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Basic Mathematics

4.7 out of 5 stars 10 customer reviews
ISBN-13: 978-0387967875
ISBN-10: 0387967877
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Editorial Reviews

From the Back Cover

The present book is intended as a text in basic mathematics. As such, it can have multiple use: for a one-year course in the high schools during the third or fourth year (if possible the third, so that calculus can be taken during the fourth year); for complementary reference in earlier high school grades (elementary algebra and geometry are covered); for a one-semester course at the college level, to review or to get a firm foundation in the basic mathematics to go ahead in calculus, linear algebra, or other topics.

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Product Details

  • Paperback: 496 pages
  • Publisher: Springer (September 1, 1998)
  • Language: English
  • ISBN-10: 0387967877
  • ISBN-13: 978-0387967875
  • Product Dimensions: 6.1 x 1.1 x 9.2 inches
  • Shipping Weight: 1.9 pounds (View shipping rates and policies)
  • Average Customer Review: 4.7 out of 5 stars  See all reviews (10 customer reviews)
  • Amazon Best Sellers Rank: #789,797 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

By N. F. Taussig on October 30, 2005
Format: Paperback
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.
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Format: Paperback
Serge Lang died September 2005, and it was a great loss for many people; he has been a prominent mathematician, who has published many book and articles. He had a very good memory, and it is said that he wrote a book in the course of one weekend on a bet. I don’t know if that’s true, but you can sense that he feels at home writing about mathematics.

Basic Mathematics is suited both for the younger readers who hasn’t begun high school yet, and for older readers who needs to refresh their skills. I believe that many people would benefit working through this book before starting in high school, as it will ease and speed up things. The book is structured in a way that it clearly brings the most important of the mathematics which later is to be used.

The book has four parts: Algebra, Intuitive Geometry, Coordinate Geometry, and Miscellanous. There are 17 chapter spread over these four parts, which each deals with an important mathematical subject. Of mention are “Functions”, “Operations on Points”, “Distance and Angles”, and “Linear Equations”. It’s mainly basic mathematical subjects, which are dealt with in an “advanced way”, so the author doesn’t look down on his reader. Nothing is dwelt upon, but nothing important isn’t absent either.

Exercises is included in nearly all sections, so that the reader can train himself in both a manipulative and a theoretical level. Som sections has many exercises (which can be tough at times), while some has only three or four. The difficulty is raised, of course, but if you just do the exercises, you’ll notice how well the book is structured in that basic techniques are used later in more advanced subjects.

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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.
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