Uncommonly interesting introduction illuminates complexities of higher mathematics while offering a thorough understanding of elementary mathematics. Covers development of complex number system and elementary theories of numbers, polynomials and operations, determinants, matrices, constructions and graphical representations. Several exercises — without solutions.

Table of Contents for Fundamental Concepts of Algebra

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 geo
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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