Fundamental Concepts of Algebra

Overview

This uncommonly interesting volume has been a favorite for 30 years, and is now available for the first time in a modestly priced paperback edition. Beginning with the fundamentals of numbers and sets, the author clearly guides the reader through the often discouraging complexities of algebra. Meserve presents the fundamental concepts of algebra illustrated by numerous examples- and, in many cases, the theory is further explicated by suitable sequences of exercises.
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Overview

This uncommonly interesting volume has been a favorite for 30 years, and is now available for the first time in a modestly priced paperback edition. Beginning with the fundamentals of numbers and sets, the author clearly guides the reader through the often discouraging complexities of algebra. Meserve presents the fundamental concepts of algebra illustrated by numerous examples- and, in many cases, the theory is further explicated by suitable sequences of exercises.
The author's aim is to introduce the concepts of higher mathematics while bringing the reader to a more thorough understanding of elementary mathematics. His lucid explanations cover these major areas:
OUR NUMBER SYSTEM: Sets, Cardinal Numbers, Order Relations, Real Numbers, Fields and Number Systems, more.
THEORY OF NUMBERS: Divisibility, Prime Numbers, Bases, Division Algorithm, Decimal Notations, Linear Congruences, more.
THEORY OF POLYNOMIALS: Rational Functions, Irreducible Polynomials, Functions, Limits, Continuity, Derivatives, more.
THEORY OF EQUATIONS: Zeros of a Polynomial, Change of a Variable, Number of Roots, Quartic Equations, more.
DETERMINANTS AND MATRICES: Historical Development, Permutations, Inversions, Transpositions, Cramer's Rule, more.
CONSTRUCTIONS: The Algebraic Viewpoint, Classical Constructions, Mechanical Angle Trisectors, Linkages, more.
GRAPHICAL REPRESENTATIONS: Euclidean and Complex Spaces, Conic Sections, Quadric Surfaces, Curve Tracing, more.
While intended for students who have previously studied college mathematics through calculus, this book has also been successfully used where calculus was not a prerequisite. It provides a very easily followed presentation and may also be used as an introductory or supplementary textbook. For prospective teachers of secondary mathematics, for students preparing for specialized advanced undergraduate courses in mathematics, and for the general reader in search of a firmer grasp of the essentials of algebra, there is no better, more helpful guide than Meserve.

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Product Details

  • ISBN-13: 9780486614700
  • Publisher: Dover Publications
  • Publication date: 10/1/1982
  • Series: Dover Books on Mathematics Series
  • Pages: 320
  • Product dimensions: 5.34 (w) x 8.30 (h) x 0.64 (d)

Table of Contents

CHAPTER 1. OUR NUMBER SYSTEM
1-1 Sets
1-2 Cardinal numbers
1-3 Equivalence relations
1-4 Peano's postulates
1-5 Addition and multiplication
1-6 Order relations
1-7 Inverses
1-8 Positive raional numbers
1-9 Negative numbers
1-10 Real numbers
*1-11 Postulates for real numbers
1-12 Properties of real numbers
1-13 Transfinite cardinal numbers
1-14 Group; number system
1-15 Complex numbers
1-16 Properties of complex numbers
1-17 De Moivre's Theorem
*1-18 Fields and number systems
CHAPTER 2. THEORY OF NUMBERS
2-1 Divisibility
2-2 Division Algorithm
2-3 Prime numbers
2-4 Unique Factorization Theorem
2-5 Euclidean Algorithm
2-6 Bases
2-7 Decimal notation
*2-8 Congruences
*2-9 Residue classes. Euler's F-function
*2-10 Evaluation of F(m)
*2-11 Linear congruences
*2-12 Diophantine problems
CHAPTER 3. THEORY OF POLYNOMIALS
3-1 Polynomials
3-2 Rings of polynomials
3-3 Rational functions
3-4 Divisibility
3-5 Division Algorithm
3-6 Irreducible polynomials
3-7 Euclidean Algorithm
3-8 Change of variable
*3-9 Ideals
3-10 Functions
3-11 Limits
3-12 Continuity
3-13 Continuous functions
3-14 Derivatives
*3-15 Taylor's series
*3-16 Analytic functions
CHAPTER 4. THEORY OF EQUATIONS
4-1 Zeros of a polynomial
4-2 Synthetic division
4-3 Change of variable
4-4 Number of roots
4-5 Determination of the roots
4-6 Conjugate imaginary roots
4-7 Elementary symmetric polynomials
4-8 Transformations of roots
*4-9 Cubic equations
*4-10 Quartic equations
4-11 Descartes' Rule of Signs
4-12 Sturm's Theorem
4-13 Multiple roots
4-14 Approximate solutions
CHAPTER 5. DETERMINANTS AND MATRICES
5-1 Historical development
5-2 Matrices
5-3 Permutations
5-4 Inversions
5-5 Transpositions
5-6 Even and odd permutations
5-7 Determinants
5-8 Properties of determinants
5-9 Expansion of determinants
5-10 Minors
5-11 Cramer's Rule
5-12 Systems of linear equations
5-13 Linear dependence
5-14 Applications in analytic geometry
5-15 Geometric transformations
CHAPTER 6.CONSTRUCTIONS
6-1 Classical constructions
6-2 Elementary classical constructions
6-3 The algebraic viewpoint
6-4 Basic classical constructions
6-5 Construction of roots of equations
6-6 Famous construction problems
6-7 Nonclassical geometric trisections
6-8 Mechanical angle trisectors
6-9 Linkages
6-10 Summary
CHAPTER 7. GRAPHICAL REPRESENTATIONS
7-1 Euclidean and complex spaces
7-2 Polynomials
7-3 Conic sections
7-4 Quadric surfaces
7-5 Higher plane surves
7-6 Rational functions
7-7 Algebraic functions
7-8 Curve tracing
7-9 Special graphs
7-10 Graphical solutions
7-11 Curve fitting
7-12 Conclusion
BIBLIOGRAPHY
SYMBOLS AND NOTATIONS
INDEX
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