Synopses & Reviews
The teaching of mathematics has undergone extensive changes in approach, with a shift in emphasis from rote memorization to acquiring an understanding of the logical foundations and methodology of problem solving. This book offers guidance in that direction, exploring arithmetic's underlying concepts and their logical development.
This volume's great merit lies in its wealth of explanatory material, designed to promote an informal and intuitive understanding of the rigorous logical approach to the number system. The first part explains and comments on axioms and definitions, making their subsequent treatment more coherent. The second part presents a detailed, systematic construction of the number systems of rational, real, and complex numbers. It covers whole numbers, hemigroups and groups, integers, ordered fields, the order relation for rationals, exponentiation, and real and complex numbers. Every step is justified by a reference to the appropriate theorem or lemma. Exercises following each chapter in Part II help readers test their progress and provide practice in using the relevant concepts.
Synopsis
This book explores arithmetic's underlying concepts and their logical development. It offers an informal and intuitive understanding of the rigorous logical approach, in addition to a detailed, systematic construction of the number systems of rational, real, and complex numbers. Numerous exercises help students test their progress and practice concepts. 1956 edition.
Synopsis
This book guides readers to the foundations of arithmetic in order to cultivate an appreciation of the concepts' logical development. Its explanatory material offers an informal and intuitive understanding of the rigorous logical approach. Numerous exercises help students test their progress and practice the concepts.1967 edition.
Synopsis
This book explores arithmetic's underlying concepts and their logical development, in addition to a detailed, systematic construction of the number systems of rational, real, and complex numbers. 1956 edition.
Table of Contents
Part I: Explanatory Treatment
1. Counting
2. Whole Numbers
3. The Laws of Arithmetic
4. Fractions
5. Negative Numbers
6. Fields
7. Irrational Numbers
8. Powers
9. Complex Numbers
10. Verification of the Axioms
11. Alternative Treatments
Part II: Systematic Treatment
1. Whole Numbers
2. Hemigroups and Groups
3. Integers
4. Fields
5. Rational Numbers
6. Ordered Fields
7. The Order-Relation for Rationals
8. Exponentiation
9. Cauchy Numbers
10. Real Numbers
11. Complex Numbers
Bibliography
Key to the Exercises
Index