图书简介
The Shape of Algebra is the authors’ attempt to share their mathematical experiences with readers who have more than a passing interest in mathematics, but have only a traditional exposure to elementary algebra. Secondary school and college teachers and students who want to expand their horizons in the field will find a fresh presentation of familiar concepts and some unexpected results.
This book serves as a text for an “appreciation” course in modern mathematics designed for non-mathematics majors or for first-year students who are considering the possibility of studying mathematics or related disciplines. It can also serve as a source of computer-supported activities that could supplement traditional courses in algebra, multivariable calculus, and complex variable.
This book gives the reader a sense of the visual nature of mathematics. Mathematical experiments with universal mapping software VisuMatica, designed by Vladimir Nodel’man, form the very core of the book. Readers are encouraged to reproduce, play with, and expand on these experiments. Numerous problems are interspersed throughout the text to guide the reader.
Our treatment of standard algebra is visual and computational. By introducing visual computational environments like VisuMatica, our book promotes this geometric approach to algebra and makes it accessible to readers a great deal earlier.
The book will enable our readers to approach its content on three levels: the first one which requires only some fluency with elementary algebraic manipulations; the second one which also presumes familiarity with the notions of derivatives and tangent lines to plane curves, and the third one which uses some basic concepts of multivariable calculus. All three levels are clearly marked in the text, and will allow for a smooth reading and virtual experiments, regardless of the level that our readers will find comfortable.
Key Features
• Promotes the visualization of elementary algebra via an original interactive software environment, thus giving the reader a taste of mathematical research
• Introduces the reader to a modern view on mathematics at large, while keeping the requirement on the readership quite modest
• Provides a uniquely fresh geometrical view on a very traditional and rather elementary body of algebraic knowledge
• Mathematics educators are the group most likely to benefit from this book
Introduction — Maps upon Maps; Maps, Models, and the Coordinate Plane; The Universe of Quadratic Polynomials; The Complex Numbers; The Geometry of Complex Linear Polynomial Maps; Complex Polynomial Maps, Closed Plane Curves and their Invariants; Experimenting with the Geometry of Complex Polynomial Maps of Degrees Two and Three; The Old Art of Solving Cubic Equations and the New Science; Complex Polynomial Maps of Higher Degrees and the Fundamental Theorem of Algebra; Local and Global Geometry of Complex Polynomial Maps; The Fundamental Theorem and the Complex Viète Map; Appendices: Estimating Polynomial Expressions; More about the Taylor Formula for Polynomials; The Geometry of Discriminants; Line Integrals and an Analytical Proof of the Loop Theorem; Deformations of Complex Polynomials and the Braids.
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