Al Cuoco
The Mathematical Association of America | 2005 | 261 páginas | pdf | 6,3 Mb
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This book is about some of the topics that form the foundations for high school mathematics. It focuses on a closely-knit collection of ideas that are at the intersection of algebra, arithmetic, combinatorics, geometry, and calculus. Most of the ideas are classical: methods for fitting polynomial functions to data, for summing powers of integers, for visualizing the iterates of a function defined on the complex plane, or for obtaining identities among entries in Pascal's triangle. Some of these ideas, previously considered quite advanced, have become tractable because of advances in computational technology. Others are just beautiful classical mathematics, topics that have fallen out of fashion and that deserve to be resurrected While the book will appeal to many audiences, one of the primary audiences is high school teachers, both practicing and prospective. It can be used as a text for undergraduate or professional courses, and the design lends itself to self-study. Of course, good mathematics for teaching is also good for many other uses, so readers of all persuasions can enjoy exploring some of the beautiful ideas presented in the pages of this book.
Contents
1. Difference tables and polynomial fits. Doing it with sums
Doing it with differences
Finding a formula: combinatorial polynomials
Making it formal: the [delta] operator
Going the other way: polynomials to tables
Conversions
From Newton to Lagrange
Agreeing to disagree
2. Form and function: the algebra of polynomials. Polynomials
The basic theorems
Coefficients and values
Up a level
Transformations
Coefficients and zeros.
3. Complex numbers, complex maps, and trigonometry. Complex numbers
The complex plane
The geometry behind multiplying
Trigonometric identities
Complex maps
Julia sets and the Mandelbrot set.
4. Combinations and locks. Combinatorial proofs and identities
The simplex lock
Some approaches to the simplex lock problem
Connections to the Mahler basis.
5. Sums of powers. Summatory polynomials
Bernoulli's method.
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