Read more
This book provides a bridge between high school and undergraduate mathematics courses on algebra and geometry.The text focuses on linear equations, polynomial equations, and quadratic forms. The first few chapters cover foundational topics. The remaining chapters form the mathematical core of the book.
List of contents
Prolegomena. Section I. Ideas. 1. The Nature of Mathematics. 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. 2. Proofs. 2.1. Mathematical Truth. 2.2. Fundamental Assumption 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. 3. Foundations. 3.1. Sets. 3.2 Boolean Proofs. 3.3. Relations. 3.4. Functions. 3.5. Equivalence Relations. 3.6. Order Relations. 3.7. Quantifiers. 3.8. Proofs by Inductions. 3.9. Counting. 3.10. Infinite Numbers. 4. Algebra Redux. 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. Characterizing Real Numbers. Section II. Theories. 5. Number Theory. 5.1. Remainder Theorem. 5.2. Greatest Common Divisors. 5.3. Fundamental Theorem of Arithmetic. 5.5. Continued Fractions. 6. Complex Numbers. 6.1. Complex Number Arithmetic. 6.2. Complex Number Geometry. 6.3 Euler’s Formula for Complex Numbers. 7. Polynomials. 7.1. Terminology. 7.2. The Remainder Theorem. 7.3. Roots of Polynomials. 7.4. Fundamental Theorem of Algebra. 7.5. Arbitrary Roots of Complex Number. 7.6. Greatest Common Divisors of Complex Numbers. 7.7. Irreducible Polynomials. 7.8 Partial Fractions. 7.9. Radical Solutions. 7.10. Algebraic and Transcendental Numbers. 7.11. Modular Arithmetic with Polynomials. 8. Matrices. 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. 9. Vectors. 9.1. Vectors Geometrically. 9.2. Vectors Algebraically. 9.3. Geometric Meaning of Determinants. 9.4. Geometry with Vectors. 9.5. Linear Functions. Algebraic Meaning of Determinants. 9.7. Quaternions. 10. The Principal Axis Theorem. 10.1. Orthogonal Matrices. 10.2. Orthogonal Diagonalization. 10.3. Conics and Quadrics. 11. What are the Real Numbers? 11.1 The Properties of the Real Numbers. 11.2. Approximating Real Numbers by Rational Numbers. 11.3. A Construction of the Real Numbers. Epilegomena. Bibliography. Index.
About the author
Mark V. Lawson is a professor in the Department of Mathematics at Heriot-Watt University. Prof. Lawson has published more than 70 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. In 2017, he was awarded the Mahoney-Neumann-Room prize by the Australian Mathematical Society for one of his papers.
Summary
This book provides a bridge between high school and undergraduate mathematics courses on algebra and geometry.The text focuses on linear equations, polynomial equations, and quadratic forms. The first few chapters cover foundational topics. The remaining chapters form the mathematical core of the book.
Additional text
"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
