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2 1 Basic Algebraic Operations Algebra is often referred to as “generalized arithmetic.” In arithmetic we deal with the basic arithmetic operations of addition, subtraction, multiplication, and division performed on specific numbers. In algebra we continue to use all that we know in arithmetic, but, in addition, we reason and work with symbols that represent one or more numbers. In this chapter we review some important basic algebraic operations usually studied in earlier courses. The material may be studied systematically before commencing with the rest of the book or reviewed as needed. SECTION 1-1 Algebra and Real Numbers • • • • • • Sets The Set of Real Numbers The Real Number Line Basic Real Number Properties Further Properties Fraction Properties The rules for manipulating and reasoning with symbols in algebra depend, in large measure, on properties of the real numbers. In this section we look at some of the important properties of this number system. To make our discussions here and elsewhere in the text clearer and more precise, we first introduce a few useful notions about sets. • Sets Georg Cantor (1845–1918) developed a theory of sets as an outgrowth of his studies on infinity. His work has become a milestone in the development of mathematics. Our use of the word “set” will not differ appreciably from the way it is used in everyday language. Words such as “set,” “collection,” “bunch,” and “flock” all convey the same idea. Thus, we think of a set as a collection of objects with the important property that we can tell whether any given object is or is not in the set. Each object in a set is called an element, or member, of the set. Symbolically, aA means “a is an element of set A” 3 {1, 3, 5} aA means “a is not an element of set A” 2 {1, 3, 5} Capital letters are often used to represent sets and lowercase letters to represent elements of a set. A set is finite if the number of elements in the set can be counted and infinite if there is no end in counting its elements. A set is empty if it contains no elements. The empty set is also called the null set and is denoted by . It is important to observe that the empty set is not written as {}. A set is usually described in one of two ways—by listing the elements between braces, { }, or by enclosing within braces a rule that determines its elements. For example, if D is the set of all numbers x such that x2 4, then using the listing method we write D {2, 2} Listing method 1-1 Algebra and Real Numbers 3 or, using the rule method we write D {x x 2 4} Rule method Note that in the rule method, the vertical bar represents “such that,” and the entire symbolic form {x x2 4} is read, “The set of all x such that x2 4.” The letter x introduced in the rule method is a variable. In general, a variable is a symbol that is used as a placeholder for the elements of a set with two or more elements. This set is called the replacement set for the variable. A constant, on the other hand, is a symbol that names exactly one object. The symbol “8” is a constant, since it always names the number eight. If each element of set A is also an element of set B, we say that A is a subset of set B, and we write AB {1, 5} {1, 3, 5} Note that the definition of a subset allows a set to be a subset of itself. Since the empty set has no elements, every element of is also an element of any given set. Thus, the empty set is a subset of every set. For example, {1, 3, 5} and {2, 4, 6} If two sets A and B have exactly the same elements, the sets are said to be equal, and we write AB {4, 2, 6} {6, 4, 2} Notice that the order of listing elements in a set does not matter. We can now begin our discussion of the real number system. Additional set concepts will be introduced as needed. • The Set of Real Numbers The real number system is the number system you have used most of your life. Informally, a real number is any number that has a decimal representation. Table 1 on the next page describes the set of real numbers and some of its important subsets. Figure 1 illustrates how these sets of numbers are related to each other. Figure 1 Real numbers and important subsets. Real numbers (R ) NZQR Rational numbers (Q ) Irrational numbers (I ) Integers (Z ) Natural numbers (N ) Zero Noninteger ratios of integers Negatives of natural numbers 4 1 Basic Algebraic Operations TABLE 1 Symbol The Set of Real Numbers Name Description Examples N Natural numbers Counting numbers (also called positive integers) 1, 2, 3, . . . Z Integers Natural numbers, their negatives, and 0 . . . , 2, 1, 0, 1, 2, . . . Q Rational numbers Numbers that can be represented as a/b, where a and b are integers and b 0; decimal representations are repeating or terminating I Irrational numbers Numbers that can be represented as nonrepeating and nonterminating decimal numbers R Real numbers Rational numbers and irrational numbers 2 4, 0, 1, 25, 3 5 , 3 , 3.67, 0.333,* 5.272727 3 2, ,7, 1.414213 . . . , 2.71828182 . . . *The overbar indicates that the number (or block of numbers) repeats indefinitely. • The Real Number Line A one-to-one correspondence exists between the set of real numbers and the set of points on a line. That is, each real number corresponds to exactly one point, and each point to exactly one real number. A line with a real number associated with each point, and vice versa, as in Figure 2, is called a real number line, or simply a real line. Each number associated with a point is called the coordinate of the point. The point with coordinate 0 is called the origin. The arrow on the right end of the line indicates a positive direction. The coordinates of all points to the right of the origin are called positive real numbers, and those to the left of the origin are called negative real numbers. The real number 0 is neither positive nor negative. Figure 2 A real number line. 27 10 • Basic Real Number Properties 4 3 Origin 5 0 7.64 5 10 We now take a look at some of the basic properties of real numbers. (See the box on the next page.) You are already familiar with the commutative properties for addition and multiplication. They indicate that the order in which the addition or multiplication of two numbers is performed doesn’t matter. For example, 4554 and 4554 Is there a commutative property relative to subtraction or division? That is, does x y y x or does x y y x for all real numbers x and y (division by 0 excluded)? The answer is no, since, for example, 7557 and 6336 1-1 Algebra and Real Numbers 5 Basic Properties of the Set of Real Numbers Let R be the set of real numbers, and let x, y, and z be arbitrary elements of R. Addition Properties Closure: x y is a unique element in R. Associative: (x y) z x (y z) Commutative: xyyx Identity: 0 is the additive identity; that is, 0 x x 0 x for all x in R, and 0 is the only element in R with this property. Inverse: For each x in R, x is its unique additive inverse; that is, x (x) (x) x 0, and x is the only element in R relative to x with this property. Multiplication Properties Closure: xy is a unique element in R. Associative: (xy)z x( yz) Commutative: xy yx Identity: 1 is the multiplicative identity; that is, for x in R, (1)x x(1) x, and 1 is the only element in R with this property. Inverse: For each x in R, x 0, 1/x is its unique multiplicative inverse; that is, x(1/x) (1/x)x 1, and 1/x is the only element in R relative to x with this property. Combined Property Distributive: x( y z) xy xz (x y)z xz yz When computing 253 or 253 why don’t we need parentheses to indicate which two numbers are to be added or multiplied first? The answer is to be found in the associative properties. These properties allow us to write (2 5) 3 2 (5 3) and (2 5) 3 2 (5 3) so it doesn’t matter how we group numbers relative to either operation. Is there an associative property for subtraction or division? The answer is no, since, for example, 6 1 Basic Algebraic Operations (8 4) 2 8 (4 2) (8 4) 2 8 (4 2) and Evaluate both sides of these equations to see why. Conclusion Relative to addition, commutativity and associativity permit us to change the order of addition at will and insert or remove parentheses as we please. The same is true for multiplication, but not for subtraction and division. What number added to a given number will give that number back again? What number times a given number will give that number back again? The answers are 0 and 1, respectively. Because of this, 0 and 1 are called the identity elements for the real numbers. Hence, for any real numbers x and y, 707 0 (x y) x y 0 is the additive identity. 166 1(x y) x y 1 is the multiplicative identity. We now consider inverses. For each real number x, there is a unique real number x such that x (x) 0. The number x is called the additive inverse of x, or the negative of x. For example, the additive inverse of 4 is 4, since 4 (4) 0. The additive inverse of 4 is (4) 4, since 4 [(4)] 0. It is important to remember: x is not necessarily a negative number; it is positive if x is negative and negative if x is positive. For each nonzero real number x there is a unique real number 1/x such that x(1/x) 1. The number 1/x is called the multiplicative inverse of x, or the reciprocal of x. For example, the multiplicative inverse of 7 is 17, since 7(17) 1. Also note that 7 is the multiplicative inverse of 17 . The number 0 has no multiplicative inverse. We now turn to a real number property that involves both multiplication and addition. Consider the two computations: 3(4 2) 3(6) 18 3(4) 3(2) 12 6 18 Thus, 3(4 2) 3(4) 3(2) and we say that multiplication by 3 distributes over the sum (4 2). In general, multiplication distributes over addition in the real number system. Two more illustrations are given below: 2(x y) 2x 2y (3 5)x 3x 5x 1-1 EXAMPLE 1 Algebra and Real Numbers 7 Using Real Number Properties Which real number property justifies the indicated statement? Statement (A) (7x)y 7(xy) (B) a(b c) (b c)a (C) (2x 3y) 5y 2x (3y 5y) (D) (x y)(a b) (x y)a (x y)b (E) If a b 0, then b a. Matched Problem 1* Which real number property justifies the indicated statement? (A) 4 (2 x) (4 2) x (C) 3x 7x (3 7)x (E) If ab 1, then b 1/a. • Further Properties DEFINITION 1 Property Illustrated Associative () Commutative () Associative () Distributive Inverse () (B) (a b) c c (a b) (D) (2x 3y) 0 2x 3y Subtraction and division can be defined in terms of addition and multiplication, respectively: Subtraction and Division For all real numbers a and b: Subtraction: a b a (b) Division: ba a b (5) (3) (5) (3) 2 a 1 a b b b0 12 323 Thus, to subtract b from a, add the negative of b to a. To divide a by b, multiply a by the reciprocal of b. Note that division by 0 is not defined, since 0 does not have a reciprocal. It is important to remember: Division by 0 is never allowed. The following properties of negatives can be proved using the preceding properties and definitions. *Answers to matched problems in a given section are found near the end of the section, before the exercise set. 8 1 Basic Algebraic Operations Theorem 1 Properties of Negatives For all real numbers a and b: 1. (a) a 2. (a)b (ab) a(b) ab 3. (a)(b) ab 4. (1)a a 5. a a a b b b 6. a a a a b b b b b0 b0 We now state an important theorem involving 0. Theorem 2 Zero Properties For all real numbers a and b: 1. a 0 0 2. ab 0 EXAMPLE 2 if and only if a 0 or b 0 or both Using Negative and Zero Properties Which real number property or definition justifies each statement? Statement (A) 3 (2) 3 [(2)] 5 (B) (2) 2 3 3 (C) 2 2 5 5 (D) 2 2 (E) If (x 3)(x 5) 0, then either x 3 0 or x 5 0. Property or Definition Illustrated Subtraction (Definition 1 and Theorem 1, part 1) Negatives (Theorem 1, part 1) Negatives (Theorem 1, part 6) Negatives (Theorem 1, part 5) Zero (Theorem 2, part 2) 1-1 Matched Problem 2 9 Which real number property or definition justifies each statement? 3 1 3 5 5 7 7 (D) 9 9 (A) EXPLORE-DISCUSS 1 Algebra and Real Numbers (B) (5)(2) (5 2) (C) (1)3 3 (E) If x 5 0, then (x 3)(x 5) 0. In general, a set of numbers is closed under an operation if performing the operation on numbers in the set always produces another number in the set. For example, the real numbers are closed under addition, multiplication, subtraction, and division, excluding division by 0. Replace each ? in the following tables with T (true) or F (false), and illustrate each false statement with an example. (See Table 1 for the definitions of the sets N, Z, I, Q, and R.) Closed under Addition Closed under Multiplication N ? ? Z ? ? Q ? ? I ? ? R T T Closed under Subtraction Closed under Division* N ? ? Z ? ? Q ? ? I ? ? R T T *Excluding division by 0. • Fraction Properties Recall that the quotient a b, b 0, written in the form a/b is called a fraction. The quantity a is called the numerator and the quantity b is the denominator. 10 1 Basic Algebraic Operations Theorem 3 Fraction Properties For all real numbers a, b, c, d, and k (division by 0 excluded): 1. a c b d 4 6 6 9 2. ad bc if and only if since ka a kb b 4966 3. 73 3 75 5 5. a c ac b b b 3 5 35 6 6 6 a c ac b d bd 4. 3 7 37 5 8 58 6. a c a d b d b c 2 5 2 7 3 7 3 5 a c ac b b b 7. 7 3 73 8 8 8 a c ad bc b d bd 2 3 2533 3 5 35 Answers to Matched Problems 1. (A) Associative () (B) Commutative () (C) Distributive (D) Identity () (E) Inverse () 2. (A) Division (Definition 1) (B) Negatives (Theorem 1, part 2) (C) Negatives (Theorem 1, part 4) (D) Negatives (Theorem 1, part 5) (E) Zero (Theorem 2, part 1) EXERCISE 1-1 All variables represent real numbers. 13. Identity property (): 0 9m ? 14. Identity property (): 1(u v) ? A In Problems 1–8, indicate true (T) or false (F). 1. 4 {3, 4, 5} 2. 6 {2, 4, 6} 3. 3 {3, 4, 5} 4. 7 {2, 4, 6} 5. {1, 2} {1, 3, 5} 6. {2, 6} {2, 4, 6} 7. {7, 3, 5} {3, 5, 7} 8. {7, 3, 5} {3, 5, 7} In Problems 15–26, each statement illustrates the use of one of the following properties or definitions. Indicate which one. Commutative (, ) Associative (, ) Distributive Identity (, ) Inverse (, ) Subtraction Division Negatives (Theorem 1) Zero (Theorem 2) 15. x ym x my 16. 7(3m) (7 3)m 17. 7u 9u (7 9)u 18. 1 19. (2)(2 )1 20. 8 12 8 (12) 10. Commutative property (): uv ? 21. w (w) 0 1 22. 5 (6) 5(6 ) 11. Associative property (): x(yz) ? 23. 3(xy z) 0 3(xy z) 12. Associative property (): 3 (7 y) ? 24. ab(c d ) abc abd In Problems 9–14, replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property. 9. Commutative property (): x 7 ? u u v v 1-1 25. x x y y 26. (x y) 0 0 B Write each set in Problems 27–32 using the listing method; that is, list the elements between braces. If the set is empty, write . { x x is a letter in “status”} { x x is a letter in “consensus”} { x x is a month starting with B} { x x is a month with 32 days} 27. {x x is an even integer between 3 and 5} 28. { x x is an odd integer between 4 and 6} 29. 30. 31. 32. 33. The set S1 {a} has only two subsets, S1 and . How many subsets does each of the following sets have? (A) S2 {a, b} (B) S3 {a, b, c} (C) S4 {a, b, c, d} 34. Based on the results in Problem 33, how many subsets do you think a set with n elements will have? In Problems 35–42, each statement illustrates the use of one of the following properties or definitions. Indicate which one. Commutative (, ) Subtraction Associative (, ) Division Distributive Negatives (Theorem 1) Identity (, ) Zero (Theorem 2) Inverse (, ) 35. (y) 2x y 2x 36. (ab)(ba) (ab)(ab) 37. (wz)(zw) w[z(zw)] Algebra and Real Numbers 11 46. Indicate which of the following are true: (A) All integers are natural numbers. (B) All rational numbers are real numbers. (C) All natural numbers are rational numbers. 47. Give an example of a rational number that is not an integer. 48. Give an example of a real number that is not a rational number. In Problems 49 and 50, list the subset of S consisting of (A) natural numbers, (B) integers, and (C) rational numbers. 49. S {3, 23, 0, 1, 3, 95, 12} 50. S {5 , 1,12, 2, 7 , 6, 25 3} In Problems 51 and 52, use a calculator* to express each number as a decimal fraction. Classify each decimal number as terminating, repeating, or nonrepeating and nonterminating. Identify the pattern of repeated digits in any repeating decimal numbers. 51. (A) 8 9 52. (A) 13 6 (B) 3 11 (B) 21 (C) 5 (C) (D) 7 16 11 8 (D) 29 111 53. Indicate true (T) or false (F), and for each false statement find real number replacements for a and b that will provide a counterexample. For all real numbers a and b: (A) a b b a (B) a b b a (C) ab ba (D) a b b a 54. Indicate true (T) or false (F), and for each false statement find real number replacements for a, b, and c that will provide a counterexample. For all real numbers a, b, and c: (A) (a b) c a (b c) (B) (a b) c a (b c) (C) a(bc) (ab)c (D) (a b) c a (b c) 38. s (t 2) (s t) 2 39. (n 2)(m 3) n(m 3) 2(m 3) C 40. p(r 1) q(r 1) (p q)(r 1) 55. If A {1, 2, 3, 4} and B {2, 4, 6}, find: (A) {x x A or x B} (B) { x x A and x B} 41. (2x 3)(3x 5) 0 if and only if 2x 3 0 or 3x 5 0 42. y y (1 y) 1 y 43. If ab 0, does either a or b have to be 0? 56. If F {2, 0, 2} and G {1, 0, 1, 2}, find: (A) {x x F or x G} (B) { x x F and x G} 44. If ab 1, does either a or b have to be 1? 45. Indicate which of the following are true: (A) All natural numbers are integers. (B) All real numbers are irrational. (C) All rational numbers are real numbers. *Later in the book you will encounter optional exercises that require a graphing calculator. If you have such a calculator, you can certainly use it here. Otherwise, any scientific calculator will be sufficient for the problems in this chapter. 12 1 Basic Algebraic Operations 57. If c 0.151515 . . . , then 100c 15.1515 . . . and 59. To see how the distributive property is behind the mechanics of long multiplication, compute each of the following and compare: 100c c 15.1515 . . . 0.151515 . . . 99c 15 Long Multiplication 23 12 5 c 15 99 33 Proceeding similarly, convert the repeating decimal 0.090909 . . . into a fraction. (All repeating decimals are rational numbers, and all rational numbers have repeating decimal representations.) 60. For a and b real numbers, justify each step using a property in this section. 58. Repeat Problem 57 for 0.181818. . . . SECTION 1-2 Use of the Distributive Property 23 12 23(2 10) 23 2 23 10 Statement 1. (a b) (a) (a) (a b) 2. [(a) a] b 3. 0b 4. b Reason 1. 2. 3. 4. Polynomials: Basic Operations • • • • • • • Natural Number Exponents Polynomials Combining Like Terms Addition and Subtraction Multiplication Combined Operations Application In this section we review the basic operations on polynomials, a mathematical form encountered frequently throughout mathematics. We start the discussion with a brief review of natural number exponents. Integer and rational exponents and their properties will be discussed in detail in subsequent sections. • Natural Number The definition of a natural number exponent is given below: Exponents DEFINITION 1 Natural Number Exponent For n a natural number and a any real number: an a a · · · a agfgbggfc n factors of a 24 2 2 2 2 4 factors of 2