Classical Algebra : Its Nature, Origins, and Uses
Roger L. Cooke
Bok Engelsk 2008 · Electronic books.
| Annen tittel | |
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| Utgitt | Hoboken : : Wiley, , 2008.
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| Omfang | 1 online resource (220 p.)
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| Opplysninger | Description based upon print version of record.. - Classical Algebra Its Nature, Origins, and Uses; Contents; Preface; Part 1. Numbers and Equations; Lesson 1. What Algebra Is; 1. Numbers in disguise; 1.1.""Classical"" and modern algebra; 2. Arithmetic and algebra; 3. The ""environment"" of algebra: Number systems; 4. Important concepts and principles in this lesson; 5. Problems and questions; 6. Further reading; Lesson 2. Equations and Their Solutions; 1. Polynomial equations, coefficients, and roots; 1.1. Geometric interpretations; 2. The classification of equations; 2.1. Diophantine equations. - 2. Ancient Chinese methods of calculating2.1. A linear problem in three unknowns; 3. Systems of linear equations; 4. Polynomial equations; 4.1. Noninteger solutions; 5. The cubic equation; 6. Problems and questions; 7. Further reading; Part 2. The Formulaic Approach to Equations; Lesson 6. Combinatoric Solutions I: Quadratic Equations; 1. Why not set up tables of solutions?; 2. The quadratic formula; 3. Problems and questions; 4. Further reading; Lesson 7. Combinatoric Solutions II: Cubic Equations; 1. Reduction from four parameters to one; 2. Graphical solutions of cubic equations. - 3. Efforts to find a cubic formula3.1. Cube roots of complex numbers; 4. Alternative forms of the cubic formula; 5. The ""irreducible case""; 5.1. Imaginary numbers; 6. Problems and questions; 7. Further reading; Part 3. Resolvents; Lesson 8. From Combinatorics to Resolvents; 1. Solution of the irreducible case using complex numbers; 2. The quartic equation; 3. Viete's solution of the irreducible case of the cubic; 3.1. Comparison of the Viète and Cardano solutions; 4. The Tschirnhaus solution of the cubic equation; 5. Lagrange's reflections on the cubic equation. - 3. Numerical and formulaic approaches to equations3.1. The numerical approach; 3.2. The formulaic approach; 4. Important concepts and principles in this lesson; 5. Problems and questions; 6. Further reading; Lesson 3. Where Algebra Comes From; 1. An Egyptian problem; 2. A Mesopotamian problem; 3. A Chinese problem; 4. An Arabic problem; 5. A Japanese problem; 6. Problems and questions; 7. Further reading; Lesson 4. Why Algebra Is Important; 1. Example: An ideal pendulum; 2. Problems and questions; 3. Further reading; Lesson 5. Numerical Solution of Equations; 1. A simple but crude method. - 5.1. The cubic formula in terms of the roots5.2. A test case: The quartic; 6. Problems and questions; 7. Further reading; Lesson 9. The Search for Resolvents; 1. Coefficients and roots; 2. A unified approach to equations of all degrees; 2.1. A resolvent for the cubic equation; 3. A resolvent for the general quartic equation; 4. The state of polynomial algebra in 1770; 4.1. Seeking a resolvent for the quintic; 5. Permutations enter algebra; 6. Permutations of the variables in a function; 6.1.Two-valued functions; 7. Problems and questions; 8. Further reading; Part 4. Abstract Algebra. - Lesson 10. Existence and Constructibility of Roots. - This insightful book combines the history, pedagogy, and popularization of algebra to present a unified discussion of the subject. Classical Algebra provides a complete and contemporary perspective on classical polynomial algebra through the exploration of how it was developed and how it exists today. With a focus on prominent areas such as the numerical solutions of equations, the systematic study of equations, and Galois theory, this book facilitates a thorough understanding of algebra and illustrates how the concepts of modern algebra originally developed from classical algebraic precurso
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| Emner | |
| Sjanger | |
| Dewey | |
| ISBN | 9780470259528
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