The Method of Coordinates [Paperback]

I.M. Gelfand (Author), E.G. Glagoleva (Author), A.A. Kirilov (Author)

Book Description

ISBN-10: 0817635335 | ISBN-13: 978-0817635336 | Publication Date: January 1, 1990 | Edition: 1

The need for improved mathematics education at the high school and college levels has never been more apparent than in the 1990s. As early as the 1960s, I. M. Gel'fand and his colleagues in the USSR thought hard about this same question and developed a style for presenting basic mathematics in a clear and simple form that engaged the curiosity and intellectual interest of thousands of high school and college students. These same ideas, this same content, unchanged by over thirty years of experience and mathematical development, are available in the present books to any student who is willing to read, to be stimulated and to learn. "The Method of Coordinates" is a way of transferring geometric images into formulas, a method for describing pictures by numbers and letters denoting constants and variables. It is fundamental to the study of calculus and other mathematical topics. Teachers of mathematics will find here a fresh understanding of the subject and a valuable path to the training of students in mathematical concepts and skills.

Editorial Reviews

Review

"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

Language Notes

Text: English (translation)
Original Language: Russian

Product Details

• Paperback: 84 pages
• Publisher: Birkhäuser Boston; 1 edition (January 1, 1990)
• Language: English
• ISBN-10: 0817635335
• ISBN-13: 978-0817635336
• Product Dimensions: 8.5 x 5.9 x 0.3 inches
• Shipping Weight: 4.8 ounces (View shipping rates and policies)
• Average Customer Review: 4.5 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #994,600 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

20 of 20 people found the following review helpful:

5.0 out of 5 stars A fascinating introduction to coordinate geometry., July 15, 2004

By 

N. F. Taussig (Bronx, NY) - See all my reviews
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This review is from: The Method of Coordinates (Dover Books on Mathematics) (Paperback)

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.

6 of 8 people found the following review helpful:

4.0 out of 5 stars An excellent primer on the Cartesian and other coordinate systems, July 12, 2006

By 

Charles Ashbacher (Marion, Iowa United States) - See all my reviews
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This review is from: The Method of Coordinates (Dover Books on Mathematics) (Paperback)

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.