| | AMS Chelsea Publishing
1961; 189 pp; hardcover
ISBN-13: 978-0-8284-0103-6 List Price: US$32
Member Price: US$28.80
Order Code: CHEL/103
This 4th edition includes additional exercises by Paul Meyer.
"The reader who masters the contents of this book will have had a stimulating intellectual experience, a good basis for understanding the mathematical manipulations of algebra, and an acquaintance with modern mathematical thought."
-- Mathematics Teacher
"This book is addressed to beginning students of mathematics ... the level of the book, however, is so unusually high, mathematically as well as pedagogically, that it merits the attention of professional mathematicians (as well as of professional pedagogues) interested in the wider dissemination of their subject among cultured people ... a closer approximation to the right way to teach mathematics to beginners than anything else now in existence."
-- Bulletin of the AMS
Table of Contents
- Sets, Statements, and Variables: Introduction; Sets and their members; Construction of sets; Variables and statement-forms; Functions; Rules of inference and proofs; Beliefs, validity, and sets; Exercises
- Cardinal Numbers: Introduction; Standard sets and cardinal numbers; Addition of cardinal numbers; The commutative and associative laws of addition; Multiplication of cardinal numbers; The commutative and associative laws of multiplication; The distributive law; Comments on the five basic laws; Cancellation laws; Special properties of zero and one; The addition and multiplication tables for cardinals; Exercises
- Expressions: Introduction; Parentheses; Expressions; Evaluation of expressions; Equal expressions; The algebra of expressions; Parentheses and the generalized addition law; Parentheses and the generalized multiplication law; Parentheses and the generalized distributive law; Exercises
- Polynomials: Introduction; Expressions with no parentheses; Exponents; Monomials; Multiplication and addition of monomials; Polynomials; Exercises
- Number Systems in General: Introduction; Operations on a set; Commutative and associative operations; Distributive operations; Numbers and number systems; Zero-element and one-element of a number system; Inverses, negatives, and reciprocals; Some remarks about numbers in general; Exercises
- Construction of the Integers: Introduction; Definition of integers; Relations among the sets \(I (x, y)\); Definition of addition for integers; The commutative and associative laws for addition of integers; Definition of multiplication for integers; The commutative and associative laws for multiplication of integers; The distributive laws for integers; Relation between cardinals and integers; Isomorphisms; Exercises
- Properties of the Integers: Introduction; The generalized laws and expressions; The zero-element and one-element of the integers; The negative of an integer; Standard notation for the integers; The cancellation laws for integers; Expressions over the integers; Subtraction; Exercises
- The Rational Number System: Introduction; The set of rational numbers; Addition and multiplication of rationals; One-element, zero-element and negatives; Relation between rationals and integers; Reciprocals; Traditional notation for rationals; Standard forms; Expressions over the rationals; Exercises
- Equations: Introduction; Definitions; Classification and solution of equations; Equations in one variable; Equations of the first degree; Quadratic equations; Equations with more than one variable; Simultaneous equations; Additional comments; Exercises
- Order: Introduction; Order for the cardinals; Order for the integers; Order in general; Order for the rationals; Exercises
- The Real Number System: Introduction; Finite decimals; The rationals as infinite decimals; Repeating decimals; Construction of the real number system; Order properties of the real number system; The reals as an extension of the rationals; Extraction of roots; Exponents; Logarithms; Exercises
- Appendix: Introduction; Infinite sets and the cancellation laws; Equal and unequal infinities; Peano axioms for natural numbers; Proof by mathematical induction; Axioms, groups, rings, and fields; Exercises