4 comments about Basic Mathematics.

1. 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.



2. Lang's Basic Mathematics is a famous mathematician's look at everything a well-prepared high school student ought to know about math before starting calculus. The exercises are thought-provoking and the solutions are enlightening. There is just enough but not too much drill on each point before moving on, and throughout there is a wonderfully mathematical attitude about the material. Recommended for anyone who has had algebra once and wants to know a lot more about what mathematics is really all about.



3. 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.



4. 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.

No comments about Commutative Algebra: Chapters 1-7.

No comments about A Course in Convexity (Graduate Studies in Mathematics, V. 54).

1 comments about Fourier Series (Mathematical Association of America Textbooks).

1. Fourier series is an area of mathematics that more students should be exposed to in depth. It began in the areas of the physics of heat transfer in solids and the movements of a vibrating string and is now used in an enormous number of applications.
   This book is designed to be an in-depth introduction to the subject for junior and senior level math or physics majors. It begins with the historical context that led Fourier to develop the series named after him as well as some of the subsequent work of additional mathematicians.
   The treatment is mathematically rigorous; Fourier series are introduced in the context of applications and then the explanations move on to a higher level of abstraction. It concludes with some applications of Fourier series in the areas of number theory, the isoperimetric problem and band matrices.
   If you are contemplating a course in Fourier series for upper level undergraduates, this book is the best available for such a course. It can also be used for self-study in the field.

2 comments about Geometry: The University of Chicago School Mathematics Project.

1. I used this book to homeschool my daughter during the year I was living in Spain, since the math course available to her in the local school duplicated what she had learned the year before in the United States. The book was everything I could have asked for, and more. My daughter worked every problem in the book. The text explanations were clear. The selection of topics was brilliant. I believe that my daughter learned both geometry and the yoga of proof, both its purpose and its delights. I have no complaints whatsoever about this text. It prepared her well for her subsequent studies in math.



2. UCSMP geometry is pretty darned good for a "contemporary style" geometry book.

   This is a decent self study book where students read and repeat. There are good tips on ideas for investigation in the "project"section at the end of each chapter.

   Chapter 1 is horrible. "Point" is defined 5 ways for 5 unique geometries until the student is left wondering literally "what is the point?" Chapter 1 requires a lot of explaining. Not a bad place to break out Euclid's Elements, to see what the "old school" was all about, notice similarity and difference.

   Ch 4- The derevation of congruence by reflection theorems in nothing short of horrible handwaving. Just suffer through it, but skipping it is rough because the book goes back to it too much in later chapters.

   Two column proofs are done in ch. 5, 6, 7, 11 and 13. Ch. 13 is circle proofs-- so nowhere are proofs longer than 6 steps ever even attempted. Maybe that's ok. Maybe skip Ch 13. I do. Why learn little kiddy proofs twice?

   OK, my big beef is with chapters 8,9, and 10. Perimeter, area, and volume. On one hand, this is a rehash of middle school basics. On the other hand, there is a bucketload of useless formulas to memorize (if you teach it that way) and on the third hand (!) memorizing the formulas destroys any chance for critical problem solving skills to develop.

   Example: surface area of a cone- here's the formula. Ask a student to MAKE a cone exactly 5 inches tall and 4 inches accross-- they are paralyzed. No idea what slant height is. Even with a square pyramid. So, hey--- make some models.

   Overall, it's a pretty good basic "contemporary" (TRADITIONAL is Euclid's Elements, still outstanding after 2300 years if you have a geometry teacher around!) book.

   If you are home schooling, you need a copy of Mike Serra's Geometry (any edition) for the great projects that helps kids figure stuff out in addition to this book.

   Hey, what do I know. I'm a constructivist.

No comments about Operator Algebras: Theory of C*-Algebras and von Neumann Algebras (Encyclopaedia of Mathematical Sciences).

No comments about Fundamentals of Algebra and Trigonometry.

No comments about A Singular Introduction to Commutative Algebra.

3 comments about Differential Geometry: Cartan's Generalization of Klein's Erlangen Program (Graduate Texts in Mathematics).

1. Sharpe's book is a detailed argument supporting the assertion that most of differential geometry can be considered the study of principal bundles and connections on them, disguised as an introductory differential geometry textbook.

   Some standard introductory material (e.g. Stokes' theorem) is omitted, as Sharpe confesses in his preface, but otherwise this is a truly wonderful place to read about the central role of Lie groups, principal bundles, and connections in differential geometry. The theme is that what one can do for Lie groups, one can do fiberwise for principal bundles, to yield information about the base.

   The informal style (just look at the table of contents) and wealth of classical examples make this book a pleasure to read. While its somewhat nonstandard approach and preference for classical terminology might confuse those who have never been introduced to the concepts, this is a perfect *second* place to read and marvel about differential geometry.




2. This is definitely a graduate school text. Though I believe the text can be read by a eager undergraduate. The text is about Differential Geometry.
   The subject matter demands that the reader read more than 1 book on the subject. This is a good introduction to a difficult but useful mathematical discipline.



3. I was fortunate enough to have Sharpe as my supervisor at University of Toronto just when his book was published. His highly abstract thinking is very impressive and I have enjoyed immensely his first chapter on differential topology, which is my specialized area. Though his book branches off into realms that don't particularly suit me, the beginnings of his book had given me great inspiration in my discipline in differential topology

5 comments about Elliptic Curves: Function Theory, Geometry, Arithmetic.

1. This book avoids the traps which would make this subject so inaccessible. Rather than frightening the reader with group theory and the sort of very advanced material that would fit it into a post graduate slot, the book starts with very little beyond geometry and complex number theory. The book carefully progresses to discussions on the projective line, and Riemann surfaces (never too much at once) to the inevitable subjects of the Icosohedral group, and invariant theory. It manages to do this almost without you noticing the depth of maths that is being covered - quite a feat!

   From here on, elliptic integrals are discussed, and the work of Jacobi, Gauss, Legendre and Abel discussed freely, with many examples and clear pictures. The text is interspersed with exercises (some of which you can do with a few moments thought, others more difficult). I enjoyed this section (and the remainder of the book) for several very interesting short accounts of subjects slightly tangential to the main material.

   [One of my favorites was the account of a letter with a amazingly strange but elegant identity with a continued fraction sent by Ramanujan to Hardy, and Hardy's subsequent absolute amazement... You MUST NOT miss reading that, even if it isn't what you picked the book up for!]

   Then the book goes into the area I bought the book for - modular groups, and the solution of the Quintic. This subject draws mostly on work by Hermite, and later, Klein, but is presented carefully and slowly.

   I was very glad to find this book. It doesn't race through the subject at breakneck speed, which is what some books on Galois Theory or Algebraic Curves do, and has illuminated quite a few additional topics for me. I guess that now I will be able to recognize the origins of so much hard maths now (and all those entries in the tables of integrals I never understood)

   After all, this subject is now very important. Elliptic curves occur in many subjects - Cryptography, Information Theory, and of course, the proof of Fermats last theorem.




2. This is a great book because it presents some of the neatest topics in mathematics, without the usual discouraging layers of abstraction and notation. It attacks the topics historically so you get some idea of the motivation and steps followed, instead of a compendium of the most general results and their most elegant proofs.

   Also, as a previous reviewer mentioned, the book derives the bizarre and amazing continued fraction formula from Ramanujan's letter to Hardy. I had always wanted to see this, ever since reading "The Man Who Knew Infinity." It is satisfying to see this demystified, even if you don't fully master the argument.

   If you literally have not seen most of these topics before, as I had not, you won't find this an easy read, but it's well worth while. I spent a long time on it, and couldn't absorb it all, but I plan to read it again one day.




3. I got this book as a gift from a long time friend. He had trouble with reading it. It is only for that reason I give it only 4 stars. These authors make others that I have read on this range of subjects look bad: Fields Medalists included! A lot of it is that they just bother to give you the real mathematics with examples. I think the initial miss definition of the Riemann surface gives a false impression, because the explanations of ramified covers and toral elliptic lattices is just wonderful. Reading this book makes Dr. Singerman's papers look so much better! I was disappointed in the treatment of triangle groups, but the treatment of modular functions and gamma1 and gamma2 makes up for that. It is a masterful work... the best I have seen by a modern author. It reminds me of books by Ulam or Russell. Sawyer's little book is not as good!



4. The popular press leaves us with the impression that math is
   intimidating. This wasn't always the case. In my time, the approach to how we teach math, and write books about it, went through a number of cycles, or trends; some of them now discredited;--or not!? Here is a sample: (1) I grew up with the boot-camp approach with its endless drills, (2) then came "The New-Math approach", followed by (3) "The back-to-basics" trend. (4)Following Eric Temple Bell, it became popular for a time to mix into the teaching of math a lot of history/ or dramatic stories about the heros in the subject. Finally, more recently:(5) "The Make-it-Seem-Easy-and Fun approach" and the motivational speakers; imitating popular TV shows.---Seriously, what I like about this lovely book is that it treats mathmatics as one unified subject, and that the authors masterfully highlight a number of unexpected connections between what otherwise are thought of as isolated specialties within math: The exciting new problems are at the same time also the old and classic problems in math: The elliptic integrals of Abel and Gauss, Jacobi's theta functions, modular functions, quadratic fields, elliptic curves, and Mordell-Weil. It is all beautifully presented. The book is selfcontained, and it is a pleasure to read. The clear and concise presentation is what makes the subject seem easy, or more importantly interesting and useful. I hope it will be a model for other math books to follow.



5. I originally bought the book for background on elliptic curves in cryptography. While this book may not be the ideal source for practical cryptography it is nevertheless a beautiful and fascinating example of how mathematics should be presented to the general reader.
   
   Please note that although the book is described as an "introduction" it presents the mature works of some of the world's greatest mathematicians. The many beautiful theorems, expressions and identities which appear on almost every page (look at Chapter 3 and weep), can only be fully appreciated if the reader has a thorough mathematical grounding.