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1. May 5, 2015

### bacte2013

Dear Physics Forum personnel,

I wrote this email because I am seeking a recommendation on selecting the pre-calculus textbooks. I have been studying the real analysis and number theory, and I felt that I need to brush up the algebra, functions, trigonometry, and geometry. So I decided to purchase a pre-calculus textbook and study it for two weeks, and return back to the real analysis and number theory. The particular pre-calculus textbooks I have in my mind are:

Allendoerfer & Oakley: Fundamentals of Freshman Mathematics & Principles of Mathematics
Serge Lang: Basic Mathematics
George Simmons: Pre-Calculus Mathematics in a Nutshell
Israel Gelfand: Algebra, Trigonometry
John Stillwell: Numbers and Geometry
Richard Courant: What is Mathematics?

Which one of them will be good for self-studying and as intensive review? Unfortunately, I forgot majority of stuffs in pre-calculus topics (particularly the trigonometry and functions) so I am planning to study from scratch and with higher level than the typical high school-level courses. If you know other good books, please inform me too!

Sincerely,

PK

2. May 6, 2015

### verty

Axler: Precalculus may be interesting to you. I have his "Linear Algebra Done Right" which is intended to be used for a second Linear Algebra course, one that is more theoretical. He says in the preface that one should pore over every page, proving every theorem and exploring every definition, and that an hour per page is about the right pace for the book. My only real complaint about his style is that his exercises can tend to be unintuitive, in the sense that finding an attack vector for the problem is the problem. But I can't fault him for writing in a very accurate and terse style.

His precalculus one is at a lower level in the sense that he does intend it to be used by enterprising high-schoolers but for example, in his "review" chapter 0 he constructs the real line and proves the irrationality of $\sqrt{2}$, hence showing why real numbers are needed, and discusses the "algebra of the real numbers". He also "reviews" inequalities and absolute value.

I see in the trig section that he includes pretty much all the formulas including half-angle formulas and the sum and difference formulas, and discusses inverse trig identities which for example I hadn't actually seen until Pranav-Arora asked about them here. So this section of the book looks very complete. I don't think he discusses power series or the complex exponential though, and while he includes matrices, it seems determinants are not included.

There is a newer edition of the book but I linked to the one that one can look inside.

Last edited: May 6, 2015