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.
Reprint of the 1967 Dover edition.
Author(s): H. A. Thurston
Series: Dover Books on Mathematics
Edition: 1
Publisher: Dover Publications
Year: 1956
Language: English
Pages: 144
City: Mineola, NY
Part 1: Explanatory Treatment
Chapter I: Counting
Chapter II: Whole numbers
Chapter III: The laws of arithmetic
chapter IV: Fractions
Chapter V: Negative numbers
Chapter VI: Fields
chapter VII: Irrational numbers
chapter VIII: Powers
Chapter IX: Complex numbers
Chapter X: Verification of the axioms
Chapter XI: Alternative treatments
Part 2: Systematic Treatment
Chapter A: Whole numbers
Chapter B: Hemigroups and groups
chapter C: Integers
Chapter D: Fields
Chapter E: Rational Numbers
Chapter F: Ordered fields
Chapter G: The order-relation for rationals
Chapter H: Exponentiation
Chapter I: Cauchy numbers
Chapter J: Real numbers
Chapter K: Complex Numbers
Bibliography
Key to the exercises.
Index
