Algebra & geometry: an introduction to university mathematics

435 Pages·2016·4.05 MB·English

by  Lawson

Algebra &
Geometry
An Introduction to
University Mathematics
Algebra &
Geometry
An Introduction to
University Mathematics
Mark V. Lawson
Heriot-Watt University
Edinburgh, UK
CRC Press
Taylor & Francis Group
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© 2016 by Taylor & Francis Group, LLC
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Version Date: 20160603
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In Memoriam
WILLIAM HENRY LAWSON
24th April 1915, Ypres
CHARLES PEIRCE ECCLESHALL
6th June 1915, Gallipoli
Contents
Preface
Prolegomena
S I IDEAS
ECTION
C 1 ▪ The nature of mathematics
HAPTER
1.1 MATHEMATICS IN HISTORY
1.2 MATHEMATICS TODAY
1.3 THE SCOPE OF MATHEMATICS
1.4 WHAT THEY (PROBABLY) DIDN’T TELL YOU IN SCHOOL
1.5 FURTHER READING
C 2 ▪ Proofs
HAPTER
2.1 MATHEMATICAL TRUTH
2.2 FUNDAMENTAL ASSUMPTIONS OF LOGIC
2.3 FIVE EASY PROOFS
2.4 AXIOMS
2.5 UN PETIT PEU DE PHILOSOPHIE
2.6 MATHEMATICAL CREATIVITY
2.7 PROVING SOMETHING FALSE
2.8 TERMINOLOGY
2.9 ADVICE ON PROOFS
C 3 ▪ Foundations
HAPTER
3.1 SETS
3.2 BOOLEAN OPERATIONS
3.3 RELATIONS
3.4 FUNCTIONS
3.5 EQUIVALENCE RELATIONS
3.6 ORDER RELATIONS
3.7 QUANTIFIERS
3.8 PROOF BY INDUCTION
3.9 COUNTING
3.10 INFINITE NUMBERS
C 4 ▪ Algebra redux
HAPTER
4.1 RULES OF THE GAME
4.2 ALGEBRAIC AXIOMS FOR REAL NUMBERS
4.3 SOLVING QUADRATIC EQUATIONS
4.4 BINOMIAL THEOREM
4.5 BOOLEAN ALGEBRAS
4.6 CHARACTERIZING REAL NUMBERS
S II THEORIES
ECTION
C 5 ▪ Number theory
HAPTER
5.1 REMAINDER THEOREM
5.2 GREATEST COMMON DIVISORS
5.3 FUNDAMENTAL THEOREM OF ARITHMETIC
5.4 MODULAR ARITHMETIC
5.5 CONTINUED FRACTIONS
C 6 ▪ Complex numbers
HAPTER
6.1 COMPLEX NUMBER ARITHMETIC
6.2 COMPLEX NUMBER GEOMETRY
6.3 EULER’S FORMULA
6.4 MAKING SENSE OF COMPLEX NUMBERS
C 7 ▪ Polynomials
HAPTER
7.1 TERMINOLOGY
7.2 REMAINDER THEOREM
7.3 ROOTS OF POLYNOMIALS
7.4 FUNDAMENTAL THEOREM OF ALGEBRA
7.5 ARBITRARY ROOTS OF COMPLEX NUMBERS
7.6 GREATEST COMMON DIVISORS OF POLYNOMIALS
7.7 IRREDUCIBLE POLYNOMIALS
7.8 PARTIAL FRACTIONS
7.9 RADICAL SOLUTIONS
7.10 ALGEBRAIC AND TRANSCENDENTAL NUMBERS
7.11 MODULAR ARITHMETIC WITH POLYNOMIALS
C 8 ▪ Matrices
HAPTER
8.1 MATRIX ARITHMETIC
8.2 MATRIX ALGEBRA
8.3 SOLVING SYSTEMS OF LINEAR EQUATIONS
8.4 DETERMINANTS
8.5 INVERTIBLE MATRICES
8.6 DIAGONALIZATION
8.7 BLANKINSHIP’S ALGORITHM
C 9 ▪ Vectors
HAPTER
9.1 VECTORS GEOMETRICALLY
9.2 VECTORS ALGEBRAICALLY
9.3 GEOMETRIC MEANING OF DETERMINANTS
9.4 GEOMETRY WITH VECTORS
9.5 LINEAR FUNCTIONS
9.6 ALGEBRAIC MEANING OF DETERMINANTS
9.7 QUATERNIONS
C 10 ▪ The principal axes theorem
HAPTER
10.1 ORTHOGONAL MATRICES
10.2 ORTHOGONAL DIAGONALIZATION
10.3 CONICS AND QUADRICS
Epilegomena
Bibliography
Index