# Customer Review

on September 11, 2005
This clearly written concise text provides the reader with a good foundation for calculus. The problems in the text are challenging and require sophisticated algebra skills. The text covers real and complex numbers, set theory, logic, algebraic proofs, polynomial and rational expressions, quadratic equations, linear and nonlinear systems of equations, inequalities, coordinate systems, equations of lines and circles, planar regions represented by inequalities, quadratic functions, rational functions, square root functions, trigonometric ratios, and applications of trigonometry.

Concepts are illustrated with examples and demonstration problems. The demonstration problems are followed by problems embedded within the text so that the reader can test her/his understanding of the material. In addition, there are exercises at the end of each section and chapter that require the reader to explore each topic further. The exercises are thought-provoking. Doing them will enhance the reader's algebra skills and provide the reader with a thorough understanding of the material.

The text was written for Japanese students, so there are a few differences in terminology and the prerequisites are not necessarily those covered in American algebra classes. The prerequisite material for this volume from grades 7, 8, and 9 of this series, also edited by Kodaira, is available through the University of Chicago School Mathematics Project.

I do have a few caveats. The authors rarely state explicitly that division by zero is not permitted. In addition, the reader may find it necessary to fill in omitted steps in the demonstration problems. While answers are provided to the chapter exercises, no answers are provided for the other problems. Consequently, the reader will have to check her/his work by making sure that the answer satisfies the given conditions, which is good practice.

That said, I think this is a valuable text. Concepts are covered in depth. For instance, rather than just giving the midpoint formula, a formula is derived for the coordinates of a point c which splits an interval [a, b] in the ratio m:n. The midpoint formula follows by setting m:n = 1:1. The demonstration problems illustrate different techniques for obtaining a solution. The exercises force the reader to decide how best to use the given information to solve the problem. In short, the book provides excellent mathematical training.

A good way to supplement this text would be to work through the texts in the Gelfand School Outreach Program. They include The Method of Coordinates,Algebra,Functions and Graphs (Dover Books on Mathematics), and Trigonometry. This text is the prerequisite for Mathematics 2: Japanese Grade 11 (Mathematical World),Algebra and Geometry: Japanese Grade 11 (Mathematical World, V. 10), and Basic Analysis: Japanese Grade 11 (Mathematical World, V. 11).
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