**Assembling Your Tools**

**In This Chapter**

* Giving names to the basic numbers

* Reading the signs - and interpreting the language

* Operating in a timely fashion

You've probably heard the word *algebra* on many occasions, and you
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 book 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 this chapter, you find some of the basics necessary to more easily find your way through the different topics in this book. I also point you toward these topics.

In a nutshell, *algebra* is a way of generalizing arithmetic.
Through the use of *variables* (letters representing numbers) and
formulas or equations involving those variables, you solve problems. The
problems may be in terms of practical applications, or they may be
puzzles for the pure pleasure of the solving. 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.

*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 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 a 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 or negatives. You could easily end up with a fraction if you're working through a problem that involves a number of cars or people. Who wants half a car or, heaven forbid, a third of a person?

Algebra uses different sets of numbers, in different circumstances. I describe the different types of numbers here.

**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 cover the gamut and can take on any form -
fractions or whole numbers, decimal numbers that can go on forever and
ever without end, positives and negatives. The variations on the theme
are endless.

*Counting on natural numbers*

A *natural number* (also called a *counting 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 the numbers starting with 1 and
going up by ones: 1, 2, 3, 4, 5, 6, 7, and so on into infinity. You'll
find lots of counting numbers in Chapter 6, where I discuss prime
numbers and factorizations.

*Wholly whole numbers**
*

*
Whole numbers* aren't a whole lot different from natural numbers.
Whole numbers are just all the natural numbers plus a 0: 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 (called their
*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 my plan in this book, too. After all, who wants a messy answer, even though, in real life, that's more often the case. I use integers in Chapters 8 and 9, where you find out how to solve equations.

*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 such as: 3.4, 5.77623, -4.5. Other rational numbers have decimals that repeat the same pattern, such as 3.164164164, or 0.666666666. 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 fractions. Each
rational number has a fraction that it's equal to. So one definition of
a *rational number* is any number that can be written as a
fraction, *p/q*, where *p* and *q* are integers (except
*q* can't be 0). If a number can't be written as a fraction, then
it isn't a rational number. Rational numbers appear in Chapter 13, where
you see quadratic equations, and in Part IV, where the applications are
presented.

*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.
Irrational numbers are often created when using the quadratic formula,
as you see in Chapter 13.

*Picking out primes and composites*

A number is considered to be *prime* if it can be divided evenly
only by 1 and by itself. The first prime numbers are: 2, 3, 5, 7, 11,
13, 17, 19, 23, 29, 31, and so on. The only prime number that's even is
2, the first prime number. Mathematicians have been studying prime
numbers for centuries, and prime numbers have them stumped. No one has
ever found a formula for producing all the primes. Mathematicians just
assume that prime numbers go on forever.

A number is *composite* if it isn't prime - if it can be divided by
at least one number other than 1 and itself. So the number 12 is
composite because it's divisible by 1, 2, 3, 4, 6, and 12. Chapter 6
deals with primes, but you also see them in Chapters 8 and 10, where I
show you how to factor primes out of expressions.

*Speaking in Algebra*

Algebra and symbols in algebra are like a foreign language. They all mean something and can be translated back and forth as needed. It's important to know the vocabulary in a foreign language; it's just as important in algebra.

*expression*is any combination of values and operations that can be used to show how things belong together and compare to one another. 2[chi square] + 4

*x*is an example of an expression. You see distributions over expressions in Chapter 7.

*term,*such as 4

*xy*, is a grouping together of one or more

*factors*(variables and/or numbers). Multiplication is the only thing connecting the number with the variables. Addition and subtraction, on the other hand, separate terms from one another. For example, the expression 3

*xy*+ 5

*x*- 6 has three

*terms*.

*equation*uses a sign to show a relationship - that two things are equal. By using an equation, tough problems can be reduced to easier problems and simpler answers. An example of an equation is 2[

*x*.sup.2] + 4

*x*= 7. See the chapters in Part III for more information on equations.

*operation*is an action performed upon one or two numbers to produce a resulting number. Operations are addition, subtraction, multiplication, division, square roots, and so on. See Chapter 5 for more on operations.

*variable*is a letter representing some unknown; a variable always represents a number, but it

*varies*until it's written in an equation or inequality. (An

*inequality*is a comparison of two values. For more on inequalities, turn to Chapter 15.) Then the fate of the variable is set - it can be solved for, and its value becomes the solution of the equation. By convention, mathematicians usually assign letters at the end of the alphabet to be variables (such as

*x, y,*and

*z*).

*constant*is a value or number that never changes in an equation - it's constantly the same. Five is a constant because it is what it is. A variable can be a constant if it is assigned a definite value. Usually, a variable representing a constant is one of the first letters in the alphabet. In the equation

*a]*

*x*.sup.2] +*bx*+*c*= 0,*a, b,*and*c*are constants and the*x*is the variable. The value of*x*depends on what*a, b,*and*c*are assigned to be.

*An**exponent*is a small number written slightly above and to the right of a variable or number, such as the 2 in the expression [3.sup.2]. It's used to show repeated multiplication. An exponent is also called the*power*of the value. For more on exponents, see Chapter 4.
*Taking Aim at Algebra Operations*

In algebra today, a variable represents the unknown. (You can see more on variables in the "Speaking in Algebra" section earlier in this chapter.) Before the use of symbols caught on, problems were written out in long, wordy expressions. Actually, using letters, signs, and operations was a huge breakthrough. First, a few operations were used, and then algebra became fully symbolic. Nowadays, you may see some words alongside the operations to explain and help you understand, like having subtitles in a movie.

By doing what early mathematicians did - letting a variable represent a value, then throwing in some operations (addition, subtraction, multiplication, and division), and then using some specific rules that have been established over the years - you have a solid, organized system for simplifying, solving, comparing, or confirming an equation. That's what algebra is all about: That's what algebra's good for.

*Deciphering the symbols*

The basics of algebra involve symbols. Algebra uses symbols for quantities, operations, relations, or grouping. The symbols are shorthand and are much more efficient than writing out the words or meanings. But you need to know what the symbols represent, and the following list shares some of that info. The operations are covered thoroughly in Chapter 5.

*+ means**add*or*find the sum, more than,*or*increased by;*the result of addition is the*sum*. It also is used to indicate a*positive number*.

*- means**subtract*or*minus*or*decreased by*or*less than;*the result is the*difference*. It's also used to indicate a*negative number*.

*x means**multiply*or*times*. The values being multiplied together are the*multipliers*or*factors;*the result is the*product*. Some other symbols meaning*multiply*can be grouping symbols: ( ), , { }, ?, *. In algebra, the x symbol is used infrequently because it can be confused with the variable*x*. The dot is popular because it's easy to write. The grouping symbols are used when you need to contain many terms or a messy expression. By themselves, the grouping symbols don't mean to multiply, but if you put a value in front of a grouping symbol, it means to multiply.

*? means**divide*. The number that's going into the*dividend*is the*divisor*. The result is the*quotient*. Other signs that indicate division are the fraction line and slash, /.

*[square root of] means to take the**square root*of something - to find the number, which, multiplied by itself, gives you the number under the sign. (See Chapter 4 for more on square roots.)

*|| means to find the**absolute value*of a number, which is the number itself or its distance from 0 on the number line. (For more on absolute value, turn to Chapter 2.)

*[pi] is the Greek letter pi that refers to the irrational number: 3.14159.... It represents the relationship between the diameter and circumference of a circle.*
*Grouping*

When a car manufacturer puts together a car, several different things
have to be done first. The engine experts have to construct the engine
with all its parts. The body of the car has to be mounted onto the
chassis and secured, too. Other car specialists have to perform the
tasks that they specialize in as well. When these tasks are all
accomplished in order, then the car can be put together. The same thing
is true in algebra. You have to do what's inside the *grouping*
symbol before you can use the result in the rest of the equation.

*Grouping* symbols tell you that you have to deal with the
*terms* inside the grouping symbols *before* you deal with the
larger problem. If the problem contains grouped items, do what's inside
a grouping symbol first, and then follow the order of operations. The
grouping symbols are

**Parentheses ():**Parentheses are the most commonly used symbols for grouping.

**Brackets and braces {}:**Brackets and braces are also used frequently for grouping and have the same effect as parentheses. Using the different types of symbols helps when there's more than one grouping in a problem. It's easier to tell where a group starts and ends.

**Radical [square root of]:**This is used for finding roots.

**Fraction line (called the**The fraction line also acts as a grouping symbol - everything above the line (in the*vinculum*):*numerator*) is grouped together, and everything below the line (in the*denominator*) is grouped together.Even though the order of operations and grouping-symbol rules are fairly straightforward, it's hard to describe, in words, all the situations that can come up in these problems. The examples in Chapters 5 and 7 should clear up any questions you may have.

*Defining relationships*

Algebra is all about relationships - not the he-loves-me-he-loves-me-not kind of relationship - but the relationships between numbers or among the terms of an equation. Although algebraic relationships can be just as complicated as romantic ones, you have a better chance of understanding an algebraic relationship. The symbols for the relationships are given here. The equations are found in Chapters 11 through 14, and inequalities are found in Chapter 15.

(Continues...)