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Preface

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:

$$x+2=3$$

$$\frac{x-2.7}{42}=\frac{2\pi}{3}$$

$$2=\frac{x+2}{x-5}$$

$$\sqrt{x+1}=3.9$$

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.¹

There is a web page for this book ( http://inneralgebra.com/help/ ). 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 (amax@hilomath.com).

Footnotes

...¹

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.