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