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22 of 22 people found the following review helpful. Legitimate book, great for self study.

By Freeze

This book is ultimately for creating a strong foundation for calculus. "Basic" mathematics can be used lightly, as it's still relatively challenging, but by no means difficult. You'll spend time trying to understand proofs (by the way, he doest an excellent job of incoorporating proofs into the learning process, which is extremely necessary if you want to fully understand math) and figuring out some of the more challenging exercise problems, but the book is by no means painful to go through.

I'll be a junior in high school next year, and was recommended studying this book by a math professor in Belgium (yes, really). The biggest thing to know about math, at least in the earlier levels, is that it is cumulative. You will be absolutely and forever f'ed if you go into Calculus with hardly any knowledge of algebra. That's why, if you find yourself struggling in math, or need to strengthen your foundations, you need to go throuh this book.

It's excellent for self studying, which is what I'm currently doing this summer. When you take this book on at your own pace, and can spend more time on each section as required, and breeze through other sections that you have a firm grasp on, you'll find yourself enjoying the book. Serge Lang was an excellent mathematician (was... only because he is dead) and his books are well written and devoted to helping you understand the material.

Remember though, that just because the book claims to be covering basic mathematics, that you won't be challenged. It can be tough, and especially when you're at the high school level, the mathematics won't seem like they're "basic", but more like, "hey, I just learned that last year/this year/whenever." It's ultimately a wonderful book if you're looking to strengthen the foundation of your mathematic ability, and Lang also includes a brief section after the first chapter about how you should logically examine math, which is invaluable.


By A Customer

Serge Lang's Basic Mathematics is an excellent overview of algebra and geometry. If you are in high school needing a tutorial primer, or an adult continuing their education after some years, this book will provide through its clarity, examples, and exercises (selected answers are in the back of the book)the refresher course you need for more advanced mathematics, such as calculus and linear algebra.

51 of 51 people found the following review helpful.

Preparation for college mathematics from a mathematician's standpoint. By N. F. Taussig

Serge Lang's text presents the topics that he feels students should understand before commencing their study of college mathematics. As such, working through this text is a good way for you to supplement what you learned in high school with material that will aid you in studying mathematics in college. Therefore, I particularly recommend it for prospective mathematics majors.

The material in the text is well motivated and clearly presented. While Lang explains how to perform routine calculations, he focuses on the underlying structure of the mathematics. The material is developed logically and results are proved throughout the text. However, the presentation of the material is marred by numerous errors, most, but not all, of which are typographical.

The problems range from routine calculations to proofs. Many of the problems are challenging and some require considerable ingenuity to solve. Answers to some of the exercises are presented in the back of the text. I should warn you that if you are used to artificial textbook problems in which the correct solution is a "nice" number, you will find that is not the case here. Also, it is useful to read through the problem sets before you begin solving them so that you can do related problems at the same time.

The first section of the book covers algebra. Properties of the integers, rational numbers, and real numbers are examined and compared. There is also more routine material on linear equations, systems of linear equations, powers and roots, inequalities, and quadratic equations.

A brief discussion of logic precedes a section on geometry. Basic assumptions about distance, angles, and right triangles are used as a starting point rather than Euclid's postulates. This leads to a discussion of isometries, including reflections, translations, and rotations. Area is discussed in terms of dilations. The treatment here is different from that in the high school text Geometry which Lang wrote with Gene Murrow. I found the material on isometries quite interesting. Be aware that the notation and some of the terminology in this section is not standard.

The third section of the book covers coordinate geometry. Distance is interpreted in terms of coordinates. This leads to a discussion of circles. Transformations are reinterpreted using coordinates. Segments, rays, and lines are presented using parametric equations. A chapter on trigonometry covers standard topics, but also includes a section on rotations. The section concludes with a chapter on conic sections. Of particular interest is a proof that all Pythagorean triples can be generated from points on the unit circle with rational coordinates.


concludes by demonstrating how determinants can be used to solve systems of linear equations.

The eminent mathematicians I. M. Gelfand and Kunihiko Kodaira have also contributed to books intended for high school students. Those of you planning to study mathematics in college would benefit from working through their texts as well.


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