Algebra & Geometry : An Introduction to University Mathematics book cover
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Algebra & Geometry
An Introduction to University Mathematics





ISBN 9781482246476
Published May 12, 2016 by Chapman and Hall/CRC
384 Pages - 89 B/W Illustrations

 
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Book Description

Algebra & Geometry: An Introduction to University Mathematics provides a bridge between high school and undergraduate mathematics courses on algebra and geometry. The author shows students how mathematics is more than a collection of methods by presenting important ideas and their historical origins throughout the text. He incorporates a hands-on approach to proofs and connects algebra and geometry to various applications.

The text focuses on linear equations, polynomial equations, and quadratic forms. The first several chapters cover foundational topics, including the importance of proofs and properties commonly encountered when studying algebra. The remaining chapters form the mathematical core of the book. These chapters explain the solution of different kinds of algebraic equations, the nature of the solutions, and the interplay between geometry and algebra

Table of Contents

IDEAS
The Nature of Mathematics
MATHEMATICS IN HISTORY
MATHEMATICS TODAY
THE SCOPE OF MATHEMATICS
WHAT THEY (PROBABLY) DIDN’T TELL YOU IN SCHOOL
FURTHER READING

Proofs
MATHEMATICAL TRUTH
FUNDAMENTAL ASSUMPTIONS OF LOGIC
FIVE EASY PROOFS
AXIOMS
UN PETIT PEU DE PHILOSOPHIE
MATHEMATICAL CREATIVITY
PROVING SOMETHING FALSE
TERMINOLOGY
ADVICE ON PROOFS

Foundations
SETS
BOOLEAN OPERATIONS
RELATIONS
FUNCTIONS
EQUIVALENCE RELATIONS
ORDER RELATIONS
QUANTIFIERS
PROOF BY INDUCTION
COUNTING
INFINITE NUMBERS

Algebra Redux
THE RULES OF THE GAME
ALGEBRAIC AXIOMS FOR REAL NUMBERS
SOLVING QUADRATIC EQUATIONS
THE BINOMIAL THEOREM
BOOLEAN ALGEBRAS
CHARACTERIZING REAL NUMBERS

THEORIES
Number Theory

THE REMAINDER THEOREM
GREATEST COMMON DIVISORS
THE FUNDAMENTAL THEOREM OF ARITHMETIC
MODULAR ARITHMETIC
CONTINUED FRACTIONS

Complex Numbers
COMPLEX NUMBER ARITHMETIC
COMPLEX NUMBER GEOMETRY
EULER’S FORMULA
MAKING SENSE OF COMPLEX NUMBERS

Polynomials
TERMINOLOGY
THE REMAINDER THEOREM
ROOTS OF POLYNOMIALS
THE FUNDAMENTAL THEOREM OF ALGEBRA
ARBITRARY ROOTS OF COMPLEX NUMBERS
GREATEST COMMON DIVISORS OF POLYNOMIALS
IRREDUCIBLE POLYNOMIALS
PARTIAL FRACTIONS
RADICAL SOLUTIONS
ALGEBRAIC AND TRANSCENDENTAL NUMBERS
MODULAR ARITHMETIC WITH POLYNOMIALS

Matrices
MATRIX ARITHMETIC
MATRIX ALGEBRA
SOLVING SYSTEMS OF LINEAR EQUATIONS
DETERMINANTS
INVERTIBLE MATRICES
DIAGONALIZATION
BLANKINSHIP’S ALGORITHM

Vectors
VECTORS GEOMETRICALLY
VECTORS ALGEBRAICALLY
THE GEOMETRIC MEANING OF DETERMINANTS
GEOMETRY WITH VECTORS
LINEAR FUNCTIONS
THE ALGEBRAIC MEANING OF DETERMINANTS
QUATERNIONS

The Principal Axes Theorem
ORTHOGONAL MATRICES
ORTHOGONAL DIAGONALIZATION
CONICS AND QUADRICS

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Author(s)

Biography

Mark V. Lawson is a professor in the Department of Mathematics at Heriot-Watt University. Dr. Lawson has published over 60 papers and has given seminars on his research work both at home and abroad. His research interests focus on algebraic semigroup theory and its applications.

Reviews

"This book is for anyone who, having had a taste of higher school mathematics, wants to see what serious math at a university level is truly like. It introduces all the important topics in algebra and algebra as it affects geometry and does so in a totally uncompromised fashion. Professor Lawson shows respect for the subject through the patience exercised in developing the material thoroughly while keeping the reader engaged with a conversational style that never loses clarity. Along the way, there are some real novelties that will surprise even seasoned readers of mathematics."
—Professor Peter M. Higgins, F.I.M.A., University of Essex