The Method of Coordinates

by I. M. Gelfand, E. G. Glagoleva, A. A. Kirillov

Overview

This introductory text explores the translation of geometric concepts into the language of numbers in order to define the position of a point in space (the orbit of a satellite, for example). The two-part treatment begins with discussions of the coordinates of points on a line, coordinates of points in a plane, and the coordinates of points in space.
Part 2 examines geometry as an aid to calculation and the necessity and peculiarities of four-dimensional space. Written for systematic study, it features a helpful series of "road signs" in the margins, alerting students to passages requiring particular attention, and an abundance of ingenious problems — with solutions, answers, and hints — promote habits of independent work

Product Details

ISBN-13:	9780486425658
Publisher:	Dover Publications
Publication date:	10/20/2011
Series:	Dover Books on Mathematics Series
Pages:	80
Product dimensions:	5.50(w) x 8.50(h) x (d)

Table of Contents

Foreword; Introduction; Part 1.
Chapter 1. The Coordinates of Points on a Line
1. The Number Axis
2. The Absolute Value of a Number
3. The Distance Between Two Points
Chapter 2. The Coordinates of Points in the Plane
4. The Coordinate Plane
5. Relations Connecting Coordinates
6. The Distance Between Two Points
7. Defining Figures
8. We Begin to Solve Problems
9. Other Systems of Coordinates
Chapter 3. The Coordinates of a Point in Space
10. Coordinate Axes and Planes
11. Defining Figures in Space
Part 2.
Chapter 1. Introduction
1. Some General Considerations
2. Geometry as an Aid in Calculation
3. The Need for Introducing Four-Dimensional Space
4. The Peculiarities of Four-Dimensional Space
5. Some Physics
Chapter 2. Four-Dimensional Space
6. Coordinate Axes and Planes
7. Some Problems
Chapter 3. The Four-Dimensional Cube
8. The Definition of the Sphere and the Cube
9. The Definition of the Sphere and the Cube
10. Problems on the Cube

Editorial Reviews

"All through both volumes [‘Functions & Graphs’ and ‘The Methods of Coordinates’], one finds a careful description of the step-by-step thinking process that leads up to the correct definition of a concept or to an argument that clinches in the proof of a theorem. We are ... very fortunate that an account of this caliber has finally made it to printed pages... Anyone who has taken this guided tour will never be intimidated ever again... High school students (or teachers) reading through these two books would learn an enormous amount of good mathematics. More importantly, they would also get a glimpse of how mathematics is done."

—- H. Wu, The Mathematical Intelligencer

"This book is a concise and compact treatment of the essential ideas of coordinate geometry. The authors demonstrate powerfully how geomtric ides may be communicated and studied effectively without the aid of pictures. Graphics are of course of vital importance int he methods of Euclidean geometry. However, the methods of coordinate geometry are able to transform pure geometric ideas into algebraic manipulations where the meaning is very clear once the formalism is learnt. In particular the book demonstrates the value of conveying information in the form of images embedded in formulas. This is very useful in the transmission of information by electronic means. . . This book is a valuable tool for teaching the redimentary concepts of analytical geometry. It contains a number of excellent examples and exercises which go further than a mere introductory programme. the exercises, while not numerous, are very thought-provoking and are bound to pose a serious challenge to the interested student."

—-The Mathematical Gazette

From the Publisher