Algebra & Geometry
An Introduction to University Mathematics
- Available for pre-order. Item will ship after June 23, 2021
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 few chapters cover foundational topics, including the importance of proofs and a discussion of the 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.
New to the second edition
- Several updated chapters, plus an all new chapter discussing the construction of the real numbers by means of approximations by rational numbers
- Includes fifteen short ‘essays’ that are accessible to undergraduate readers, but which direct interested students to more advanced developments of the material
- Expanded references.
Table 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.
Mark V. Lawson is a professor in the Department of Mathematics at Heriot-Watt University. Prof. Lawson has published over 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.