Review. A Look Ahead. Outline

Transcription

1 SULLMC00_057759X.QXD /0/06 :8 PM Page R Review A Look Ahead Outline Chapter R, as the title states, contains review material. Your instructor may choose to cover all or part of it as a regular chapter at the beginning of your course or later as a just-in-time review when the content is required. Regardless, when information in this chapter is needed, a specific reference to this chapter will be made so you can review. R. R. R. R.4 R.5 R.6 R.7 R.8 Real Numbers Algebra Essentials Geometry Essentials Polynomials Factoring Polynomials Synthetic Division Rational Epressions nth Roots; Rational Eponents Chapter Review Chapter Test

2 CHAPTER R Review R. Real Numbers PREPARING FOR THIS BOOK Before getting started, read To the Student on page vii at the front of this book. OBJECTIVES Work with Sets (p. ) Classify Numbers (p. 4) Evaluate Numerical Epressions (p. 8) 4 Work with Properties of Real Numbers (p. 9) Work with Sets A set is a well-defined collection of distinct objects. The objects of a set are called its elements. By well-defined, we mean that there is a rule that enables us to determine whether a given object is an element of the set. If a set has no elements, it is called the empty set, or null set, and is denoted by the symbol. For eample, the set of digits consists of the collection of numbers 0,,,, 4, 5, 6, 7, 8, and 9. If we use the symbol D to denote the set of digits, then we can write D = 50,,,, 4, 5, 6, 7, 8, 96 In this notation, the braces 5 6 are used to enclose the objects, or elements, in the set. This method of denoting a set is called the roster method. A second way to denote a set is to use set-builder notation, where the set D of digits is written as D { is a digit} Read as "D is the set of all such that is a digit." EXAMPLE Using Set-builder Notation and the Roster Method (a) E = 5ƒ is an even digit6 = 50,, 4, 6, 86 (b) O = 5ƒ is an odd digit6 = 5,, 5, 7, 96 Because the elements of a set are distinct, we never repeat elements. For eample, we would never write 5,,, 6; the correct listing is 5,, 6. Because a set is a collection, the order in which the elements are listed is immaterial. 5,, 6, 5,, 6, 5,, 6,and so on, all represent the same set. If every element of a set A is also an element of a set B, then we say that A is a subset of B and write A 8 B. If two sets A and B have the same elements, then we say that A equals B and write A = B. For eample, 5,, 6 8 5,,, 4, 56 and 5,, 6 = 5,, 6. DEFINITION If A and B are sets, the intersection of A with B, denoted A B, is the set consisting of elements that belong to both A and B.The union of A with B, denoted A B, is the set consisting of elements that belong to either A or B, or both. EXAMPLE Finding the Intersection and Union of Sets Let A = 5,, 5, 86, B = 5, 5, 76, and C = 5, 4, 6, 86. Find: (a) A B (b) A B (c) B A C

3 SECTION R. Real Numbers Solution (a) (b) (c) A B = 5,, 5, 86 5, 5, 76 = 5, 56 A B = 5,, 5, 86 5, 5, 76 = 5,, 5, 7, 86 B A C = 5, 5, 76 5,, 5, 86 5, 4, 6, 864 = 5, 5, 76 5,,, 4, 5, 6, 86 = 5, 56 Now Work PROBLEM Usually, in working with sets, we designate a universal set U, the set consisting of all the elements that we wish to consider. Once a universal set has been designated, we can consider elements of the universal set not found in a given set. DEFINITION If A is a set, the complement of A, denoted A, is the set consisting of all the elements in the universal set that are not in A.* EXAMPLE Finding the Complement of a Set If the universal set is U = 5,,, 4, 5, 6, 7, 8, 96 and if A = 5,, 5, 7, 96, then A = 5, 4, 6, 86. It follows that A A = U and A A =. Do you see why? Now Work PROBLEM 7 Figure A B C Universal set It is often helpful to draw pictures of sets. Such pictures, called Venn diagrams, represent sets as circles enclosed in a rectangle, which represents the universal set. Such diagrams often help us to visualize various relationships among sets. See Figure. If we know that A 8 B, we might use the Venn diagram in Figure (a). If we know that A and B have no elements in common, that is, if A B =, we might use the Venn diagram in Figure (b). The sets A and B in Figure (b) are said to be disjoint. Figure Universal set Universal set A B A B (a) A B subset (b) A B disjoint sets Figures (a), (b), and (c) use Venn diagrams to illustrate the definitions of intersection, union, and complement, respectively. Figure Universal set Universal set Universal set A B A B A A (a) A B intersection (b) A B union (c) A complement * Some books use the notation A for the complement of A.

4 4 CHAPTER R Review Classify Numbers It is helpful to classify the various kinds of numbers that we deal with as sets. The counting numbers, or natural numbers, are the numbers in the set 5,,, 4 Á 6. (The three dots, called an ellipsis, indicate that the pattern continues indefinitely.) As their name implies, these numbers are often used to count things. For eample, there are 6 letters in our alphabet; there are 00 cents in a dollar. The whole numbers are the numbers in the set 50,,,, Á 6, that is, the counting numbers together with 0. DEFINITION The integers are the set of numbers 5 Á, -, -, -, 0,,,, Á 6. These numbers are useful in many situations. For eample, if your checking account has $0 in it and you write a check for $5, you can represent the current balance as -$5. Notice that the set of counting numbers is a subset of the set of whole numbers. Each time we epand a number system, such as from the whole numbers to the integers, we do so in order to be able to handle new, and usually more complicated, problems. The integers allow us to solve problems requiring both positive and negative counting numbers, such as profit/loss, height above/below sea level, temperature above/below 0 F, and so on. But integers alone are not sufficient for all problems. For eample, they do not answer the question What part of a dollar is 8 cents? To answer such a question, 8 we enlarge our number system to include rational numbers. For eample, answers the question What part of a dollar is 8 cents? 00 a DEFINITION A rational number is a number that can be epressed as a quotient of two b integers.the integer a is called the numerator, and the integer b, which cannot be 0, is called the denominator. The rational numbers are the numbers in the set 5 ƒ = a where a, b are integers and b Z 06. b, 0 00 a Eamples of rational numbers are and Since for any 4, - 4, 5,,. = a integer a, it follows that the set of integers is a subset of the set of rational numbers. Rational numbers may be represented as decimals. For eample, the rational 7 numbers - and may be represented as decimals by merely carrying out 4, 5,, 66 the indicated division: 4 = =.5 - = Á = = Á = Notice that the decimal representations of and terminate, or end. The decimal 4 7 representations of - and do not terminate, but they do ehibit a pattern of repetition. For - 66 the 6 repeats indefinitely, as indicated by the bar over the 6; for, 7 the block 06 repeats indefinitely, as indicated by the bar over the 06. It can be 66, shown that every rational number may be represented by a decimal that either terminates or is nonterminating with a repeating block of digits, and vice versa. On the other hand, some decimals do not fit into either of these categories. Such decimals represent irrational numbers. Every irrational number may be represented by a decimal that neither repeats nor terminates. In other words, irrational numbers a cannot be written in the form where a, b are integers and b Z 0. b,

5 SECTION R. Real Numbers 5 Irrational numbers occur naturally. For eample, consider the isosceles right triangle whose legs are each of length. See Figure 4. The length of the hypotenuse is, an irrational number. Also, the number that equals the ratio of the circumference C to the diameter d of any circle, denoted by the symbol p (the Greek letter pi), is an irrational number. See Figure 5. Figure 4 Figure 5 p = C d C d DEFINITION The set of real numbers is the union of the set of rational numbers with the set of irrational numbers. Figure 6 shows the relationship of various types of numbers.* Figure 6 Irrational numbers Rational numbers Integers Whole numbers Natural or counting numbers Real numbers EXAMPLE 4 Classifying the Numbers in a Set List the numbers in the set 5-, 4, 0.,, p, 0,.555 Á where the block 5 repeats6 that are (a) Natural numbers (b) Integers (c) Rational numbers (d) Irrational numbers (e) Real numbers Solution (a) 0 is the only natural number. (b) - and 0 are integers. (c) -, 0, 4, 0., and.555 Á are rational numbers. (d) and p are irrational numbers. (e) All the numbers listed are real numbers. Now Work PROBLEM * The set of real numbers is a subset of the set of comple numbers. We discuss comple numbers in Chapter, Section..

6 6 CHAPTER R Review Approimations Every decimal may be represented by a real number (either rational or irrational), and every real number may be represented by a decimal. In practice, the decimal representation of an irrational number is given as an approimation. For eample, using the symbol L (read as approimately equal to ), we can write L.44 p L.46 In approimating decimals, we either round off or truncate to a given number of decimal places.* The number of places establishes the location of the final digit in the decimal approimation. Truncation: Drop all the digits that follow the specified final digit in the decimal. Rounding: Identify the specified final digit in the decimal. If the net digit is 5 or more, add to the final digit; if the net digit is 4 or less, leave the final digit as it is. Then truncate following the final digit. EXAMPLE 5 Solution Approimating a Decimal to Two Places Approimate to two decimal places by (a) Truncating (b) Rounding For , the final digit is 8, since it is two decimal places from the decimal point. (a) To truncate, we remove all digits following the final digit 8. The truncation of to two decimal places is (b) The digit following the final digit 8 is the digit 7. Since 7 is 5 or more, we add to the final digit 8 and truncate. The rounded form of to two decimal places is EXAMPLE 6 Approimating a Decimal to Two and Four Places Rounded Rounded Truncated Truncated to Two to Four to Two to Four Decimal Decimal Decimal Decimal Number Places Places Places Places (a) (b) (c) Now Work PROBLEM 7 Calculators Calculators are finite machines. As a result, they are incapable of displaying decimals that contain a large number of digits. For eample, some calculators are capable of displaying only eight digits. When a number requires more than eight digits, * Sometimes we say correct to a given number of decimal places instead of truncate.

7 SECTION R. Real Numbers 7 the calculator either truncates or rounds. To see how your calculator handles decimals, divide by. How many digits do you see? Is the last digit a 6 or a 7? If it is a6, your calculator truncates; if it is a 7, your calculator rounds. There are different kinds of calculators. An arithmetic calculator can only add, subtract, multiply, and divide numbers; therefore, this type is not adequate for this course. Scientific calculators have all the capabilities of arithmetic calculators and also contain function keys labeled ln, log, sin, cos, tan, y, inv, and so on. As you proceed through this tet, you will discover how to use many of the function keys. Graphing calculators have all the capabilities of scientific calculators and contain a screen on which graphs can be displayed. For those who have access to a graphing calculator, we have included comments, eamples, and eercises marked with a, indicating that a graphing calculator is required. We have also included an appendi that eplains some of the capabilities of a graphing calculator. The comments, eamples, and eercises may be omitted without loss of continuity, if so desired. Operations In algebra, we use letters such as, y, a, b, and c to represent numbers. The symbols used in algebra for the operations of addition, subtraction, multiplication, and division are +, -, #, and >. The words used to describe the results of these operations are sum, difference, product, and quotient. Table summarizes these ideas. Table Operation Symbol Words Addition Subtraction Multiplication Division a + b a - b a # b, (a) # b, a # (b), (a) # (b), ab, (a)b, a(b), (a)(b) a>b or a b Sum: a plus b Difference: a minus b Product: a times b Quotient: a divided by b In algebra, we generally avoid using the multiplication sign * and the division sign, so familiar in arithmetic. Notice also that when two epressions are placed net to each other without an operation symbol, as in ab, or in parentheses, as in ab, it is understood that the epressions, called factors, are to be multiplied. We also prefer not to use mied numbers in algebra. When mied numbers are used, addition is understood; for eample, means + In algebra, use of a 4 4. mied number may be confusing because the absence of an operation symbol between two terms is generally taken to mean multiplication. The epression is 4 therefore written instead as.75 or as 4. The symbol =, called an equal sign and read as equals or is, is used to epress the idea that the number or epression on the left of the equal sign is equivalent to the number or epression on the right. EXAMPLE 7 Writing Statements Using Symbols (a) The sum of and 7 equals 9. In symbols, this statement is written as + 7 = 9. (b) The product of and 5 is 5. In symbols, this statement is written as # 5 = 5. Now Work PROBLEM 9

8 8 CHAPTER R Review Evaluate Numerical Epressions Consider the epression + # 6. It is not clear whether we should add and to get 5, and then multiply by 6 to get 0; or first multiply and 6 to get 8, and then add to get 0. To avoid this ambiguity, we have the following agreement. In Words Multiply first, then add. We agree that whenever the two operations of addition and multiplication separate three numbers, the multiplication operation always will be performed first, followed by the addition operation. For + # 6, we have + # 6 = + 8 = 0 EXAMPLE 8 Solution Finding the Value of an Epression Evaluate each epression. (a) + 4 # 5 (b) 8 # + (c) (a) (c) + # + 4 # 5 = + 0 = (b) c Multiply first + # = + 4 = 6 8 # + = 6 + = 7 c Multiply first Now Work PROBLEM 5 To first add and 4 and then multiply the result by 5, we use parentheses and write + 4 # 5. Whenever parentheses appear in an epression, it means perform the operations within the parentheses first! EXAMPLE 9 Finding the Value of an Epression (a) (b) 5 + # 4 = 8 # 4 = # 8 - = 9 # 6 = 54 When we divide two epressions, as in it is understood that the division bar acts like parentheses; that is, = The following list gives the rules for the order of operations. Rules for the Order of Operations. Begin with the innermost parentheses and work outward. Remember that in dividing two epressions the numerator and denominator are treated as if they were enclosed in parentheses.. Perform multiplications and divisions, working from left to right.. Perform additions and subtractions, working from left to right.

9 SECTION R. Real Numbers 9 EXAMPLE 0 Solution Finding the Value of an Epression Evaluate each epression. (a) 8 # + (c) # 7 (a) 8 # + = 6 + = 9 c Multiply first (b) c Parentheses first + 5 (c) + 4 # = = 7 0 (d) (b) (d) 5 # = 5 # 7 + = 5 + = 7 c Multiply before adding # = # 64 5 # # = = + 64 = 8 Figure 7 Be careful if you use a calculator. For Eample 0(c), you need to use parentheses. See Figure 7.* If you don t, the calculator will compute the epression giving a wrong answer # 7 = =.5 Now Work PROBLEMS 57 AND 65 4 Work with Properties of Real Numbers We have used the equal sign to mean that one epression is equivalent to another. Four important properties of equality are listed net. In this list, a, b, and c represent real numbers.. The refleive property states that a number always equals itself; that is, a = a.. The symmetric property states that if a = b then b = a.. The transitive property states that if a = b and b = c then a = c. 4. The principle of substitution states that if a = b then we may substitute b for a in any epression containing a. Now, let s consider some other properties of real numbers. We begin with an eample. EXAMPLE Commutative Properties (a) + 5 = = = 5 + (b) # = 6 # = 6 # = # This eample illustrates the commutative property of real numbers, which states that the order in which addition or multiplication takes place will not affect the final result. * Notice that we converted the decimal to its fraction form. Consult your manual to see how your calculator does this.

10 0 CHAPTER R Review Commutative Properties a + b = b + a a # b = b # a (a) (b) Here, and in the properties listed net and on pages, a, b, and c represent real numbers. EXAMPLE Associative Properties (a) = + 7 = = = = (b) # # 4 = # = 4 # # 4 = 6 # 4 = 4 # # 4 = # # 4 The way we add or multiply three real numbers will not affect the final result. Epressions such as and # 4 # 5 present no ambiguity, even though addition and multiplication are performed on one pair of numbers at a time. This property is called the associative property. Associative Properties a + b + c = a + b + c = a + b + c a # b # c = a # b # c = a # b # c (a) (b) The net property is perhaps the most important. Distributive Property a # b + c = a # b + a # c a + b # c = a # c + b # c (a) (b) The distributive property may be used in two different ways. EXAMPLE Distributive Property (a) (b) (c) # + = # + # = = + 5 = 8 Use to remove parentheses. Use to combine two epressions. + + = = = + ( + ) + 6 = Now Work PROBLEM 87 The real numbers 0 and have unique properties. EXAMPLE 4 Identity Properties (a) = = 4 (b) # = # = The properties of 0 and illustrated in Eample 4 are called the identity properties.

11 SECTION R. Real Numbers Identity Properties 0 + a = a + 0 = a a # = # a = a (4a) (4b) We call 0 the additive identity and the multiplicative identity. For each real number a, there is a real number -a, called the additive inverse of a, having the following property: Additive Inverse Property a + -a = -a + a = 0 (5a) EXAMPLE 5 Finding an Additive Inverse (a) The additive inverse of 6 is -6, because = 0. (b) The additive inverse of -8 is --8 = 8, because = 0. The additive inverse of a, that is, -a, is often called the negative of a or the opposite of a.the use of such terms can be dangerous, because they suggest that the additive inverse is a negative number, which may not be the case. For eample, the additive inverse of -, or --, equals, a positive number. For each nonzero real number a, there is a real number called the multiplicative inverse of a, having the following a, property: Multiplicative Inverse Property a # a = a # a = if a Z 0 (5b) The multiplicative inverse a reciprocal of a. of a nonzero real number a is also referred to as the EXAMPLE 6 Finding a Reciprocal (a) The reciprocal of 6 is because 6 # 6, 6 =. (b) The reciprocal of - is because -, (c) The reciprocal of is because, - # - =. # =. With these properties for adding and multiplying real numbers, we can now define the operations of subtraction and division as follows: DEFINITION The difference a - b, also read a less b or a minus b, is defined as a - b = a + -b (6)

12 CHAPTER R Review To subtract b from a, add the opposite of b to a. a DEFINITION If b is a nonzero real number, the quotient also read as a divided by b b, or the ratio of a to b, is defined as a b = a # b if b Z 0 (7) EXAMPLE 7 Working with Differences and Quotients (a) 8-5 = = (b) 4-9 = = -5 (c) 5 8 = 5 # 8 In Words The result of multiplying by zero is zero. For any number a, the product of a times 0 is always 0; that is, Multiplication by Zero a # 0 = 0 (8) For a nonzero number a, Division Properties 0 a = 0 a a = if a Z 0 (9) NOTE Division by 0 is not defined. One reason is to avoid the following difficulty: means to 0 find such that 0 # But 0 # = =. equals 0 for all, so there is no unique number such that 0 =. Rules of Signs a-b = -ab --a = a -ab = -ab a -b = -a b = - a b -a-b = ab -a -b = a b (0) EXAMPLE 8 Applying the Rules of Signs (a) (c) (e) - = - # = -6 - = - = - - = # = - - (b) (d) --5 = # 5 = = 4 9

13 SECTION R. Real Numbers Cancellation Properties ac = bc implies a = b if c Z 0 ac bc = a if b Z 0, c Z 0 b () EXAMPLE 9 NOTE We follow the common practice of using slash marks to indicate cancellations. Using the Cancellation Properties (a) If = 6, then (b) 8 = # 6 # 6 = c Cancel the 6 s. = 6 = # = Factor 6. Cancel the s. In Words If a product equals 0, then one or both of the factors is 0. Zero-Product Property If ab = 0, then a = 0, or b = 0, or both. () EXAMPLE 0 Using the Zero-Product Property If = 0, then either = 0 or = 0. Since Z 0, it follows that = 0. Arithmetic of Quotients a b + c d = ad bd + bc bd a # c b d = ac a b c d = ad + bc bd if b Z 0, d Z 0 bd if b Z 0, d Z 0 = a b # d c = ad bc if b Z 0, c Z 0, d Z 0 () (4) (5) EXAMPLE Adding, Subtracting, Multiplying, and Dividing Quotients (a) + 5 = # c By equation () (b) 5 - = 5 + a - b = c c By equation (6) By equation (0) = # + 5 # - c # + # 5 # = # + # 5 5 # = By equation () # = = = - 5 = - 5

14 4 CHAPTER R Review NOTE Slanting the cancellation marks in different directions for different factors, as shown here, is a good practice to follow, since it will help in checking for errors. (c) (d) 8 # 5 4 = 8 # = 0 c c By equation (4) By equation () # = # 4 # # 5 4 # 4 # = # 5 = 5 # 9 7 = # 9 æ By equation (5) c 5 # = By equation (4) EXAMPLE NOTE In writing quotients, we shall follow the usual convention and write the quotient in lowest terms. That is, we write it so that any common factors of the numerator and the denominator have been removed using the cancellation properties, equation (). As eamples, 90 4 = 5 # 6 4 # = = 4 # 6 # # # 6 # = 4 Z 0 Now Work PROBLEMS 67, 7, AND 8 Sometimes it is easier to add two fractions using least common multiples (LCM). The LCM of two numbers is the smallest number that each has as a common multiple. Finding the Least Common Multiple of Two Numbers Find the least common multiple of 5 and. Solution To find the LCM of 5 and, we look at multiples of 5 and. 5, 0, 45, 60, 75, 90, 05, 0, Á, 4, 6, 48, 60, 7, 84, 96, 08, 0, Á The common multiples are in blue. The least common multiple is 60. EXAMPLE Solution Using the Least Common Multiple to Add Two Fractions Find: We use the LCM of the denominators of the fractions and rewrite each fraction using the LCM as a common denominator. The LCM of the denominators ( and 5) is 60. Rewrite each fraction using 60 as the denominator = 8 # # 5 5 = = 60 = = 9 0 Now Work PROBLEM 75

15 SECTION R. Real Numbers 5 Historical Feature The real number system has a history that stretches back at least to the ancient Babylonians (800 BC). It is remarkable how much the ancient Babylonian attitudes resemble our own. As we stated in the tet, the fundamental difficulty with irrational numbers is that they cannot be written as quotients of integers or, equivalently, as repeating or terminating decimals. The Babylonians wrote their numbers in a system based on 60 in the same way that we write ours based on 0. They would carry as many places for p as the accuracy of the problem demanded, just as we now use p L 7 depending on how accurate we need to be. Things were very different for the Greeks, whose number system allowed only rational numbers. When it was discovered that was not a rational number, this was regarded as a fundamental flaw in the number concept. So serious was the matter that the Pythagorean Brotherhood (an early mathematical society) is said to have drowned one of its members for revealing this terrible secret. Greek mathematicians then Historical Problems or p L.46 or p L.459 or p L The Babylonian number system was based on 60. Thus,0 means + 0 and 4,5,4 means 60 =.5, = = Á. What are the following numbers in Babylonian notation? 5 (a) (b) 6 turned away from the number concept, epressing facts about whole numbers in terms of line segments. In astronomy, however, Babylonian methods, including the Babylonian number system, continued to be used. Simon Stevin (548 60), probably using the Babylonian system as a model, invented the decimal system, complete with rules of calculation, in 585. [Others, for eample, al-kashi of Samarkand (d. 49), had made some progress in the same direction.] The decimal system so effectively conceals the difficulties that the need for more logical precision began to be felt only in the early 800s. Around 880, Georg Cantor (845 98) and Richard Dedekind (8 96) gave precise definitions of real numbers. Cantor s definition, although more abstract and precise, has its roots in the decimal (and hence Babylonian) numerical system. Sets and set theory were a spin-off of the research that went into clarifying the foundations of the real number system. Set theory has developed into a large discipline of its own, and many mathematicians regard it as the foundation upon which modern mathematics is built. Cantor s discoveries that infinite sets can also be counted and that there are different sizes of infinite sets are among the most astounding results of modern mathematics.. What are the following Babylonian numbers when written as fractions and as decimals? (a),0 (b) 4,5,0 (c),8,9,44 R. Assess Your Understanding Concepts and Vocabulary. The numbers in the set and b Z 06, are called numbers. 5ƒ = a, where a, b are integers 5. True or False Rational numbers have decimals that either b. The value of the epression # 6 - is.. The fact that + = + is a consequence of the Property. 4. The product of 5 and + equals 6 may be written as. terminate or are nonterminating with a repeating block of digits. 6. True or False The Zero-Product Property states that the product of any number and zero equals zero. 7. True or False The least common multiple of and 8 is True or False No real number is both rational and irrational. Skill Building In Problems 9 0, use U universal set 50,,,, 4, 5, 6, 7, 8, 96, A = 5,, 4, 5, 96, B = 5, 4, 6, 7, 86, and C = 5,, 4, 66 to find each set. 9. A B 0. A C. A B. A C. A B C 4. A B C 5. A 6. C 7. A B 8. B C 9. A B 0. B C

16 6 CHAPTER R Review In Problems 6, list the numbers in each set that are (a) Natural numbers, (b) Integers, (c) Rational numbers, (d) Irrational numbers, (e) Real numbers.. A = e -6,, -. Á the s repeat, p,, 5 f. B = e - 5, Á the block 06 repeats,.5, 0,, 5 f. C = e0,,,, f 4. D = 5-, -., -., E = e, p, +, p + f 6. F = e -, p +, + 0. f In Problems 7 8, approimate each number (a) rounded and (b) truncated to three decimal places In Problems 9 48, write each statement using symbols. 9. The sum of and equals The product of 5 and equals The sum of and is the product of and The sum of and y is the sum of and. 4. The product of and y is the sum of and. 44. The product of and is the product of 4 and The difference less equals The difference less y equals The quotient divided by is The quotient divided by is 6. In Problems 49 86, evaluate each epression # # # 5 + # # # # # # # # 4-6 # # # # 86. # # In Problems 87 98, use the Distributive Property to remove the parentheses a 9. a b 6 b

17 SECTION R. Algebra Essentials 7 Discussion and Writing 99. Eplain to a friend how the Distributive Property is used to justify the fact that + = Eplain to a friend why whereas + # 4 = 0. + # 4 = 4, # # # Eplain why # 4 is not equal to Eplain why is not equal to Is subtraction commutative? Support your conclusion with an eample. 04. Is subtraction associative? Support your conclusion with an eample. 05. Is division commutative? Support your conclusion with an eample. 06. Is division associative? Support your conclusion with an eample. 07. If =, why does =? 08. If = 5, why does + = 0? 09. Are there any real numbers that are both rational and irrational? Are there any real numbers that are neither? Eplain your reasoning. 0. Eplain why the sum of a rational number and an irrational number must be irrational.. A rational number is defined as the quotient of two integers. When written as a decimal, the decimal will either repeat or terminate. By looking at the denominator of the rational number, there is a way to tell in advance whether its decimal representation will repeat or terminate. Make a list of rational numbers and their decimals. See if you can discover the pattern. Confirm your conclusion by consulting books on number theory at the library. Write a brief essay on your findings.. The current time is noon CST. What time (CST) will it be,997 hours from now? a 0. Both a Z 0 and are undefined, but for different reasons. 0 0 Write a paragraph or two eplaining the different reasons. R. Algebra Essentials OBJECTIVES Graph Inequalities (p. 8) Find Distance on the Real Number Line (p. 9) Evaluate Algebraic Epressions (p. 0) 4 Determine the Domain of a Variable (p. ) 5 Use the Laws of Eponents (p. ) 6 Evaluate Square Roots (p. ) 7 Use a Calculator to Evaluate Eponents (p. 4) 8 Use Scientific Notation (p. 4) The Real Number Line Figure 8 Real number line 0 units Scale unit O The real numbers can be represented by points on a line called the real number line. There is a one-to-one correspondence between real numbers and points on a line. That is, every real number corresponds to a point on the line, and each point on the line has a unique real number associated with it. Pick a point on the line somewhere in the center, and label it O. This point, called the origin, corresponds to the real number 0. See Figure 8. The point unit to the right of O corresponds to the number. The distance between 0 and determines the scale of the number line. For eample, the point associated with the number is twice as far from O as. Notice that an arrowhead on the right end of the line indicates the direction in which the numbers increase. Points to the left of the origin correspond to the real numbers -, -, and so on. Figure 8 also shows the points associated with the rational numbers - and with the irrational and numbers and p. DEFINITION The real number associated with a point P is called the coordinate of P, and the line whose points have been assigned coordinates is called the real number line. Now Work PROBLEM

18 8 CHAPTER R Review Figure 9 0 Negative real numbers O Zero Positive real numbers The real number line consists of three classes of real numbers, as shown in Figure 9.. The negative real numbers are the coordinates of points to the left of the origin O.. The real number zero is the coordinate of the origin O.. The positive real numbers are the coordinates of points to the right of the origin O. Negative and positive numbers have the following multiplication properties: Multiplication Properties of Positive and Negative Numbers. The product of two positive numbers is a positive number.. The product of two negative numbers is a positive number.. The product of a positive number and a negative number is a negative number. Graph Inequalities Figure 0 a b (a) a b a b (b) a b (c) a b b a An important property of the real number line follows from the fact that, given two numbers (points) a and b, either a is to the left of b, or a is at the same location as b, or a is to the right of b. See Figure 0. If a is to the left of b, we say that a is less than b and write a 6 b. If a is to the right of b, we say that a is greater than b and write a 7 b. If a is at the same location as b, then a = b. If a is either less than or equal to b, we write a b. Similarly, a Ú b means that a is either greater than or equal to b. Collectively, the symbols 6, 7,, and Ú are called inequality symbols. Note that a 6 b and b 7 a mean the same thing. It does not matter whether we write 6 or 7. Furthermore, if a 6 b or if b 7 a, then the difference b - a is positive. Do you see why? EXAMPLE Using Inequality Symbols (a) 6 7 (b) (c) (d) (e) (f) In Eample (a), we conclude that 6 7 either because is to the left of 7 on the real number line or because the difference, 7 - = 4, is a positive real number. Similarly, we conclude in Eample (b) that either because -8 lies to the right of -6 on the real number line or because the difference, = = 8, is a positive real number. Look again at Eample. Note that the inequality symbol always points in the direction of the smaller number. Statements of the form a 6 b or b 7 a are called strict inequalities, whereas statements of the form a b or b Ú a are called nonstrict inequalities. An inequality is a statement in which two epressions are related by an inequality symbol. The epressions are referred to as the sides of the inequality. Based on the discussion so far, we conclude that a 7 0 is equivalent to a is positive a 6 0 is equivalent to a is negative

19 SECTION R. Algebra Essentials 9 We sometimes read a 7 0 by saying that a is positive. If a Ú 0, then either a 7 0 or a = 0, and we may read this as a is nonnegative. Now Work PROBLEMS 5 AND 5 We shall find it useful in later work to graph inequalities on the real number line. EXAMPLE Graphing Inequalities (a) On the real number line, graph all numbers for which 7 4. (b) On the real number line, graph all numbers for which 5. Figure 0 4 Solution (a) See Figure. Notice that we use a left parenthesis to indicate that the number 4 is not part of the graph. (b) See Figure. Notice that we use a right bracket to indicate that the number 5 is part of the graph. Figure Now Work PROBLEM Figure 4 units units Find Distance on the Real Number Line The absolute value of a number a is the distance from 0 to a on the number line. For eample, -4 is 4 units from 0, and is units from 0. See Figure. Thus, the absolute value of -4 is 4, and the absolute value of is. A more formal definition of absolute value is given net. DEFINITION The absolute value of a real number a, denoted by the symbol ƒaƒ, is defined by the rules ƒaƒ = a if a Ú 0 and ƒaƒ = -a if a 6 0 For eample, since --4 = , the second rule must be used to get ƒ -4ƒ = EXAMPLE Computing Absolute Value (a) ƒ8ƒ = 8 (b) ƒ0ƒ = 0 (c) ƒ -5ƒ = --5 = 5 Look again at Figure.The distance from -4 to is 7 units.this distance is the difference - -4, obtained by subtracting the smaller coordinate from the larger. However, since ƒ - -4ƒ = ƒ7ƒ = 7 and ƒ -4 - ƒ = ƒ -7ƒ = 7, we can use absolute value to calculate the distance between two points without being concerned about which is smaller. DEFINITION If P and Q are two points on a real number line with coordinates a and b respectively, the distance between P and Q, denoted by dp, Q, is dp, Q = ƒb - aƒ Since ƒb - aƒ = ƒa - bƒ, it follows that dp, Q = dq, P.

20 0 CHAPTER R Review EXAMPLE 4 Finding Distance on a Number Line Let P, Q, and R be points on a real number line with coordinates respectively. Find the distance (a) between P and Q (b) between Q and R Solution See Figure 4. -5, 7, and -, Figure 4 P R Q d(p, Q) 7 ( 5) d(q, R) 7 0 (a) (b) dp, Q = ƒ7 - -5ƒ = ƒƒ = dq, R = ƒ - - 7ƒ = ƒ -0ƒ = 0 Now Work PROBLEM 7 Evaluate Algebraic Epressions Remember, in algebra we use letters such as, y, a, b, and c to represent numbers. If the letter used is to represent any number from a given set of numbers, it is called a variable. A constant is either a fied number, such as 5 or, or a letter that represents a fied (possibly unspecified) number. Constants and variables are combined using the operations of addition, subtraction, multiplication, and division to form algebraic epressions. Eamples of algebraic epressions include y - t To evaluate an algebraic epression, substitute for each variable its numerical value. EXAMPLE 5 Evaluating an Algebraic Epression Evaluate each epression if = and y = -. y (a) + y (b) 5y (c) (d) ƒ -4 + yƒ - Solution (a) Substitute for and - for y in the epression + y. (b) If = and y = -, then (c) If = and y = -, then (d) If = and y = -, then + y = + - = + - = 0 q =, y = - 5y = 5- = -5 y - = - - = = - -4 = 4 ƒ yƒ = ƒ ƒ = ƒ ƒ = ƒ - ƒ = Now Work PROBLEMS 9 AND 47

21 4 Determine the Domain of a Variable SECTION R. Algebra Essentials In working with epressions or formulas involving variables, the variables may be allowed to take on values from only a certain set of numbers. For eample, in the formula for the area A of a circle of radius r, A = pr, the variable r is necessarily restricted to the positive real numbers. In the epression the variable cannot, take on the value 0, since division by 0 is not defined. DEFINITION The set of values that a variable may assume is called the domain of the variable. EXAMPLE 6 Finding the Domain of a Variable The domain of the variable in the epression 5 - is 5ƒ Z 6, since, if =, the denominator becomes 0, which is not defined. EXAMPLE 7 Circumference of a Circle In the formula for the circumference C of a circle of radius r, C = pr the domain of the variable r, representing the radius of the circle, is the set of positive real numbers. The domain of the variable C, representing the circumference of the circle, is also the set of positive real numbers. In describing the domain of a variable, we may use either set notation or words, whichever is more convenient. Now Work PROBLEM 57 5 Use the Laws of Eponents Integer eponents provide a shorthand device for representing repeated multiplications of a real number. For eample, 4 = # # # = 8 Additionally, many formulas have eponents. For eample, The formula for the horsepower rating H of an engine is H = D N.5 where D is the diameter of a cylinder and N is the number of cylinders. A formula for the resistance R of blood flowing in a blood vessel is L R = C r 4 where L is the length of the blood vessel, r is the radius, and C is a positive constant.

22 CHAPTER R Review DEFINITION If a is a real number and n is a positive integer, then the symbol a n represents the product of n factors of a.that is, a n a a... a n factors () Here it is understood that a = a. Then a = a # a, a = a # a # a, and so on. In the epression a n, a is called the base and n is called the eponent, or power. We read a n as a raised to the power n or as a to the nth power. We usually read a as a squared and a as a cubed. In working with eponents, the operation of raising to a power is performed before any other operation. As eamples, 4 # = 4 # 9 = 6-4 = -6 + = = 5 # + # 4 = 5 # 9 + # 4 = = 5 Parentheses are used to indicate operations to be performed first. For eample, - 4 = ---- = 6 + = 5 = 5 DEFINITION If a Z 0, we define a 0 = if a Z 0 DEFINITION If a Z 0 and if n is a positive integer, then we define a -n = a n if a Z 0 Whenever you encounter a negative eponent, think reciprocal. EXAMPLE 8 Evaluating Epressions Containing Negative Eponents (a) (b) (c) a - 5 b - = -4 = = a = = b 5 = 5 Now Work PROBLEMS 75 AND 95 The following properties, called the Laws of Eponents, can be proved using the preceding definitions. In the list, a and b are real numbers, and m and n are integers. THEOREM Laws of Eponents a m a n = a m + n a m n = a mn ab n = a n b n a m a n = am - n = a n - m if a Z 0 a a n b b = an b n if b Z 0

23 SECTION R. Algebra Essentials EXAMPLE 9 Using the Laws of Eponents (a) - # 5 = = Z 0 (b) (c) (d) (e) - = - # = -6 = 6 Z 0 = # = 8 a b 4 - = 4 4 = = = Z 0 Now Work PROBLEM 77 EXAMPLE 0 Using the Laws of Eponents Write each epression so that all eponents are positive. (a) 5 y - y Z 0, y Z 0 (b) - y - - Z 0, y Z 0 Solution (a) (b) 5 y - y - y - - = 5 # y- y = 5 - # y - - = y - = # y = y = - - y - - = y - = - 9 y = 96 y Now Work PROBLEM 87 6 Evaluate Square Roots A real number is squared when it is raised to the power. The inverse of squaring is finding a square root. For eample, since 6 = 6 and -6 = 6, the numbers 6 and -6 are square roots of 6. The symbol, called a radical sign, is used to denote the principal, or nonnegative, square root. For eample, 6 = 6. DEFINITION If a is a nonnegative real number, the nonnegative number b, such that is the principal square root of a, is denoted by b = a. b = a The following comments are noteworthy:. Negative numbers do not have square roots (in the real number system), because the square of any real number is nonnegative. For eample, -4 is not a real number, because there is no real number whose square is -4.. The principal square root of 0 is 0, since 0 = 0. That is, 0 = 0.. The principal square root of a positive number is positive. 4. If c Ú 0, then c = c. For eample, = and =. EXAMPLE Evaluating Square Roots (a) 64 = 8 (b) (c) A.4B =.4 A 6 = 4

24 4 CHAPTER R Review Eamples (a) and (b) are eamples of square roots of perfect squares, since and 6 = a 64 = 8 4 b. Consider the epression a. Since a Ú 0, the principal square root of a is defined whether a 7 0 or a 6 0. However, since the principal square root is nonnegative, we need an absolute value to ensure the nonnegative result. That is, 4 a = ƒaƒ a any real number () EXAMPLE Using Equation () (a) (b) (c) 4. = ƒ.ƒ = = ƒ -.ƒ =. 4 = ƒƒ Now Work PROBLEM 8 7 Use a Calculator to Evaluate Eponents Your calculator has either a caret key, ^ or an y key, which is used for computations involving eponents. Figure 5 EXAMPLE Solution Eponents on a Graphing Calculator Evaluate:. 5 Figure 5 shows the result using a TI-84 graphing calculator. Now Work PROBLEM 8 Use Scientific Notation Measurements of physical quantities can range from very small to very large. For eample, the mass of a proton is approimately kilogram and the mass of Earth is about 5,980,000,000,000,000,000,000,000 kilograms. These numbers obviously are tedious to write down and difficult to read, so we use eponents to rewrite each. DEFINITION When a number has been written as the product of a number, where 6 0, times a power of 0, it is said to be written in scientific notation. In scientific notation, Mass of a proton =.67 * 0-7 kilogram Mass of Earth = 5.98 * 0 4 kilograms

25 SECTION R. Algebra Essentials 5 Converting a Decimal to Scientific Notation To change a positive number into scientific notation:. Count the number N of places that the decimal point must be moved to arrive at a number, where If the original number is greater than or equal to, the scientific notation is * 0 N. If the original number is between 0 and, the scientific notation is * 0 -N. EXAMPLE 4 Solution Using Scientific Notation Write each number in scientific notation. (a) 958 (b).45 (c) 0.85 (d) (a) The decimal point in 958 follows the. We count left from the decimal point stopping after three moves, because 9.58 is a number between and 0. Since 958 is greater than, we write 958 = 9.58 * 0 (b) The decimal point in.45 is between the and. Since the number is already between and 0, the scientific notation for it is.45 * 0 0 =.45. (c) The decimal point in 0.85 is between the 0 and the. We count stopping after one move, because.85 is a number between and 0. Since 0.85 is between 0 and, we write 0.85 =.85 * 0 - (d) The decimal point in is moved as follows: As a result, = 5.6 * 0-4 Now Work PROBLEM 9 EXAMPLE 5 Solution Changing from Scientific Notation to Decimals Write each number as a decimal. (a). * 0 4 (b).6 * 0-5 (c) * 0 - (a) ,000 4 (b) (c) Now Work PROBLEM 7

26 6 CHAPTER R Review EXAMPLE 6 Using Scientific Notation (a) The diameter of the smallest living cell is only about centimeter (cm).* Epress this number in scientific notation. (b) The surface area of Earth is about.97 * 0 8 square miles. Epress the surface area as a whole number. Solution (a) cm = * 0-5 cm because the decimal point is moved five places and the number is less than. (b).97 * 0 8 square miles = 97,000,000 square miles. Now Work PROBLEM 5 COMMENT On a calculator, a number such as.65 * 0 is usually displayed as.65e. *Powers of Ten, Philip and Phylis Morrison. 998 Information Please Almanac. Historical Feature The word algebra is derived from the Arabic word al-jabr. This word is a part of the title of a ninth century work, Hisâb al-jabr w al-muqâbalah, written by Mohammed ibn Músâ al-khowârizmî. The word al-jabr means a restoration, a reference to the fact that, if a number is added to one side of an equation, then it must also be added to the other side in order to restore the equality. The title of the work, freely translated, is The Science of Reduction and Cancellation. Of course, today, algebra has come to mean a great deal more. R. Assess Your Understanding Concepts and Vocabulary. A(n) is a letter used in algebra to represent any number from a given set of numbers.. On the real number line, the real number zero is the coordinate of the.. An inequality of the form a 7 b is called a(n) inequality. 4. In the epression 4, the number is called the and 4 is called the. 5. In scientific notation, =. 6. True or False The product of two negative real numbers is always greater than zero. 7. True or False The distance between two distinct points on the real number line is always greater than zero. 8. True or False The absolute value of a real number is always greater than zero. 9. True or False When a number is epressed in scientific notation,it is epressed as the product of a number, 0 6, and a power of True or False To multiply two epressions having the same base, retain the base and multiply the eponents. Skill Building. On the real number line, label the points with coordinates 0,, -, 5 and 0.5., -.5, 4,. Repeat Problem for the coordinates 0, -,, -.5, and,,. In Problems, replace the question mark by 6, 7, or =, whichever is correct ? ? ? p?.4? 0 8.? ? 0.5? 0.? ? 0.5

27 SECTION R. Algebra Essentials 7 In Problems 8, write each statement as an inequality.. is positive 4. z is negative 5. is less than 6. y is greater than is less than or equal to 8. is greater than or equal to In Problems 9, graph the numbers on the real number line. 9. Ú In Problems 8, use the given real number line to compute each distance. A B C D E dc, D 4. dc, A 5. dd, E 6. dc, E 7. da, E 8. dd, B In Problems 9 46, evaluate each epression if = - and y = y y 4. 5y y + y + y y - y + y - y In Problems 47 56, find the value of each epression if = and y = ƒ + yƒ 48. ƒ - yƒ 49. ƒƒ + ƒyƒ 50. ƒƒ - ƒyƒ 5. ƒyƒ ƒ4-5yƒ 54. ƒ + yƒ 55. ƒƒ4ƒ - ƒ5yƒƒ 56. ƒƒ + ƒyƒ y In Problems 57 64, determine which of the value(s) (a) through (d), if any, must be ecluded from the domain of the variable in each epression: (a) = (b) = (c) = 0 (d) = ƒƒ In Problems 65 68, determine the domain of the variable in each epression In Problems 69 7, use the formula C = 5 F - for converting degrees Fahrenheit into degrees Celsius to find the Celsius measure 9 of each Fahrenheit temperature. 69. F = 70. F = 7. F = F = -4 In Problems 7 84, simplify each epression # 4-6 # In Problems 85 94, simplify each epression. Epress the answer so that all eponents are positive. Whenever an eponent is 0 or negative, we assume that the base is not 0. y y y 89. y 4 - y - 4 yz 4 - yz y 4y 6y y z 4 y - -

28 8 CHAPTER R Review In Problems 95 06, find the value of each epression if = and y = y y y 98. y 99. y y y y 05. y 06. y 07. Find the value of the epression if =. What is the value if =? 08. Find the value of the epression if =. What is the value if =? What is the value of 0. What is the value of 0. 0? 4? In Problems 8, use a calculator to evaluate each epression. Round your answer to three decimal places In Problems 9 6, write each number in scientific notation ,55 4., In Problems 7 4, write each number as a decimal * * * * * * * * 0 - Applications and Etensions In Problems 5 44, epress each statement as an equation involving the indicated variables. 5. Area of a Rectangle The area A of a rectangle is the product of its length l and its width. w l 9. Area of an Equilateral Triangle The area A of an equilateral triangle is times the square of the length of one 4 side. A w 6. Perimeter of a Rectangle The perimeter P of a rectangle is twice the sum of its length l and its width. w 7. Circumference of a Circle The circumference C of a circle is the product of p and its diameter d. d C 40. Perimeter of an Equilateral Triangle The perimeter P of an equilateral triangle is times the length of one side Volume of a Sphere The volume V of a sphere is times p times the cube of the radius r. 8. Area of a Triangle The area A of a triangle is one-half the product of its base b and its height h. r h b 4. Surface Area of a Sphere The surface area S of a sphere is 4 times p times the square of the radius r.