From the Publisher

One of the most commonly asked questions in a mathematics classroom is, "Will I ever use this stuff in real life?" Some teachers can give a good, convincing answer; others hem and haw and stare at the floor. The real response to the question should be, "Yes, you will, because algebra gives you power" – the power to help your children with their math homework, the power to manage your finances, the power to be successful in your career (especially if you have to manage the company budget). The list goes on.

Algebra is a system of mathematical symbols and rules that are universally understood, no matter what the spoken language. Algebra provides a clear, methodical process that can be followed from beginning to end to solve complex problems. There's no doubt that algebra can be easy to some while extremely challenging to others. For those of you who are challenged by working with numbers, Algebra For Dummies can provide the help you need.

This easy-to-understand reference not only explains algebra in terms you can understand, but it also gives you the necessary tools to solve complex problems. But rest assured, this book is not about memorizing a bunch of meaningless steps; you find out the whys behind algebra to increase your understanding of how algebra works.

In Algebra For Dummies, you'll discover the following topics and more:

• All about numbers – rational and irrational, variables, and positive and negative
  
  
• Figuring out fractions and decimals
  
  
• Explaining exponents and radicals
  
  
• Solving linear and quadratic equations
  
  
• Understanding formulasand solving story problems
  
  
• Having fun with graphs
  
  
• Top Ten lists on common algebraic errors, factoring tips, and divisibility rules.
  
  

No matter if you're 16 years old or 60 years old; no matter if you're learning algebra for the first time or need a quick refresher course; no matter if you're cramming for an algebra test, helping your kid with his or her homework, or coming up with next year's company budget, Algebra For Dummies can give you the tools you need to succeed.

Product Details

• ISBN: 0764553259
  
• ISBN-13: 9780764553257
• Format: Paperback, 384pp
• Publisher: Wiley, John & Sons, Incorporated
• Pub. Date: September 2001
• Edition Number: 1
• Sales Rank: 6,857
• Series: For Dummies Series

• Table of Contents
• Read a Sample Chapter

Table of Contents

Read a Sample Chapter

Algebra For Dummies

John Wiley & Sons

Chapter One

In This Chapter

* Nailing down the basics: Numbers

* Recognizing the players: Variables and signs

* Grouping terms and operations together

* Playing the game and following the rules

You probably have heard the word algebra on many occasions and knew that it had something to do with mathematics. Perhaps you remember that algebra has enough information to require taking two separate high school algebra classes - Algebra I and Algebra II. But what exactly is algebra? What is it really used for?

This chapter answers these questions and more, providing the straight scoop on some of the contributions to algebra's development, what it's good for, how algebra is used, and what tools you need to make it happen.

In a nutshell, algebra is a way of generalizing arithmetic. Through the use of variables that can generally represent any value in a given formula, general formulas can be applied to all numbers. Algebra uses positive and negative numbers, integers, fractions, operations, and symbols to analyze the relationships between values. It's a systematic study of numbers and their relationship, and it uses specific rules.

For example, the formula a × 0 = 0 shows that any real number, represented here by the a, multiplied by zero always equals zero. (For more information on themultiplication property of zero, see Chapter 14.)

In algebra, by using an x to represent the number two, for example in x + x + x = 6, you can generalize with the formula 3x = 6.

You may be thinking, "That's great and all, but come on. Is it really necessary to do that - to plop in letters in place of numbers and stuff?" Well, yes. Early mathematicians found that using letters to represent quantities simplified problems. In fact, that's what algebra is all about - simplifying problems.

The basic purpose of algebra has been the same for thousands of years: to allow people to solve problems with unknown answers.

Beginning with the Basics: Numbers

Where would mathematics and algebra be without numbers? A part of everyday life, numbers are the basic building blocks of algebra. Numbers give you a value to work with.

Where would civilization be today if not for numbers? Without numbers to figure the total cubits, Noah couldn't have built his ark. Without numbers to figure the distances, slants, heights, and directions, the pyramids would never have been built. Without numbers to figure out navigational points, the Vikings would never have left Scandinavia. Without numbers to examine distance in space, humankind could not have landed on the moon.

Even the simple tasks and the most common of circumstances require a knowledge of numbers. Suppose that you wanted to figure the amount of gasoline it takes to get from home to work and back each day. You need a number for the total miles between your home and business and another number for the total miles your car can run on one gallon of gasoline.

The different sets of numbers are important because what they look like and how they behave can set the scene for particular situations or help to solve particular problems. It's sometimes really convenient to declare, "I'm only going to look at whole-number answers," because whole numbers do not include fractions. This may happen if you're working through a problem that involves a number of cars. Who wants half a car?

Algebra uses different sets of numbers, such as whole numbers and those that follow here, to solve different problems.

Really real numbers

Real numbers are just what the name implies. In contrast to imaginary numbers, they represent real values - no pretend or make-believe. Real numbers, the most inclusive set of numbers, comprise the full spectrum of numbers; they cover the gamut and can take on any form - fractions or whole numbers, decimal points or no decimal points. The full range of real numbers includes decimals that can go on forever and ever without end. The variations on the theme are endless.

For the purposes of this book, I always refer to real numbers.

Counting on natural numbers

A natural number is a number that comes naturally. What numbers did you first use? Remember someone asking, "How old are you?" You proudly held up four fingers and said, "Four!" The natural numbers are also counting numbers: 1, 2, 3, 4, 5, 6, 7, and so on into infinity.

You use natural numbers to count items. Sometimes the task is to count how many people there are. A half-person won't be considered (and it's a rather grisly thought). You use natural numbers to make lists.

Wholly whole numbers

Whole numbers aren't a whole lot different from the natural numbers. The whole numbers are just all the natural numbers plus a zero: 0, 1, 2, 3, 4, 5, and so on into infinity.

Whole numbers act like natural numbers and are used when whole amounts (no fractions) are required. Zero can also indicate none. Algebraic problems often require you to round the answer to the nearest whole number. This makes perfect sense when the problem involves people, cars, animals, houses, or anything that shouldn't be cut into pieces.

Integrating integers

Integers allow you to broaden your horizons a bit. Integers incorporate all the qualities of whole numbers and their opposites, or additive inverses of the whole numbers (refer to the "Operating with opposites" section in this chapter for information on additive inverses). Integers can be described as being positive and negative whole numbers: ... -3, -2, -1,0,1,2,3 ....

Integers are popular in algebra. When you solve a long, complicated problem and come up with an integer, you can be joyous because your answer is probably right. After all, it's not a fraction! This doesn't mean that answers in algebra can't be fractions or decimals. It's just that most textbooks and reference books try to stick with nice answers to increase the comfort level and avoid confusion. This is the plan in this book, too. After all, who wants a messy answer, even though, in real life, that's more often the case.

Being reasonable: Rational numbers

Rational numbers act rationally! What does that mean? In this case, acting rationally means that the decimal equivalent of the rational number behaves. The decimal ends somewhere, or it has a repeating pattern to it. That's what constitutes "behaving." Some rational numbers have decimals that end in 2, 3.4, 5.77623, -4.5. Other rational numbers have decimals that repeat the same pattern, such as 3.164164164 ... = 3.[bar.164], or .666666666 .[bar.6]. The horizontal bar over the 164 and the 6 lets you know that these numbers repeat forever.

In all cases, rational numbers can be written as a fraction. They all have a fraction that they are equal to. So one definition of a rational number is any number that can be written as a fraction.

Restraining irrational numbers

Irrational numbers are just what you may expect from their name - the opposite of rational numbers. An irrational number cannot be written as a fraction, and decimal values for irrationals never end and never have a nice pattern to them. Whew! Talk about irrational! For example, pi, with its never-ending decimal places, is irrational.

Evening out even and odd numbers

An even number is one that divides evenly by two. "Two, four, six, eight. Who do we appreciate?"

An odd number is one that does not divide evenly by two. The even and odd numbers alternate when you list all the integers.

Varying Variables

Variable is the most general word for a letter that represents the unknown, or what you're solving for in an algebra problem. A variable always represents a number.

Algebra uses letters, called variables, to represent numbers that correspond to specific values. Usually, if you see letters toward the beginning of the alphabet in a problem, such as a, b, or c, they represent known or set values, and the letters toward the end of the alphabet, such as x, y, or z, represent the unknowns, things that can change, or what you're solving for.

The following list goes through some of the more commonly used variables.