# Good Pre-Calculus book?

I was reading this article and the author mentioned I should come here and get some advice. I'm 17, currently taking Pre-Calc in high schooling doing really good, but I feel like I'm not getting the most out of it. The teacher feels like he's more interested in covering chapters than getting us to understand things deeply and that worries me. The article says:

Try to find a book where the author treats you as the intelligent, independent person you are, not as someone who has to take a course for a degree requirement...go to some math forums (like Math Overflow) and ask for book recommendations, telling them you want to become good at math and not just pass a required course; give them specific details and they can help find a book perfect for you.

So yeah asked on Math Overflow and was suggested to come here. I want to get better at math and really understand the concepts deeply and appreciate it like it was intended to. Any help I can get will be appreciated. Thanks!

• +1 for effort any way when you say pre-calculus what are you studying (topics) in high-school it will help me answer better – MRK Dec 14 '14 at 0:14
• Racional expressions, linear equations, Inequations,and functions are some of the topics with we've touched in class. – Wil Dec 14 '14 at 0:21
• Here are some names of good authors: Howard ANTON, and RON LARSON, they have (each one) many books. On the other hand Russian books are good too, and there is a better (mybe the best) one of DEMIDOVICH, it contains more that 3000 exercises in the calculus field. HInts or indications are given for each exercise. – Idris Dec 14 '14 at 0:49
• @Idris Demidovich is a book of calculus problems. The question is about pre-calculus. – user147263 Dec 14 '14 at 1:00
• Possible duplicate of what is the best book for Pre-Calculus? – Greek - Area 51 Proposal May 28 '18 at 4:20

The series of books Algebra, Functions and Graphs, Trigonometry, and The Method of Coordinates by I. M. Gelfand and various co-authors is an excellent way to supplement a pre-calculus course. The books were written for advanced high school students taking correspondence courses with professors in the Soviet Union and are available in English translation. The books are clearly written, supplement topics found in the typical pre-calculus text, and provide challenging problems.

Another good source is a series of Japanese books edited by Kunihiko Kodaira. They include Mathematics I: Japanese Grade 10, Basic Analysis: Japanese Grade 11, and Algebra and Geometry: Japanese Grade 11. These books are also available in English translation. The grade 10 book is for a required course roughly equivalent to pre-calculus. Regular track students then take a course based on Mathematics II: Japanese grade 11. Mathematically inclined students take courses based on both the Algebra and Geometry and Basic Analysis texts. The texts are a good source of challenging problems and contain material that will supplement what you would learn in a pre-calculus course.

I'm considerably older than you and failed miserably at math in high school so this may not apply to your case, but I found "Precalculus Mathematics in a Nutshell" by George F. Simmons to be a fantastic encapsulation of pre-calc topics when studying math as an adult. He really boils it down to the essentials. E.g. here's how he opens his chapter on Trig:

Most trigonometry textbooks have been written by people who appear to believe that the importance of the subject lies in its applications to surveying and navigation. Even though very few people become surveyors or navigators, the students who study these books are expected to undertake many lengthy calculations about the heights of flagpoles, the widths of rivers and the positions of ships at sea.

The truth is that the primary importance of trigonometry lies in a completely different direction - in the mathematical description of vibrations, rotations, and periodic phenomena of all kinds, including light, sound, alternating currents and the orbits of the planets around the sun. What matters most in the subject is not making computations about triangles, but grasping the trigonometric functions as indispensable tools in science, engineering and higher mathematics. These functions and their properties are the sole subject matter of this chapter.

The entire book has that vibe. It's wonderful.

I can recommend the Precalculus volume of a series called the CME Project. It's a high school textbook written by a team of thoughtful and savvy mathematicians. It works to make connections between topics, emphasizes making use of structure in calculation, and builds generalizations from concrete cases. It's a "habits of mind" approach that focuses on mathematical thinking and not just rote processes. I think you'll find in this book what is lacking from your class. Enjoy!

there is a series of books written by gelfand and shen, i believe, is very nice. in particular used on of their books called algebra. it teaches you mainly through solving lots and lots of problems. i don't have at hand but it has hundreds of problems.

• Actually, the name is Shen – Artem Dec 14 '14 at 3:10

Get this book Higher Math for Beginning Physicists and Engineers and enjoy the authors treating you as "an intelligent independent person".

There are many good textbooks for Calculus, such as

[1] Walter Rudin. Principles in Mathematical Analysis. Academic Press.

[2] Zorich. Mathematical Analysis, Vol. 1-2. Springer-Verlag.

[3] Григорий Михайлович Фихтенгольц. Course of Calculus. ?

• These are not even close to appropriate suggestions. They are several years of study beyond the poster's level. – Nate C-K Sep 11 '17 at 15:57

"Pre-calculus" is not a subject that exists for any intellectually legitimate reason. So my suggestion would be either to simply start learning calculus or to explore wider mathematical horizons, for instance, number theory.

For calculus, practically the only modern book that treats the reader as a reader, let alone as an "intelligent, independent person," is Michael Spivak's Calculus. This requires no previous knowledge of the material of precalculus-in fact, Spivak will start much farther back, and you won't even define such functions as $e^x$ and $\sin$ until well into the book (of course, in your current course, these functions were never properly defined at all.) That said, you will almost certainly find his problems an order of magnitude more challenging than what you've seen 'till now, but the solutions manual is readily available, and, of course, so are the members of this site!

I don't have any particularly specific suggestions for number theory, but there are several Dover books with titles like "elementary number theory" with good reviews (stay away from "analytic" or "algebraic" number theory for now.) The great thing about Dover books is you can buy three for half the price of an ordinary book and compare. Best of luck!

• Any comment regarding the downvote? – Kevin Carlson Dec 14 '14 at 0:50
• I'm not the one who downvoted, but I do disagree with your answer a bit. You're essentially saying that life starts at calculus, when in fact even "basic algebra" didn't exist until the 10th century, roughly 200,000 years after modern humans are believed to have come into being. – Matt Samuel Dec 14 '14 at 2:22
• @MattSamuel I certainly didn't intend to imply that there's no reason for any math before calculus, if I'm correctly reading your purple prose (a fortiori, I made no comment about where human life itself begins...) I just find that precalculus is a waste of time-a motivated student can move from algebra to calculus with no problems. – Kevin Carlson Dec 14 '14 at 2:36
• I suppose part of the problem is with the definition of Pre-calculus. Most texts I've come across group Geometry, Algebra and Trig in the category "Pre-calc". I can't imagine the frustration one would experience trying to tackle calculus without being adept at manipulating symbols, understanding geometric relationships (e.g. when doing related rates in Differential Calc) or understanding the trig functions. You'd constantly be taking steps backwards to learn the specifics of those pre-calc domains and risk losing sight of the beauty of calculus. – Nick Dec 14 '14 at 20:56
• @Nick thanks for your thoughts. I certainly wouldn't recommend calculus to someone without some algebra and geometry preparation, but at least for American students precalc is usually the fourth consecutive year in this category, which is much more preparation than an engaged student needs. I'm on the fence about whether people really need to know trig before calculus. – Kevin Carlson Dec 15 '14 at 1:13