Get started with a FREE account. Networking for Dummies --For Dummies ; 7th Ed. Load more similar PDF files. Chapter 1: Putting a Name to Linear Algebra 13 Notice that I placed a 0 where there was a missing term in an equation.
The coefficient matrix is so much easier to look at than the equation. But you have to follow the rules of order. And I named the matrix — nothing glamorous like Angelina, but something simple, like A.
When using coefficient matrices, you usually have them accompanied by two vectors. A vector is just a one-dimensional matrix; it has one column and many rows or one row and many columns.
See Chapters 2 and 3 for more on vectors. The vectors that correspond to this same system of equations are the vector of variables and the vector of constants. I name the vectors X and C. Once in matrix and vector form, you can perform operations on the matrices and vectors individually or perform operations involving one operating on the other. All that good stuff is found beginning in Chapter 2. Let me show you, though, a more practical application of matrices and why putting the numbers coefficients into a matrix is so handy.
Consider an insurance agency that keeps track of the number of policies sold by the dif- ferent agents each month. Betty sold. Also, the commissions to agents can be computed by performing matrix multiplication. This matrix addition and matrix multiplication business is found in Chapter 3. Other processes for the insurance company that could be performed using matrices are figuring the percent increases or decreases of sales of the whole company or individual salespersons by performing operations on summary vectors, determining commissions by multiplying totals by their respective rates, setting percent increase goals, and so on.
The possibilities are limited only by your lack of imagination, determination, or need. Valuating Vector Spaces In Part IV of this book, you find all sorts of good information and interesting mathematics all homing in on the topic of vector spaces.
In other chapters, I describe and work with vectors. But the words vector space are really just a mathematical expression used to define a par- ticular group of elements that exist under a particular set of conditions. You can find information on the properties of vector spaces in Chapter Think of a vector space in terms of a game of billiards.
You have all the ele- ments the billiards balls that are confined to the top of the table well, they stay there if hit properly. Even when the billiard balls interact bounce off one another , they stay somewhere on the tabletop. So the billiard balls are the elements of the vector space and the table top is that vector space.
You have operations that cause actions on the table — hitting a ball with a cue stick or a ball being hit by another ball. And you have rules that govern how all the actions can occur. The actions keep the billiard balls on the table in the vector space. Other areas in mathematics have similar entities classifications and designs. The common theme of such designs is that they contain a set or grouping of objects that all have something in common. Certain properties are attached to the plan — properties that apply to all the members of the grouping.
Vector spaces contain vectors, which really take on many different forms. The easiest form to show you is an actual vector, but the vectors may actu- ally be matrices or polynomials. As long as these different forms follow the rules, then you have a vector space. In Chapter 14, you see the rules when investigating the subspaces of vector spaces. The rules regulating a vector space are highly dependent on the operations that belong to that vector space.
You find some new twists to some famil- iar operation notation. With vector spaces, the operation of addition may be defined in a completely different way. Does that rule work in a vector space?
Determining Values with Determinants A determinant is tied to a matrix, as you see in Chapter The determi- nant incorporates all the elements of a matrix into its grand plan. You have a few qualifications to meet, though, before performing the operation determinant.
Square matrices are the only candidates for having a determinant. Let me show you just a few examples of matrices and their determinants. The determinants of the respective matrices go from complicated to simple to compute. For example, the matrix D, that I show you here, has a determinant of 0 and, consequently, no inverse. Matrix D looks perfectly respectable on the surface, but, lurking beneath the surface, you have what could be a big problem when using the matrix to solve problems.
You need to be aware of the consequences of the determi- nant being 0 and make arrangements or adjustments that allow you to pro- ceed with the solution. The values of the variables are ratios of dif- ferent determinants computed from the coefficients in the equations.
Zeroing In on Eigenvalues and Eigenvectors In Chapter 16, you see how eigenvalues and eigenvectors correspond to one another in terms of a particular matrix. Each eigenvalue has its related eigen- vector.
So what are these eigen-things? An eigen- value is a number, called a scalar in this linear algebra setting. For example, let me reach into the air and pluck out the number For now, just trust me on this. The resulting vector is the same whether I multiply the vector by 13 or by the matrix.
You can find the hocus-pocus needed to do the multiplication in Chapter 3. I just want to make a point here: Sometimes you can find a single number that will do the same job as a complete matrix. I actually peeked. Every matrix has its own set of eigenvalues the numbers and eigenvectors that get multiplied by the eigenvalues. In Chapter 16, you see the full treatment — all the steps and procedures needed to discover these elusive entities.
A vector is a special type of matrix rectangular array of numbers. The vectors in this chapter are columns of numbers with brack- ets surrounding them. Two-space and three-space vectors are drawn on two axes and three axes to illustrate many of the properties, measurements, and operations involving vectors.
You may find the discussion of vectors to be both limiting and expanding — at the same time. Vectors seem limiting, because of the restrictive structure. As with any mathematical presentation, you find very specific meanings for otherwise everyday words and some not so everyday. Keep track of the words and their meanings, and the whole picture will make sense. Lose track of a word, and you can fall back to the glossary or italicized definition.
Describing Vectors in the Plane A vector is an ordered collection of numbers. Vectors containing two or three numbers are often represented by rays a line segment with an arrow on one end and a point on the other end. Representing vectors as rays works with two or three numbers, but the ray loses its meaning when you deal with larger vectors and numbers.
The properties that apply to smaller vectors also apply to larger vectors, so I introduce you to the vectors that have pictures to help make sense of the entire set. When you create a vector, you write the numbers in a column surrounded by brackets.
Vectors have names no, not John Henry or William Jacob. The names of vectors are usually written as single, boldfaced, lowercase letters. You often see just one letter used for several vectors when the vectors are related to one another, and subscripts attached to distinguish one vector from another: u1, u2, u3, and so on.
Here, I show you four of my favorite vectors, named u, v, w, and x: The size of a vector is determined by its rows or how many numbers it has. Technically, a vector is a column matrix matrices are covered in great detail in Chapter 3 , meaning that you have just one column and a certain number of rows. Vectors in two-space are represented on the coordinate x,y plane by rays. In standard position, the ray representing a vector has its endpoint at the origin and its terminal point or arrow at the x,y coordinates designated by the column vector.
The x coordinate is in the first row of the vector, and the y coordinate is in the second row. The coordinate axes are used, with the horizontal x-axis and ver- tical y-axis. Figure shows the six vectors listed in the preceding section, drawn in their standard positions.
The coordinates of the terminal points are indicated on the graph. The following vector is just as correctly drawn by starting with the point —1,4 as an endpoint, and then drawing the vector by moving two units to the right and three units down, ending up with the terminal point at 1,1. Both the length and the direction uniquely determine a vector and allow you to tell if one vector is equal to another vector.
Vectors can actually have any number of rows. Also, the applica- tions for vectors involving hundreds of entries are rather limited and difficult to work with, except on computers. Adding a dimension with vectors out in space Vectors in R3 are said to be in three-space. The vectors representing three- space are column matrices with three entries or numbers in them. The R part of R3 indicates that the vector involves real numbers. Three-space vectors are represented by three-dimensional figures and arrows pointing to positions in space.
Picture a vector drawn in three- space as being a diagonal drawn from one corner of a box to the opposite corner. A ray representing the following vector is shown in Figure with the endpoint at the origin and the terminal point at 2,3,4. Vectors are groupings of numbers just waiting to have operations performed on them — ending up with predictable results.
The different geometric transformations performed on vectors include rota- tions, reflections, expansions, and contractions. You find the rotations and reflections in Chapter 8, where larger matrices are also found.
As far as oper- ations on vectors, you add vectors together, subtract them, find their oppo- site, or multiply by a scalar constant number. You can also find an inner product — multiplying each of the respective elements together. Swooping in on scalar multiplication Scalar multiplication is one of the two basic operations performed on vec- tors that preserves the original format. You may not be all that startled by this revelation, but you really should appreciate the fact that the scalar main- tains its original dimension.
Reading the recipe for multiplying by a scalar A scalar is a real number — a constant value. Multiplying a vector by a scalar means that you multiply each element in the vector by the same constant value that appears just outside of and in front of the vector. Chapter 2: The Value of Involving Vectors 25 Opening your eyes to dilation and contraction of vectors Vectors have operations that cause dilations expansions and contractions shrinkages of the original vector. Both operations of dilation and contrac- tion are accomplished by multiplying the elements in the vector by a scalar.
If the scalar, k, that is multiplying a vector is greater than 1, then the result is a dilation of the original vector. If the scalar, k, is a number between 0 and 1, then the result is a contraction of the original vector.
In Figure , you see the results of the dilation and contraction on the origi- nal vector. You also may have wondered why I only multiplied by numbers greater than 0. The rule for contractions of vectors involves numbers between 0 and 1, nothing smaller. In the next section, I pursue the negative numbers and 0. The illustration for multiplying by 0 in two-space is a single point or dot. Not unexpected. The zero vector is the identity for vector addition, just as the number 0 is the identity for the addition of real numbers.
When you multiply a vector by —2, as shown with the following vector, each element in the vector changes and has a greater absolute value: In Figure , you see the original vector as a diagonal in a box moving upward and away from the page and the resulting vector in a larger box moving downward and toward you.
Chapter 2: The Value of Involving Vectors 27 z —2,3,5 5 —6 y 3 —10 Figure Multiplying x a vector by 4,—6,—10 a negative scalar.
Adding and subtracting vectors Vectors are added to one another and subtracted from one another with just one stipulation: The vectors have to be the same size.
The process of adding or subtracting vectors involves adding or subtracting the corresponding elements in the vectors, so you need to have a one-to-one match-up for the operations. Figure shows all three vectors. So, if you want to change a subtraction problem to an addition problem perhaps to change the order of the vectors in the operation , you rewrite the second vector in the problem in terms of its opposite. For example, changing the following subtraction problem to an addition problem, and rewriting the order, you have: Yes, of course the answers come out the same whether you subtract or change the second vector to its opposite.
The maneuvers shown here are for the structure or order of the problem and are used in various applications of vectors. Vectors with more than three rows also have magnitude, and the computation is the same no matter what the size of the vector.
The magnitude of vector v is designated with two sets of vertical lines, v , and the formula for computing the magnitude is where v1, v2 ,. The box measures 3 x 2 x 4 feet. How long a rod can you fit in the box, diagonally?
According to the formula for the magnitude of the vector whose numbers are the dimensions of the box, you can place a rod measuring about 5. Adjusting magnitude for scalar multiplication The magnitude of a vector is determined by squaring each element in the vector, finding the sum of the squares, and then computing the square root of that sum.
What happens to the magnitude of a vector, though, if you mul- tiply it by a scalar? Can you predict the magnitude of the new vector without going through all the computation if you have the magnitude of the original vector? Chapter 2: The Value of Involving Vectors 31 The magnitude of the new vector is three times that of the original. So it looks like all you have to do is multiply the original magnitude by the scalar to get the new magnitude.
Careful there! In mathematics, you need to be suspicious of results where someone gives you a bunch of numbers and declares that, because one example works, they all do. So, if you multiply a vector by a negative number, the value of the magnitude of the resulting vector is still going to be a positive number. Making it all right with the triangle inequality When dealing with the addition of vectors, a property arises involving the sum of the vectors. The theorem involving vectors, their magnitudes, and the sum of their magnitudes is called the triangle inequality or the Cauchy- Schwarz inequality named for the mathematicians responsible.
For any vectors u and v, the following, which says that the magnitude of the sum of vectors is always less than or equal to the sum of the magnitudes of the vectors, holds: Showing the inequality for what it is In Figure , you see two vectors, u and v, with terminal points x1,y1 and x2,y2 , respectively. The triangle inequality theorem says that the magnitude of the vector resulting from adding two vectors together is either smaller or sometimes the same as the sum of the magnitudes of the two vectors being added together.
Then I compare the magni- tude to the sum of the two separate magnitudes. The sums are mighty close, but the magnitude of the sum is smaller, as expected. You find the average of two numbers by adding them together and dividing by two. To find a geometric mean of two numbers, you just determine the square root of the product of the numbers. The geometric mean of a and b is while the arithmetic mean is For an example of how the arithmetic and geometric means of two numbers compare, consider the two numbers 16 and The geometric mean is the square root of the product of the numbers.
In this example, the geometric mean is slightly smaller than the arithmetic mean. In fact, the geometric mean is never larger than the arithmetic mean — the geometric mean is always smaller than, or the same as, the arithmetic mean.
I show you why this is so by using two very carefully selected vectors, u and v, which have elements that illustrate my statement. First, let Assume, also, that both a and b are positive numbers. To get to the last step, I used the commutative property of addition on the left changing the order and found that I had two of the same term. Now I square both sides of the inequality, divide each side by 2, square the binomial, distribute the 2, and simplify by subtracting a and b from each side: See!
The geometric mean of the two numbers, a and b, is less than or equal to the arithmetic mean of the same two numbers. Getting an inside scoop with the inner product The inner product of two vectors is also called its dot product. His birth, during a time of At one point, Cauchy responded to a request political upheaval, seemed to set the tone for from the then-deposed king, Charles X, to tutor the rest of his life.
His family was often visited by mathematicians Cauchy was raised in a political environment of the day — notably Joseph Louis Lagrange and was rather political and opinionated. He and Pierre-Simon Laplace — who encouraged was, more often than not, rather difficult in the young prodigy to be exposed to languages, his dealings with other mathematicians.
No first, and then mathematics. Cauchy was quick to publish his find- abandoned engineering for mathematics. At ings unlike some mathematicians, who tended a time when most jobs or positions for math- to sit on their discoveries , perhaps because of ematicians were as professors at universities, an advantage that he had as far as getting his Cauchy found it difficult to find such a position work in print.
He was married to Aloise de Bure, because of his outspoken religious and political the close relative of a publisher. The superscript T in the notation uTv means to transpose the vector u, to change its orientation. The reason should become crystal clear in the following section. Consider the two vectors shown in Figure , which are drawn perpendicular to one another and form a degree angle or right angle where their endpoints meet.
You can confirm that the rays forming the vectors are perpendicular to one another, using some basic algebra, because their slopes are negative reciprocals. The slope of a line is determined by finding the difference between the y-coor- dinates of two points on the line and dividing that difference by the difference between the corresponding x-coordinates of the points on that line. And, further, two lines are perpendicular form a right angle if the product of their slopes is —1.
So what does this have to do with vectors and their orthogonality? Read on. If the inner product of vectors u and v is equal to 0, then the vectors are perpendicular. Referring to the two vectors in Figure , you have Now, finding their inner product, Since the inner product is equal to 0, the rays must be perpendicular to one another and form a right angle. In Figure , you see the two vectors whose terminal points are 2,6 and —1,5.
Then put the numbers in their respective places in the formula: Using either a calculator or table of trigonometric functions, you find that the angle whose cosine is closest to 0. The angle formed by the two vectors is close to a degree angle. Matrices have their own arithmetic. What you think of when you hear multiplication has just a slight resemblance to matrix mul- tiplication.
Matrix algebra has identities, inverses, and operations. Getting Down and Dirty with Matrix Basics A matrix is made up of some rows and columns of numbers — a rectangular array of numbers. You have the same number of numbers in each row and the same number of numbers in each column.
The number of rows and col- umns in a matrix does not have to be the same. A vector is a matrix with just one column and one or more rows; a vector is also called a column vector. Matrices are generally named so you can distinguish one matrix from another in a discussion or text.
Nice, simple, capital letters are usually the names of choice for matrices: Matrix A has two rows and two columns, and Matrix B has four rows and six columns. The rectangular arrays of numbers are surrounded by a bracket to indicate that this is a mathematical structure called a matrix. The different positions or values in a matrix are called elements. The elements themselves are named with lowercase letters with subscripts.
Documents can only be sent to your Kindle devices from e-mail accounts that you added to your Approved Personal Document E-mail List. What's the problem with this file? Promotional spam Copyrighted material Offensive language or threatening Something else. Notify me of new posts via email. Menu Skip to content. Home About. There are everyday formulas and not-so-everyday formulas. There are familiar situations and situations that may be totally unfamiliar.
I also give you some graphing basics in this part. A picture is truly worth a thousand words, or, in the case of mathematics, a graph is worth an infinite number of points. You also find my choice for the ten most famous equations.
You may have other favorites, but these are my picks. Icons Used in This Book The little drawings in the margin of the book are there to draw your attention to specific text. Here are the icons I use in this book: To make everything work out right, you have to follow the basic rules of algebra or mathematics in general. Whenever I give you an algebra rule, I mark it with this icon.
An explanation of an algebraic process is fine, but an example of how the process works is even better. Paragraphs marked with the Remember icon help clarify a symbol or process.
I may discuss the topic in another section of the book, or I may just remind you of a basic algebra rule that I discuss earlier.
The Technical Stuff icon indicates a definition or clarification for a step in a process, a technical term, or an expression. The Warning icon alerts you to something that can be particularly tricky. Part IV is where the good stuff is — applications — things to do with all those good solutions. When the first edition of this book came out, my mother started by reading all the sidebars.
Why not? Studying algebra can give you some logical exercises. What a good place to use it, right here! Yes, you read that right. Algebra is poetry, deep meaning, and artistic expression. Welcome to algebra! Enjoy the adventure! Traveling abroad takes preparation and planning: You need to get your passport renewed, apply for a visa, pack your bags with the appropriate clothing, and arrange for someone to take care of your pets.
In order for the trip to turn out well and for everything to go smoothly, you need to prepare. The same is true of algebra: It takes preparation for the algebraic experience to turn out to be a meaningful one. Careful preparation prevents problems along the way and helps solve problems that crop up in the process. In this part, you find the essentials you need to have a successful algebra adventure.
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? 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. 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 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.
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. Aha algebra Dating back to about B. The Babylonians were solving three-term quadratic equations, while the Egyptians were more concerned with linear equations. The Hindus made further advances in about the sixth century A. In the seventh century, Brahmagupta of India provided general solutions to quadratic equations and had interesting takes on 0.
The Hindus regarded irrational numbers as actual numbers — although not everybody held to that belief. The sophisticated communication technology that exists in the world now was not available then, but early civilizations still managed to exchange information over the centuries.
Known as the Rhind Mathematical Papyrus after the Scotsman who purchased the 1-foot-wide, foot-long papyrus in Egypt in , the artifact is preserved in the British Museum — with a piece of it in the Brooklyn Museum.
Scholars determined that in B. The word aha designated the unknown. Can you solve this early Egyptian problem? It would be translated, using current. 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? 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. 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:. 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. This is my plan in this book, too. 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. Other rational numbers have decimals that repeat the same pattern, such , or. The horizontal bar over the and the 6 lets as you know that these numbers repeat forever.
In all cases, rational numbers can be written as fractions. 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. 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 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. 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. 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. You see distributions over expressions in Chapter 7. Addition and subtraction, on the other hand, separate terms from one another. By using an equation, tough problems can be reduced to easier problems and simpler answers.
See the chapters in Part III for more information on equations. Operations are addition, subtraction, multiplication, division, square roots, and so on. See Chapter 5 for more on operations.
An inequality is a comparison of two values. For more on inequalities, turn to Chapter 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.
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. The value of x depends on what a, b, and c are assigned to be. 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.
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. 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.
It also is used to indicate a positive number. The values being multiplied together are the multipliers or factors; the result is the product. The grouping symbols are used when you need to contain many terms or a messy expression. The result is the quotient. See Chapter 4 for more on square roots. For more on absolute value, turn to Chapter 2.
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. Grouping symbols tell you that you have to deal with the terms inside the grouping symbols before you deal with the larger problem. The examples in Chapters 5 and 7 should clear up any questions you may have. 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 Taking on algebraic tasks Algebra involves symbols, such as variables and operation signs, which are the tools that you can use to make algebraic expressions more usable and readable.
These things go hand in hand with simplifying, factoring, and solving problems, which are easier to solve if broken down into basic parts. Using symbols is actually much easier than wading through a bunch of words.
See Part II for more on factoring. In algebra, it means to figure out what the variable stands for. But solving these equations is just a means to an end. The real beauty of algebra shines when you solve some problem in real life — a practical application. Are you ready for these two words: story problems? Story problems are the whole point of doing algebra. Yes, some folks are like that.
Simplify, factor, solve, check. Lucky you. This chapter tells you how to add, subtract, multiply, and divide signed numbers, no matter whether all the numbers are all the same sign or a combination of positive and negative. Showing Some Signs Early on, mathematicians realized that using plus and minus signs and making rules for their use would be a big advantage in their number world.
After all, positive and negative numbers are related to one another, and inserting a minus sign in front of a number works well.
Negative numbers have positive counterparts and vice versa. Additive inverses are always the same distance from 0 in opposite directions on the number line. For example, the. If you were to arrange a tug-of-war between positive and negative numbers, the positive numbers would line up on the right side of 0, as shown in Figure Figure Positive numbers 0 getting larger to the right.
Check out the difference between freezing water and boiling water to see how much more positive a number can be! Making the most of negative numbers The concept of a number less than 0 can be difficult to grasp.
Think of entering the ground floor of a large government building. You go to the elevator and have to choose between going up to the first, second, third, or fourth floors, or going down to the first, second, third, fourth, or fifth subbasement down where all the secret stuff is. The farther you are from the ground floor, the farther the number of that floor is from 0.
The second subbasement could be called floor —2, but that may not be a good number for a floor. On a line with 0 in the middle, negative numbers line up on the left, as shown in Figure Figure Negative numbers getting smaller to the left.
This situation can get confusing because you may think that — is bigger than — When comparing negative numbers, the number closer to 0 is the bigger or greater number.
Comparing positives and negatives Although my mom always told me not to compare myself to other people, comparing numbers to other numbers is often useful. Remember: Positive numbers are always bigger than negative numbers. Figure Positive and negative numbers on -5 a number line. I keep comparing numbers to see how far they are from 0. Is 0 positive or negative? Zero has the unique distinction of being neither positive nor negative.
Zero separates the positive numbers from the negative ones — what a job! Going In for Operations Operations in algebra are nothing like operations in hospitals. Well, you get to dissect things in both, but dissecting numbers is a whole lot easier and a lot less messy than dissecting things in a hospital. Algebra is just a way of generalizing arithmetic, so the operations and rules used in arithmetic work the same for algebra.
Some new operations do crop up in algebra, though, just to make things more interesting than adding, subtracting, multiplying, and dividing. I introduce three of those new operations after explaining the difference between a binary operation and a non-binary operation. Breaking into binary operations Bi means two.
A bicycle has two wheels. A bigamist has two spouses. A binary operation involves two numbers. Addition, subtraction, multiplication, and division are all binary operations because you need two numbers to perform them. You need another number. A non-binary operation performs a task and spits out the answer. Square roots are non-binary operations. You find by performing this operation on just one number see Chapter 4 for more on square roots.
In the following sections, I show you three non-binary operations. Getting it absolutely right with absolute value One of the most frequently used non-binary operations is the one that finds the absolute value of a number — its value without a sign.
The absolute value tells you how far a number is from 0. The symbol for absolute value is two vertical bars:. Getting the facts straight with factorial The factorial operation looks like someone took you by surprise.