2nd Edition

# Algebra & Geometry An Introduction to University Mathematics

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** Algebra & Geometry: An Introduction to University Mathematics, Second Edition** 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 solutions 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
- Contains chapter exercises with solutions provided online at www.routledge.com/9780367563035

**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.**

### Biography

**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.

"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 geometrysuccessfully 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 ofMathematics for the Curious