
75(3)
3.5 Evaluating Variables, Functions, and Expressions
78(7)
3.5.1 Complete (Total) Evaluation
79(1)
3.5.2 Partial Evaluation
80(2)
3.5.3 Undefined Values of Functions and Expressions
82(3)
3.6 Representations of Values vs. Names of Variables
85(2)
4 Important Structures and Concepts in the Language of Mathematics
87(46)
4.1 Common Structures of Values
87(15)
4.1.1 Sets
88(4)
4.1.2 Arrays (Indexed Variables), Subscripted Variables, and Matrices
92(2)
4.1.3 Sequences
94(2)
4.1.4 The Equivalence of Array Variables, Functions, Sequences, and Variables
96(1)
4.1.5 Direct Correspondence of Other Mathematical Objects and Structures
96(3)
4.1.6 Relations
99(1)
4.1.7 Finite State Machines
100(2)
4.2 Infinity
102(1)
4.3 Iterative Definitions and Recursion
103(2)
4.4 Convergence, Limits, and Bounds
105(6)
4.5 Calculus
111(6)
4.6 Probability Theory
117(13)
4.6.1 Mathematical Model of a Probabilistic Process
118(3)
4.6.2 Mean, Median, Variance, and Deviation
121(5)
4.6.3 Independent Probabilistic Processes
126(2)
4.6.4 Dependent Probabilistic Processes and Conditional Probabilities
128(2)
4.7 Theorems
130(1)
4.8 Symbols and Notation
131(2)
5 Solving Problems Mathematically
133(26)
5.1 Manipulating Expressions
134(6)
5.2 Proving Theorems
140(10)
5.2.1 Techniques and Guidelines for Proving Theorems
141(1)
5.2.2 Notation for Proofs
142(1)
5.2.3 Lemmata and Examples of Proofs
143(5)
5.2.4 Additional Useful Identities
148(2)
5.3 Solving Equations and Other Boolean Expressions
150(3)
5.4 Solving Optimization Problems
153(6)
PART C English, the Language of Mathematics, and Translating between them
6 Linguistic Characteristics of English and the Language of Mathematics
159(50)
6.1 Universe of Discourse
159(3)
6.2 Linguistic Elements in the Language of Mathematics and in English
162(14)
6.2.1 Verbs, Clauses, and Phrases
164(2)
6.2.2 Nouns and Pronouns
166(1)
6.2.3 Adjectives, Adverbs, and Prepositional Phrases
166(1)
6.2.4 Conjunctions
166(1)
6.2.5 Negation
167(2)
6.2.6 Parts of Speech and Naming Conventions for Functions and Variables
169(7)
6.3 Cause and Effect
176(1)
6.4 Word Order
177(3)
6.5 Grammatical Agreement
180(4)
6.6 Verbs: Tense, Mood, Voice, Action vs. State or Being, Stative
184(3)
6.7 Ambiguity
187(2)
6.8 Style
189(3)
6.9 Limitations and Extendability of the Language of Mathematics
192(1)
6.10 The Languages Used in Mathematical Text
193(2)
6.11 Evaluating Statements in English and Expressions in the Language of Mathematics
195(1)
6.12 Meanings of Boolean Expressions in an English Language Context
195(2)
6.13 Mathematical Models and Their Interpretation
197(12)
6.13.1 Dimensions of Numerical Variables
199(1)
6.13.2 An Example of a Mathematical Model and Its Interpretation
200(9)
7 Translating English to Mathematics
209(36)
7.1 General Considerations
210(5)
7.2 Sentences of the Form "... Is (a) ..." (Singular Forms)
215(3)
7.3 Sentences of the Form "... s Are ...s" (Plural Forms)
218(3)
7.4 Percent, Per ..., and Other Low-Level Equivalences
221(2)
7.5 Modeling Time and Dynamic Processes in the Language of Mathematics
223(11)
7.5.1 Dynamic Processes in Continuous Time
223(2)
7.5.2 Dynamic Processes in Discrete Time Steps
225(9)
7.6 Questions in Translations from English to Mathematics
234(2)
7.7 Summary of Guidelines for Translating English to the Language of Mathematics
236(2)
7.8 Accuracy, Errors, and Discrepancies in Mathematical Models
238(7)
7.8.1 Errors Translating the Actual Problem into English
239(1)
7.8.2 Errors Translating the English Text into a Mathematical Model
240(1)
7.8.3 Errors Transforming the Mathematical Model into a Mathematical Solution
241(2)
7.8.4 Errors Translating the Mathematical Solution into an English Specification
243(1)
7.8.5 Errors Implementing the English Specification of the Solution
243(2)
8 Examples of Translating English to Mathematics
245(120)
8.1 Students with the Same Birthday
246(2)
8.2 Criterion for Searching an Array
248(8)
8.2.1 Search for Any Occurrence of a Value in an Array
248(4)
8.2.2 Search for the First Occurrence of a Value in an Array
252(4)
8.3 Specifying the Initial State of a Board Game
256(8)
8.3.1 Initialization of a Game Board: A Correct Solution
256(6)
8.3.2 Initialization of a Game Board: A Wrong "Solution"
262(2)
8.4 Price Discounts
264(4)
8.4.1 Flat Discounts
264(1)
8.4.2 Discount Rates Depending on Quantity
265(2)
8.4.3 Buy 2, Get 1 Free
267(1)
8.5 Model of a Very Small Economy
268(5)
8.6 A Logical Puzzle
273(14)
8.6.1 English Statement of the Puzzle
273(1)
8.6.2 Restatement of the Puzzle
273(1)
8.6.3 General Assumptions
274(1)
8.6.4 The Values, Variables, and Functions in the Mathematical Model
275(2)
8.6.5 The Interpretation of Values, Variables, Functions, and Sets
277(1)
8.6.6 The Mathematical Model
278(3)
8.6.7 Solving the Puzzle
281(6)
8.7 Covering a Modified Chess Board with Dominoes
287(2)
8.8 Validity of a Play in a Card Game
289(5)
8.8.1 The Rules of Play
289(1)
8.8.2 Translating the Rules of Play
289(1)
8.8.3 Identifying the Noun Phrases in the English Text
290(1)
8.8.4 Developing the Mathematical Model
291(3)
8.9 The Logical Paradox of the Barber of Seville
294(3)
8.9.1 English Statement of the Paradox
295(1)
8.9.2 Mathematical Model
296(1)
8.10 Controlling the Water Level in a Reservoir: Simple On/Off Control
297(2)
8.10.1 English Statement of the Requirements
298(1)
8.10.2 The Mathematical Variables and Their Interpretation
298(1)
8.10.3 The Mathematical Model
298(1)
8.10.4 Shortcomings of the Simple On/Off Control
299(1)
8.11 Controlling the Water Level in a Reservoir: Two-Level On/Off Control
299(2)
8.11.1 English Statement of the Requirements
299(1)
8.11.2 Interpretation
299(1)
8.11.3 The Mathematical Model
300(1)
8.12 Reliable Combinations of Less Reliable Components
301(8)
8.12.1 A Door Closure Sensor
301(1)
8.12.2 Increased Reliability with Additional Redundant Door Sensors
302(4)
8.12.3 The Complete Mathematical Model for the Redundant Door Sensing Systems
306(3)
8.13 Shopping Mall Door Controller
309(56)
8.13.1 Persons' View of the Door
309(1)
8.13.2 Physical Devices Associated with the Door
309(1)
8.13.3 The Door Controller's Inputs and Outputs
310(1)
8.13.4 Required Responses of the Door Controller
310(1)
8.13.5 Method of Operation of the Controller
310(1)
8.13.6 The Variables
311(2)
8.13.7 Interpretation of the Variables
313(1)
8.13.8 The Mathematical Model
314(1)
8.13.9 The Controller Function
315(1)
8.13.10 Constructing the Controller Function Table
316(41)
8.13.11 The Complete Controller Function Table
357(8)
PART D Conclusion
9 Summary
365(6)
9.1 Transforming English to Mathematics: A Language---Not a Mathematical---Problem
365(1)
9.2 Advantages of the Language of Mathematics for Reasoning and Analyzing
366(1)
9.3 Comparison of Key Characteristics of English and the Language of Mathematics
366(2)
9.4 Translating from English to the Language of Mathematics: Interpretation
368(1)
9.5 Translating from English to the Language of Mathematics: Approach and Strategy
369(2)
Appendix A Representing Numbers 371(5)
Appendix B Symbols in the Language of Mathematics 376(3)
Appendix C Sets of Numbers 379(3)
Appendix D Special Structures in Mathematics 382(3)
Appendix E Mathematical Logic 385(4)
Appendix F Waves and the Wave Equation 389(6)
Appendix G Glossary: English to the Language of Mathematics 395(3)
Appendix H Programming Languages and the Language of Mathematics 398(2)
Appendix I Other Literature 400(7)
Index 407
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.