Preface for Students

Yes, it’s made difficult, but I didn’t make it that way. The difficulty is intrinsic. It is also surmountable. It requires no special intelligence to master the material in this book, but it does require a commitment: a couple of hours a day, every day, for the better part of a year. Fortunately, this material (algebra, coordinate geometry, trigonometry) is genuinely engaging if you approach it in the right way. The right way is to understand it. If you understand mathematics, you’ll experience the pleasure of feeling its pieces lock logically together in your mind in an aesthetically satisfying way. $ ^{*} $

On the other hand, if you approach mathematics – as so many people do – in the wrong way (memorizing procedures without understanding why they work), then the subject will be sheer tedium. Don’t let this happen. The difficulty is intrinsic. The tedium is not. The choice is yours.

Precalculus Made Difficult is meant to be read slowly and carefully. Strive to read the relevant sections in the text before your teacher lectures on them. The lectures will then reinforce what you’ve understood, and clarify what you haven’t. Read with pencil and paper at the ready. $ ^{†} $ When I omit details, you should supply them. When I use a phrase such as “as you should verify”, I am not being facetious. Verify it. Only after reading a section should you attempt to solve the problems with which it concludes. Whenever you encounter something in the text that you do not understand (even an individual algebraic step), you should mark the relevant passage and try to clear it up, which may involve discussing it with your classmates or teacher or reviewing earlier material.

Hundreds of thousands of people succeed in learning this material every year. You can be one of them. But it will require hard work, and at times, you may wonder whether it is worthwhile. It is.

Let’s begin.