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The Method of Coordinates

I. M. Gelfand, E. G. Glagoleva, and A. A. Kirillov
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Charles Ashbacher
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Given that we use coordinate geometry so routinely in mathematics, it is easy to lose our perspective on how powerful a concept it is. By converting equations into images and giving us a way to represent every position in a graph, we can go nearly seamlessly from equations to graphical representations. There are also times when a change of coordinate representations can convert a difficult problem into one accurately described as trivial.

This book is a short introduction to the basics of coordinate geometry that can serve as a text for a class or for review. The explanations are clear, easily understandable by anyone having fundamental knowledge of algebra, from the high school student up through the interested non-mathematician. There are many explanatory diagrams and several small sets of exercises, although solutions are not included.

Some books are measured by the degree to which you think beyond the contents to the consequences of the topics. In this case I constantly thought about how useful the comparatively simple topics of coordinate geometry are.

Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.

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