Algebra Know-It-ALL Beginner to Advanced, and Everything in Between

Book Preface

If you want to improve your understanding of algebra, then this book is for you. It can supplement standard texts at the middle-school and high-school levels. It can also serve as a self-teaching or home-schooling supplement. The essential prerequisite is a solid background in arithmetic. It will help if you’ve had some pre-algebra as well. This book contains three major sections. Part 1 involves numbers, sets, arithmetic operations, and basic equations. Part 2 is devoted to first-degree equations, relations, functions, and systems of linear equations. Part 3 deals with quadratic, cubic, and higher-degree equations, and introduces you to logarithms, exponentials, and systems of nonlinear equations. Chapters 1 through 9, 11 through 19, and 21 through 29 end with practice exercises. You may (and should) refer to the text as you solve these problems. Worked-out solutions appear in Apps. A, B, and C. Often, these solutions do not represent the only way a problem can be figured out. Feel free to try alternatives! Chapters 10, 20, and 30 contain question-and-answer sets that finish up Parts 1, 2, and 3, respectively. These chapters will help you review the material. A multiple-choice final exam concludes the course. Don’t refer to the text while taking the exam. The questions in the exam are more general (and less time consuming) than the practice exercises at the ends of the chapters. The final exam is designed to test your grasp of the concepts, not to see how well you can execute calculations. The correct answers are listed in App. D. In my opinion, middle-school and high-school students aren’t sufficiently challenged in mathematics these days. I think that most textbooks place too much importance on “churning out answers,” and often fail to explain how and why you get those answers. I wrote this book to address these problems. The presentation sometimes gets theoretical, but I’ve tried to introduce the language gently so you won’t get lost in a wilderness of jargon. Many of the examples and problems are easy, some take work, and a few are designed to make you think hard. If you complete one chapter per week, you’ll get through this course in a school year. But don’t hurry. Proceed at your own pace. When you’ve finished this book, I highly recommend McGraw-Hill’s Algebra Demystified and College Algebra Demystified, both by Rhonda Huettenmueller, for further study.
Stan Gibilisco

Algebra is a science of numbers. To work with numbers, you need symbols to represent them. The way these symbols relate to actual quantities is called a numeration system. In this chapter, you’ll learn about numeration systems for whole-unit quantities such as 4, 8, 1,509, or 1,580,675. Fractions, negative numbers, and more exotic numbers will come up later.
Fingers and Sticks Throughout history, most cultures developed numeration systems based on the number of fingers and thumbs on human hands. The word digit derives from the Latin word for “finger.” This is no accident. Fingers are convenient for counting, at least when the numbers are small!

Number or numeral? The words number and numeral are often used as if they mean the same thing. But they’re different. A number is an abstraction. You can’t see or feel a number. A numeral is a tangible object, or a group of objects, that represents a number. Suppose you buy a loaf of bread cut into eighteen slices. You can consider the whole sliced-up loaf as a numeral that represents the number eighteen, and each slice as a digit in that numeral. You can’t eat the number eighteen, but you can eat the bread. In this chapter, when we write about numbers as quantities, let’s write them out fully in words, like eighteen or forty-five or three hundred twenty-one. When we want to write down a numeral, it’s all right to put down 18 or 45 or 321, but we have to be careful about this sort of thing. When you see a large quantity written out in full here, keep this in mind: It means we’re dealing with a number, not a numeral.
Figuring with fingers Imagine it’s the afternoon of the twenty-fourth day of July. You have a doctor’s appointment for the afternoon of the sixth of August. How many days away is your appointment?

A calculator won’t work very well to solve this problem. Try it and see! You can’t get the right answer by any straightforward arithmetic operation on twenty-four and six. If you attack this problem as I would, you’ll count out loud starting with tomorrow, July twenty-fifth (under your breath): “twenty-five, twenty-six, twenty-seven, twenty-eight, twenty-nine, thirty, thirty-one, one, two, three, four, five, six!” While jabbering away, I would use my fingers to count along or make “hash marks” on a piece of paper (Fig. 1-1). You might use a calendar and point to the days one at a time as you count them out. However you do it, you’ll come up with thirteen days if you get it right. But be careful! This sort of problem is easy to mess up. Don’t be embarrassed if you find yourself figuring out simple problems like this using your fingers or other convenient objects. You’re making sure that you get the right answer by using numerals to represent the numbers. Numerals are tailor-made for solving number problems because they make abstract things easy to envision.

Toothpicks on the table Everyone has used “hash marks” to tally up small numbers. You can represent one item by a single mark and five items by four marks with a long slash. You might use objects such as toothpicks to create numerals in a system that expands on this idea, as shown in Fig. 1-2. You can represent ten by making a capital letter T with two toothpicks. You can represent fifty by using three toothpicks to make a capital letter F. You can represent a hundred by making a capital letter H with three toothpicks. This lets you express rather large numbers such as seventy-four or two hundred fifty-three without having to buy several boxes of toothpicks and spend a lot of time laying them down. In this system, any particular arrangement of sticks is a numeral. You can keep going this way, running an F and H together to create a symbol that represents five hundred. You can run a T and an H together to make a symbol that represents a thousand. How about ten thousand? You could stick another T onto the left-hand end of the symbol for a thousand, or you could run two letters H together to indicate that it’s a hundred hundred! Use your imagination. That’s what mathematicians did when they invented numeration systems in centuries long past.

Are you confused? If the toothpick numeral system puzzles you, don’t feel bad. It’s awkward. It’s impractical for expressing gigantic numbers. People aren’t used to counting in blocks of five or fifty or five hundred. It’s easier to go straight from blocks of one to blocks of ten, and then from ten to a hundred, then to a thousand, then to ten thousand, and so on. But using blocks of five, fifty, five hundred and so on, in addition to the traditional multiples of ten, conserves toothpicks. Here’s a challenge! Using toothpick numerals represent the number seven hundred seventy-seven in two different ways. Make sure one of your arrangements is the most “elegant” possible way to represent seven hundred seventyseven, meaning that it uses the smallest possible number of toothpicks. Solution Figure 1-3 shows two ways you can represent this number. In order to represent five hundred, you build the F and the H together so they’re a single connected pattern of sticks. The arrangement on top is the most “elegant” possible numeral. You can represent seven hundred seventy-seven in more ways than just the two shown here. You can make numerals that are far more “inelegant” than the bottom arrangement. The worst possible approach is to lay down seven hundred seventy-seven toothpicks side-by-side.