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Important Note for Students

  1. If your instructor asks you to write out all the steps you take to solve an exercise, please do that... even if (because you read this book) you can solve the whole thing in your head.
  2. For best results, only use the methods in this book to do math that you can already do "normally" (by writing it out). If you have trouble doing a certain calculation on paper, wait until you CAN do it that way before using the methods in this book to solve it.
If you are in school and taking a math course right now, your instructor will probably want you to "show your work" on homework and tests. This book will teach you how to correctly do those exercises while showing less of your work. If you practice what's in here, you will probably be able to solve some exercises entirely in your head. That's really great, except that your teacher wants you to write out intermediate steps.

Please make sure you actually do write out everything the way your instructors would like. Your instructors are responsible for making sure you get a certain level of math knowledge (assuming you do your part); that's their job. Writing out everything helps them do that. More importantly, it also helps you learn the math deeply. Of course, you are the one who benefits from all this.

This book assumes you already know algebra, and can use it to solve equations. If you are still learning algebra, that's okay - you can still benefit from this book. The only requirement is that you have some experience solving algebra equations. As you read, if there are parts that talk about math that you haven't learned yet, just skip over those sections for now.

There is a difference between mathematical knowledge, and what goes on inside of you when you apply that knowledge. This book is only about the second part. There is a bit of trickiness here. Some of the techniques in this book are, in a sense, shortcuts. You can use them to mentally bypass steps in a calculation. But if you use them to do math that you don't yet understand how to do normally, then it's possible that you will apply the techniques incorrectly. You will still get an answer, but it will probably not be mathematically correct. Even worse, you may "learn" faulty math habits.

Fortunately, there is an easy way to handle this: only use the techniques in this book to do math that you already know how to solve normally (on paper). If you are not sure, attempt to solve it on paper first. As you do this, you will get better and better at knowing when it is safe to use the techniques in this book. Following this guideline will allow you to safely get the greatest long-term value from this book.


Many mathematicians and engineers can look at an equation, think a moment, and know its solution. Some can do very complex math this way. Most people can learn to do this. By using certain mental abilities you have in ways described in this book, you can do things mathematically that may have not seemed possible before.

In this book, we focus on algebra; you learn to solve algebra equations mentally. It is necessary that you have already learned algebra in some way (probably through a math course). What you learn here is to do certain things that allow you to do algebra internally. If you are just learning algebra (for instance, if you are now taking a course in school), you will begin to find this book useful as soon as you have some experience solving equations. (You will mainly be working with what are called singular equations, which are explained below. For now you'll ignore the other kind, called plural equations.)

The skills taught in this book come in several stages. In the first, you learn to solve algebra equations much faster. For simple problems, this means you will know the solution just by looking at it and thinking a moment. If the equation is more complex, you will solve it with fewer written steps than before. This first stage will take place after you read through the end of chapter [*], working the exercises along the way.

As mentioned, we can categorize algebra equations as singular and plural. An equation is usually singular if it can be reduced to the form ``x = <some number>''. A singular equation has one variable and a single solution. Here are some singular equations:

\begin{displaymath} x+2=3\end{displaymath}

\begin{displaymath} \frac{x-2.7}{42}=\frac{2\pi}{3}\end{displaymath}

\begin{displaymath} 2=\frac{x+2}{x-5}\end{displaymath}

\begin{displaymath} \sqrt{x+1}=3.9\end{displaymath}

An equation is plural if it is not singular. There are several kinds. Examples of plural equations are those with more than one solution, like quadratic or higher-order polynomial equations; or equations like $\sin x=0.7$.

(By the way, if you are wondering why the first type of equation is called singular and the second type is called plural, it has to do with how you will learn to solve them. This is explained more fully in chapter [*].)

In the second stage, you start to solve singular equations without writing intermediate steps. You will look at it, do some things mentally, and know the solution. You get to the second stage by practicing what you learn in stage 1. If your job or school work involves algebra, getting this practice is simple. Since you'll be manipulating equations anyway, you just practice these techniques while doing it. As you practice, you get better, and naturally need to write fewer steps down. (It may or may not be easy - since you are relearning and reshaping habits, it will take some dedication on your part.) There will also be equations you can solve by just looking at them and thinking for a moment, whereas before you started this book, you would have needed to write a lot to solve it.

In the third stage, you will start to solve plural equations in your head. These equations have some special qualities. The skills for solving them build greatly on those for solving singular equations.0.1

There is a web page for this book ( ). Any additions, corrections, FAQS, or other helpful information will be put on this page. Also, feel free to contact me by email if you have questions or comments (


Thanks to my friends Colby Lemon and John Chodera, and my sister Annette Maxwell, for encouragement and feedback. Thanks to my longtime friend, Scott Tabbert, for his enthusiastic support, despite him not being a math person at all. And thanks to my housemate Kate McLaughlin-Williamson for valuable suggestions on the cover design. (Be glad I asked her for feedback. If you can stand to look at it for very long, you have her to thank.) These people only helped improve it, of course - if anything is less than ideal, that is my doing.

This book has been written entirely using open source software. The platform was a Debian GNU/Linux system, using either the icewm or the KDE window manager environments. Some other software used includes LYX and the LATEX typesetting system, The Gimp (for image creation and editing), the ImageMagick image processing tools, Perl (for making various support software), Subversion for document version control, the use-for-all-sorts-of-things tools such as emacs, vi, konsole, konqueror... and many others.

All of this software exists because, worldwide, developers I never met invested their time and energy to write software, with the purpose of freely giving it to anyone who wants to use it. This book simply would not exist in this form were it not for those developers' generosity. My tremendous thanks and appreciation to all of you.


Some plural equations are easier to solve than others. Quadratic equations are plural equations, and are relatively easy to solve. You will be able to get them soon after you gain some experience working singular equations.

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