Customer Reviews

The Method of Coordinates

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The authors of this slim volume demonstrate the power of coordinate geometry, which they describe as a means of translating geometric figures into algebraic formulas, through their lucid exposition, interesting examples, and well-chosen exercises.

The authors begin with the coordinate geometry of the real line. They discuss absolute value and define what distance means. Next the authors examine the coordinate geometry of the plane. They define distance in the plane, show how relations among the coordinates define geometric figures, and discuss different coordinate systems that can be used in the plane. Their examples illustrate how algebraic methods developed by Rene Descartes make it possible to solve geometric problems efficiently that would be quite difficult to solve using synthetic geometry. The authors then treat the coordinate geometry of three-dimensional space in a similar manner.

The second part of the book begins with a problem concerning lattice points in the plane. The authors use this example and its generalizations to justify exploring the coordinate geometry of four-dimensional space. They carefully treat the example of a four-dimensional unit hypercube, examining its properties by considering its analogues in lower dimensions: the segment [0, 1] of the real number line, the unit square in the coordinate plane, and the unit cube in space.

Since the book was initially written for a correspondence course for high school students in the Soviet Union, it is designed for self-study and accessible to students who have had high school courses in algebra and geometry. Since students in the Soviet Union were able to mail their solutions to the exercises to the authors when the authors were professors at the University of Moscow, answers to most of the exercises are not provided. The exercises are thought-provoking and some are quite challenging.

I also highly recommend that you explore the other volumes in the Gelfand School Outreach Program. They include Algebra, Functions and Graphs (Dover Books on Mathematics), and Trigonometry.

To succeed in mathematics, it is necessary to understand the different coordinate systems. The Cartesian coordinate system, where algebra and geometry are combined into a single synergistic operation, is one of the greatest of all mathematical achievements. Most students at the lower levels are exposed to the Cartesian system, but unfortunately not to the other coordinate systems.
The authors develop a full explanation of the basic Cartesian system by starting with the linear coordinate system. They then expand it out to two, three and four dimensions. The transition is easy and understandable. They also briefly cover the various forms of polar coordinates in two and three dimensions. With thorough and complete explanations of the basic coordinate systems, this book is an excellent primer on this fundamental concept of mathematics. All students should be exposed to coordinate systems other than the Cartesian, humans use the three-dimensional coordinate system more than we use the two dimensional coordinate system.