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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. 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. 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 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...)
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Uspoređivanje racionalnih brojeva Zbrajanje i oduzimanje racionalnih brojeva Dijeljenje racionalnih brojeva Racionalni brojevi - uvod Racionalni brojevi su svi negativni razlomci, nula i pozitivni razlomci. Skup racionalnih brojeva označava se s Q. Svaki razlomak možemo zapisati u obliku , gdje je a cijeli broj, a b prirodni broj. Oblik razlomka je standardni oblik racionalnog broja. Za svaki racionalni broj r vrijedi: Racionalne brojeve možemo pridruživati točkama pravca. Racionalni brojevi su suprotni brojevi. Modul racionalnog broja (apsolutna vrijednost) je racionalni broj koji za brojnik ima modul brojnika, a za nazivnik modul nazivnika zadanog racionalnog broja. Primjer: Racionalni brojevi mogu se proširivati i skraćivat, a pri tom vrijedi: