Fundamentals of Elementary Mathematics (eBook)

Merlyn J. Behr, Dale G. Jungst (Autoren)

eBook Download: PDF

2014 | 1. Auflage
440 Seiten
Elsevier Science (Verlag)
978-1-4832-7779-0 (ISBN)

Infos

Fundamentals of Elementary Mathematics provides an understanding of the fundamental aspects of elementary mathematics. This book presents the relevance of the mathematical concepts, which are also demonstrated in numerous exercises. Organized into 10 chapters, this book begins with an overview of the study of logic to understand the nature of mathematics. This text then discusses mathematics as a system of structure or as a collection of substructures. Other chapters consider the four essential components in a mathematical or logical system or structure, namely, undefined terms, defined terms, postulates, and theorems. This book discusses as well several principles used in numeration systems and provides examples of some numeration systems that are in use to illustrate these principles. The final chapter deals with the classification of certain mathematical systems as groups, fields, or rings to demonstrate some abstract mathematics. This book is a valuable resource for students and teachers in elementary mathematics.

Inhalt

Front Cover 1
Fundamentals of Elementary Mathematics 4
Copyright Page 5
Table of Contents 8
Dedication 6
PREFACE 12
ACKNOWLEDGMENTS 16
GLOSSARY OF SYMBOLS 18
CHAPTER 1. AN INTRODUCTION TO LOGIC AND MATHEMATICAL REASONING 24
1.1 Basic ideas 24
1.2 Forming new statements from given statements 26
1.3 Negation, disjunction, and conjunction 28
1.4 Conditional statements 33
1.5 The biconditional, logical equivalence, tautologies, and contradictions 34
1.6 Statements related to a conditional 36
1.7 Some important properties 38
1.8 Open sentences and quantifiers 40
1.9 The negation of universally and existentially quantified statements 46
1.10 Some rules of inference-the tools of proof 48
1.11 Summary of Chapter 1 51
CHAPTER 2. SETS, RELATIONS, FUNCTIONS AND OPERATIONS 55
2.1 The development of mathematics 55
2.2 More on the set concept and set notation 57
2.3 Relations on sets 59
2.4 Venn diagrams 61
2.5 One-to-one correspondence 61
2.6 Operations on sets 66
2.7 Element tables 70
2.8 Algebra of sets 72
2.9 Relations 78
2.10 Reflexive, symmetric, transitive, and equivalence relations 82
2.11 Functions 86
2.12 Operations 89
2.13 Mathematical systems and isomorphism 96
2.14 Summary of Chapter 2 100
CHAPTER 3. THE SYSTEM OF WHOLE NUMBERS 103
3.1 Introduction 103
3.2 Construction of the nonempty set… the set of whole numbers 104
3.3 Equality and addition in W 109
3.4 Structural properties of+ 112
3.5 Multiplication in Wand structural properties of X 114
3.6 Order in the whole numbers 119
3.7 Subtraction and division in W 122
3.8 Some comments on multiplication and division in W 130
3.9 Counting, cardinal and ordinal use of whole numbers 131
3.10 Elementary number theory 132
3.11 The whole-number line, open sentences, graphs 147
3.12 The whole-number plane, relations on W, open sentences in two variables 155
3.13 Summary of Chapter 3 161
CHAPTER 4. NUMERATION SYSTEMS 165
4.1 Introduction 165
4.2 The additive and positional principles 165
4.3 Examples of numeration systems 166
4.4 Positional systems and the Hindu-Arabic system 170
4.5 Positional numeration systems with bases other than ten 173
4.6 The Babylonian numeration system… a sexagesimal system 177
4.7 Arithmetic in other bases 178
4.8 Positional numerals for fractional numbers 191
4.9 Summary of Chapter 4 193
CHAPTER 5. ALGORITHMS FOR COMPUTATION WITH WHOLE NUMBERS 195
5.1 Introduction 195
5.2 Addition algorithms 195
5.3 Subtraction algorithms 198
5.4 Multiplication algorithms 201
5.5 Division algorithms 206
5.6 Summary of Chapter 5 208
CHAPTER 6. THE SYSTEM OF FRACTIONAL NUMBERS 210
6.1 Introduction 210
6.2 Construction of the fractional-number system 212
6.3 Addition and multiplication in F 215
6.4 Structural properties of + and . 219
6.5 Fractions and standard names for fractional numbers 223
6.6 An isomorphism between the system of whole numbers and a subsystem of the fractional numbers 229
6.7 Mixed numerals 231
6.8 Order, subtraction, and division in F 234
6.9 Density in F, a one-to-one correspondence between W and F 242
6.10 Interpretations of fractional numbers 245
6.11 The fractional-number line and plane, open fractional-number sentences, graphs of subsets of F and F x F 248
6.12 Summary of Chapter 6 252
CHAPTER 7. THE SYSTEM OF INTEGERS 257
7.1 Introduction 257
7.2 Construction of the system of integers 259
7.3 Addition and multiplication in I 264
7.4 Positive and negative integers, standard names 265
7.5 Structural properties of + and x 270
7.6 An isomorphism between W and the nonnegative integers 273
7.7 Subtraction, division, order, absolute value in I 275
7.8 Elementary number theory 281
7.9 A brief section on open integer sentences 288
7.10 Summary of Chapter 7 292
CHAPTER 8. THE SYSTEM OF RATIONAL NUMBERS 296
8.1 Introduction 296
8.2 Construction of the rational-number system 297
8.3 Addition and multiplication in R 300
8.4 Structural properties of . and . 302
8.5 Positive and negative rational numbers, rational numerals…the standard names for rational numbers 305
8.6 An isomorphism between the system of integers and a subsystem of the rationals 310
8.7 Subtraction, division, order in R 312
8.8 An interpretation of rational numbers 320
8.9 Open number sentences, graphs 321
CHAPTER 9. DECIMAL NUMERALS, INTRODUCTION: REAL NUMBERS 331
9.1 Introduction 331
9.2 Decimal numerals for fractional numbers and computational algorithms using decimal numerals 332
9.3 A one-to-one correspondence between fractional numbers and repeating decimals 336
9.4 Repeating decimals which correspond to rational numbers 341
9.5 Infinite decimals and the real numbers 343
9.6 A brief overview of the number systems 349
9.7 Mathematical sentences 351
9.8 Summary of Chapter 9 371
CHAPTER 10. ABSTRACT SYSTEMS 375
10.1 Introduction 375
10.2 Groups 375
10.3 Functions and groups 378
10.4 Definition of ring and field 384
10.5 Congruence modulo b…finite rings and fields 386
10.6 Open linear congruences 395
10.7 Summary of Chapter 10 398
Answers and Suggestions for Selected Exercises 402
INDEX 436

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
• Themenwelt: Technik
• ISBN-10: 1-4832-7779-8 / 1483277798
• ISBN-13: 978-1-4832-7779-0 / 9781483277790