Thurston H.A. The Number System

Dover Publications, 2007 (first published in 1967). — 144 p. — eISBN-13 978-0-486-15494-7.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.Explanatory treatment
Counting
Whole numbers
The laws of arithmetic
Fractions
Negative numbers
Fields
Chapterirrational numbers
Powers
Complex numbers
Verification of the axioms
Alternative treatments
Systematic treatment
Whole numbers
Hemigroups and groups
Rational numbers
Ordered fields
The order-relation for rationals
Exponentiation
Cauchy numbers
Real numbers
Complex numbers