Algebra & Geometry

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.

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.