This text in basic mathematics is ideal for high school or college students. It provides a firm foundation in basic principles of mathematics and thereby acts as a springboard into calculus, linear algebra and other more advanced topics. The information is clearly presented, and the author develops concepts in such a manner to show how one subject matter can relate and evolve into another.
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30 of 30 people found the following review helpful
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. 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.
The final section of miscellaneous topics addresses functions, more generalized mappings, complex numbers, proofs by mathematical induction, summations, geometric series, and determinants. The text 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.
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.
The final section of miscellaneous topics addresses functions, more generalized mappings, complex numbers, proofs by mathematical induction, summations, geometric series, and determinants. The text 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.
24 of 25 people found the following review helpful
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.
Recommended.
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.
Recommended.
17 of 18 people found the following review helpful
Format:Paperback
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.
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21
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- Undergraduate Algebra (Undergraduate Texts in Mathematics) by Serge Lang in Front Matter, and Back Matter
- Fundamentals of Differential Geometry (Graduate Texts in Mathematics) by Serge Lang in Front Matter
- Introduction to Diophantine Approximations by Serge Lang in Front Matter
- Undergraduate Analysis (Undergraduate Texts in Mathematics) by Serge Lang in Front Matter
- Math Talks for Undergraduates by Serge Lang in Front Matter
See all 21 books citing this book
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