Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry

I am a guy who learned to love mathematics in college, but believe it or not, I was kind of a clown in high school. It is in this way that my pre-calculus background sucked. I got really good grades in my calculus and differential equations classes on college, but the sad thing is that I had to learn pre-calculus in a brute force approach: when trying to understand some topic, if that topic required an important pre-calculus concept (e.g., something as simple as factorization during some techniques for integration) I had to learn that concept on-the-fly. You can imagine then how horribly dispersed were my ideas of pre-calculus. In an effort to correct this I bought Precalculus in a Nutshell, and the results were spectacular. In just a week I was able to finish the book and work on 95% of the problems (there are many!). Simmons goes a long way in removing any useless additional information from his book while keeping the explanations fresh. I've seen huge pre-calculus textbooks that seriously don't teach as much and as well as Simmons does: they are verbose, dry and dull. In less than 120 pages this book covers Geometry, Algebra and Trigonometry. These three parts are independent of each other, so you can read then in any order you want. Even if Simmons aims for brevity, he always gives good examples (and solutions) to the topic being covered. Also, on each topic, he gives proofs for most formulas and concepts. And his proofs are so intuitive (but correct!) that when one understands one has to smile of the satisfaction. Of course, Simmons does not prove obvious things. For example, he himself argues that proving that one point is always in the middle of three points that lay in a line segment is painful to discuss, and says "...when examining a proof, the natural reaction of an intelligent student is irritation and impatience, and he is right." One word of advice though: some proofs are obscure in the sense that they are not completely laid out in just one block of the text. Simmons sometimes assumes you have a COMPLETE understanding of all the topics before the proof, so he goes over some details hoping you know what is going on. But this is not really bad, because he will always tell you for example: "Because of (a) above..." and this will be you hint to discover the influence of the topic (a) in that proof. It must be said: the explanations of some topics are really brief (though not incomplete), so if you are in a hurry you can finish this book in about three days. I don't think you could work on all the exercises in three days, but anyhow if you are very short on time you could maybe do like 5 or 6 exercises per topic. That seems like a feasible goal for a three day limit. Finally, Simmons does not cover every tiny little detail of pre-calculus in his book. He covers what he thinks are the most important topics. I agree with his choice, because looking back, these are the topics that I needed the most for my calculus, linear algebra, statistics and differential equations classes.

This book is an excellent source for review and reference. It contains three chapters covering the important concepts from middle school and high school geometry, algebra, and trigonometry. It is clear and concise yet surprisingly thorough. The author focuses on giving a conceptual understanding of the topics omitting unnecessary details. His explanations and derivations are clear and understandable with helpful illustrations when necessary. This doesn't necessarily mean that it is all an easy read; many concepts will require a good amount of focus and thinking to properly understand. There are example problems and at the end of each section there are practice problems. At the ends of the geometry and trigonometry chapters are all of the important formulas so they can all be seen together for easy reference. The chapters are completely independent of each other and the sections within each chapter are mostly independent of each other so it is easy and convenient to skip to whatever section is necessary to review or reference without the need to study background material elsewhere in the book. That being said, this book definitely covers the topics "in a nutshell". It would probably not be great for learning precalculus topics for the first time and I don't think that is what the author intended it to be used for. It could be an effective supplement to a class or another more thorough textbook, but alone it would probably be insufficient. The examples and practice problems are sufficient for someone who already has a basic understanding of the concepts, but a new student will almost certainly need more guidance and practice.

At 80, I decided that it was time to renew my math, and this book is just what I needed. It is like new and a good place to start.

I bought this book for the last minute review of my daughter who heads to engineering college in three weeks. Firmly believing that foundation in precalculus or algebra II along with geometry is the key for science/engineering application of math, I find this little book a very handy and high quality book. The author is well known for pedagogy at college and through a few college level math books. The book contains three parts: geometry, algebra, and trigonometry. No spilled words, every sentence crisp and clear. Problems are a good mixture from basic level to reasonably sophisticated level. Overall, very good. One personal view. If I were to write a book of this sort, I would consider complex numbers an integral part of the algebra II or precalculus. It can be a part of the algebra section, and then the trigonometry part would be super-easy since demouvre theorem would make all kinds of identity formulas a matter of triviality. I found my kids losing all their focus when it comes to traditional way of teaching trigonometry (pythagorean theorem part ok, but complicated derivation (almost memorizing) of trigonometric identities and grinding exercises based on that formula). So, what I did to my daughter is to let her review complex number part and demouvre theorem in her precalculus textbook and come back to the trigonometry part of this little book. She appreciated this route very much. I also tried this book to my son who just starts 9th grade honors geometry. He previewed the material during the summer, and now he found this little book a real "fun" --- I didn't ask him, but he read it over and got motivated by himself to sit down and attack all problems one by one. He even asked me whether there would be an alternative, shorter proof of heron's area formula! There are other important parts of precalculus. For the reason that this book does not even mention a sentence on them, I rated 4 stars. Otherwise, this is an excellent book, especially, for review and parent-guided introduction purposes.

As many say here: this is not a textbook, it’s a resource for reviewing Algebra 1/2, Geometry and Trig. Basically, it replaces four years of lost or low quality class notes with 50-odd pages of correct, legible and well-written text and some exercises. (If you think that’s depressing, one popular calculus guide summarizes those four years in five pages.) That said, this is so well-written (Simmons is famous for the clarity of his undergrad texts, such as Topology and Modern Analysis) that I’m planning on using it as guided tutoring material for my kid - I can explain it and he can then use the book as his notes.

As a professor who teaches Capital Markets in a Masters program in Finance, I only wish that all my students came prepared with this much math (let alone, basic calculus). To answer criticisms first. This is not a book to learn pre-calculus math from, with no previous exposure -- the explanations are too condensed; the reader is expected to fill-in key steps from his/her previous knowledge. However, for those reviewing topics for which they already have some understanding, I cannot think of a better book. It cuts down to the essence what often takes 500, verbose pages. It is a excellent summary that requires the reader to be active and think through the steps along the way. Again, for those who have previously studied the topic, what would be a good follow-on for elementary calculus? Probably, either Spivak: http://www.amazon.com/Hitchhikers-Guide-Calculus-Michael-Spivak/dp/0883858126/ref=sr_1_17?ie=UTF8&s=books&qid=1257190231&sr=1-17 or Kleppner and Ramsey: http://www.amazon.com/Quick-Calculus-Self-Teaching-Guide-2nd/dp/0471827223/ref=sr_1_1?ie=UTF8&s=books&qid=1257190721&sr=1-1 Personally, I would slightly prefer Kleppner and Ramsey (because it goes a little further, without sacrificing understanding).

The book is definitely not what I expected based on reviews. The book offers views, interpretations and insights on how to understand calculus. It doesn’t actually give instructions or ways to remember formulas. This book might be helpful to others but not so much for me.

I love this little book. I've been away from teaching for a few years and have forgotten/never known many of the things Simmons makes clear. For example: "The system of assigning coordinate points in a line enable us to study geometry with the tools of algebra, and this is what analytic geometry is all about." No techno-gibberish. It's all plain English which makes the book brilliant, deep, and simple all at once. This was written by a master teacher and masterful communicator.