Algebra & Geometry: An Introduction to University Mathematics

by Mark V. Lawson

Overview

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

Product Details

ISBN-13:	9781482246506
Publisher:	CRC Press
Publication date:	11/25/2016
Sold by:	Barnes & Noble
Format:	eBook
Pages:	384
File size:	2 MB

About the Author

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.

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

Editorial Reviews

"Algebra and geometry successfully meets its aims. It has a reassuringly large overlap with familiar ideas from school mathematics but reappraises them in a
readable yet rigorous manner. It introduces readers to the style of abstract reasoning that will be the staple of pure mathematics courses at university. It also includes
plenty of nuggets that can be savoured after a first reading (such as the construction of the real numbers via equivalence classes of Cauchy sequences of rationals, and the proofs of the generalised associativity and Cantor-Schröder-Bernstein theorems). I shall happily recommend this book to prospective undergraduate mathematicians and warmly welcome it to the growing shelf of recent bridging texts."
- The Mathematical Gazette, July 2023 

"This book aims to provide a bridge between school mathematics and university mathematics centred on themes from algebra and geometry. In doing so, it seeks to build on intuitions already developed, making them rigorous through an introduction to formal proofs, as well as pointing the way ahead to new ideas that will be met in the years ahead.
[. . .] Algebra and geometry successfully meets its aims. It has a reassuringly large overlap with familiar ideas from school mathematics but reappraises them in a readable yet rigorous manner. It introduces readers to the style of abstract reasoning that will be the staple of pure mathematics courses at university. It also includes plenty of nuggets that can be savoured after a first reading (such as the construction of the real numbers via equivalence classes of Q-Cauchy sequences of rationals, and the proofs of the generalised associativity and Cantor-Schröder-Bernstein theorems). As such, I shall happily recommend this book to prospective undergraduate mathematicians and warmly welcome it to the growing shelf of recent bridging texts."
- The Mathematical Gazette

"This is an excellent mathematics book on introductory algebra and geometry. It's written for early year university students, but it's not your dull everyday textbook. It's both an easy and an enjoyable read, almost like a book on popular science, but all the while actually teaching you the material. What'struck me most, in addition to the broad perspective on mathematics and clear eyed view of the material presented, was the way it brought out wider vistas to ponder over. These things along with the links made between the topics covered will give students a feeling of real accomplishment and, dare I say it, power. This is really a fine book for students and self-learners alike."
–Samuel L. Braunstein, Professor at University of York

"This book introduces the basic ideas that underpin algebra and geometry at degree level.It rewards the student with a true feel for university mathematics and so, as the student progresses through the subject, it is likely to acquire the status of an old and trusted friend. As its contents become ever more familiar, the owner will value it as a prized possession."
–Peter Higgins, Professor at Essex University and the author of Mathematics for the Curious

From the Publisher