1.3. THE RATIONAL NUMBERS 1.3 31 The Rational Numbers We begin with the definition of a rational number. Rational Numbers. Any number that can be expressed in the form p/q, where p and q are integers, q 6= 0, is called a rational number. The letter Q is used to represent the set of rational numbers. That is: p : p and q are integers, q 6= 0 Q= q Because −2/3, 4/5, and 123/(−12) have the form p/q, where p and q are integers, each is an example of a rational number. If you think you hear the word “fraction” when we say “rational number,” you are correct in your thinking. Any number that can be expressed as a fraction, where the numerator and denominator are integers, is a rational number. Every integer is also a rational number. Take, for example, the integer −12. There are a number of ways we can express −12 as a fraction with integer numerator and denominator, −12/1, 24/(−2), and −36/3 being a few. Reducing Fractions to Lowest Terms First, we define what is meant by the greatest common divisor of two integers. The Greatest Common Divisor. Given two integers a and b, the greatest common divisor of a and b is the largest integer that divides evenly (with no remainder) into both a and b. The notation GCD(a, b) is used to represent the greatest common divisor of a and b. For example, GCD(12, 18) = 6, GCD(32, 40) = 8, and GCD(18, 27) = 9. We can now state when a fraction is reduced to lowest terms. Lowest Terms. A fraction a/b is said to be reduced to lowest terms if and only if GCD(a, b) = 1. A common technique used to reduce a fraction to lowest terms is to divide both numerator and denominator by their greatest common divisor. You Try It! EXAMPLE 1. Reduce 8/12 to lowest terms. Reduce: −48/60 32 CHAPTER 1. THE ARITHMETIC OF NUMBERS Solution: Note that GCD(8, 12) = 4. Divide both numerator and denominator by 4. 8 8÷4 = 12 12 ÷ 4 Divide numerator and denominator by GCD(8, 12) = 4. 2 = 3 Answer: −4/5 Simplify numerator and denominator. Thus, 8/12 = 2/3. Recall the definition of a prime number. Prime Number. A natural number greater than one is prime if and only if its only divisors are one and itself. For example, 7 is prime (its only divisors are 1 and 7), but 14 is not (its divisors are 1, 2, 7, and 14). In like fashion, 2, 3, and 5 are prime, but 6, 15, and 21 are not prime. You Try It! Reduce 18/24 to lowest terms. EXAMPLE 2. Reduce 10/40 to lowest terms. Solution: Note that GCD(10, 40) = 10. Divide numerator and denominator by 10. 10 10 ÷ 10 = 40 40 ÷ 10 = Divide numerator and denominator by GCD(10, 40) = 10. 1 4 Simplify numerator and denominator. Alternate solution: Use factor trees to express both numerator and denominator as a product of prime factors. 10 2 40 5 10 4 2 2 2 5 Hence, 10 = 2 · 5 and 40 = 2 · 2 · 2 · 5. Now, to reduce 10/40 to lowest terms, replace the numerator and denominator with their prime factorizations, then cancel factors that are in common to both numerator and denominator. 1.3. THE RATIONAL NUMBERS 10 2·5 = 40 2·2·2·5 2· 5 = 2 · 2 · 2·5 1 = 4 33 Prime factor numerator and denominator. Cancel common factors. Simplify numerator and denominator. When we cancel a 2 from both the numerator and denominator, we’re actually dividing both numerator and denominator by 2. A similar statement can be made about canceling the 5. Canceling both 2 and a 5 is equivalent to dividing both numerator and denominator by 10. This explains the 1 in the numerator when all factors cancel. Answer: 3/4 Example 2 demonstrates an important point. When all factors cancel. When all of the factors cancel in either numerator or denominator, the resulting numerator or denominator is equal to one. Multiplying Fractions First, the definition. Multiplication of Fractions. If a/b and c/d are two fractions, then their product is defined as follows: a c ac · = b d bd Thus, to find the product of a/b and c/d, simply multiply numerators and multiply denominators. For example: 1 3 3 2 7 14 5 1 5 · = and − · =− and − · − = 2 4 8 5 3 15 8 6 48 Like integer multiplication, like signs yield a positive answer, unlike signs yield a negative answer. Of course, when necessary, remember to reduce your answer to lowest terms. EXAMPLE 3. Simplify: − 14 10 · 20 21 You Try It! 27 8 Simplify: − · − 9 20 34 CHAPTER 1. THE ARITHMETIC OF NUMBERS Solution: Multiply numerators and denominators, then reduce to lowest terms. − 14 10 140 · =− 20 21 420 Multiply numerators and denominators. 2·2·5·7 2·2·3·5·7 2· 2· 5· 7 =− 2 · 2 · 3 · 5 ·7 1 =− 3 =− Answer: 6/5 Prime factor. Cancel common factors. Simplify. Note that when all the factors cancel from the numerator, you are left with a 1. Thus, (−14/20) · (10/21) = −1/3. Cancellation Rule. When multiplying fractions, cancel common factors according to the following rule: “Cancel a factor in a numerator for an identical factor in a denominator.” The rule is “cancel something on the top for something on the bottom.” Thus, an alternate approach to multiplying fractions is to factor numerators and denominators in place, then cancel a factor in a numerator for an identical factor in a denominator. You Try It! Simplify: 35 6 − · − 45 14 EXAMPLE 4. Simplify: 15 14 · − 8 9 Solution: Factor numerators and denominators in place, then cancel common factors in the numerators for common factors in the denominators. 14 2·7 3·5 15 · − · − = Factor numerators 8 9 2·2·2 3·3 and denominators. 3·5 2·7 · − = Cancel a factor in a 2·2·2 3·3 numerator for a common. factor in a denominator. 35 =− Multiply numerators and. 12 denominators. Answer: 1/3 Note that unlike signs yield a negative product. Thus, (15/8) · (−14/9) = −35/12. 35 1.3. THE RATIONAL NUMBERS Dividing Fractions Every nonzero rational number has was it called a multiplicative inverse or reciprocal. The Reciprocal. If a is any nonzero rational number, then 1/a is called the multiplicative inverse or reciprocal of a, and: a· 1 =1 a Note that: 2· 1 =1 2 and 3 5 · =1 5 3 and − 4 7 · − = 1. 7 4 Thus, the reciprocal of 2 is 1/2, the reciprocal of 3/5 is 5/3, and the reciprocal of −4/7 is −7/4. Note that to find the reciprocal of a number, simply invert the number (flip it upside down). Now we can define the quotient of two fractions. Division of Fractions. If a/b and c/d are two fractions, then their quotient is defined as follows: c a d a ÷ = · b d b c That is, dividing by c/d is the same as multiplying by the reciprocal d/c. The above definition of division is summarized by the phrase “invert and multiply.” You Try It! 35 10 4 27 EXAMPLE 5. Simplify: − ÷ − Simplify: − ÷ 21 12 9 81 Solution: Invert and multiply, then factor in place and cancel common factors in a numerator for common factors in a denominator. 35 10 12 35 − ÷ − =− · − Invert and multiply. 21 12 21 10 5·7 2·2·3 =− · − Prime factor. 3·7 2·5 2 5 ·2·3 ·7 · − Cancel common factors. =− 3· 7 2· 5 2 = Multiply numerators and denominators. 1 =2 Simplify. 36 Answer: −4/3 CHAPTER 1. THE ARITHMETIC OF NUMBERS Note that when all the factors in a denominator cancel, a 1 remains. Thus, (−35/21) ÷ (−10/12) = 2. Note also that like signs yield a positive result. Adding Fractions First the definition. Addition of Fractions. If two fractions have a denominator in common, add the numerators and place the result over the common denominator. In symbols: a b a+b + = c c c For example: 3 7 4 − + = 5 5 5 and 4 7 11 − + − =− 3 3 3 and 4 5 1 + − =− 7 7 7 If the fractions do not posses a common denominator, first create equivalent fractions with a least common denominator, then add according to the rule above. Least Common Denominator. If the fractions a/b and c/d do not share a common denominator, the least common denominator for b and d, written LCD(b, d), is defined as the smallest number divisible by both b and d. You Try It! Simplify: 5 1 − + 6 9 Answer: −13/18 3 5 EXAMPLE 6. Simplify: − + 8 12 Solution: The least common denominator in this case is the smallest number divisible by both 8 and 12. In this case, LCD(8, 12) = 24. We first need to make equivalent fractions with a common denominator of 24. 3 5 3 3 5 2 − + =− · + · Make equivalent fraction with 8 12 8 3 12 2 a common denominator of 24. 9 10 =− + Multiply numerators and denominators. 24 24 1 = Add: −9 + 10 = 1. 24 Note how we add the numerators in the last step, placing the result over the common denominator. Thus, −3/8 + 5/12 = 1/24. 37 1.3. THE RATIONAL NUMBERS Order of Operations Rational numbers obey the same Rules Guiding Order of Operations as do the integers. Rules Guiding Order of Operations. When evaluating expressions, proceed in the following order. 1. Evaluate expressions contained in grouping symbols first. If grouping symbols are nested, evaluate the expression in the innermost pair of grouping symbols first. 2. Evaluate all exponents that appear in the expression. 3. Perform all multiplications and divisions in the order that they appear in the expression, moving left to right. 4. Perform all additions and subtractions in the order that they appear in the expression, moving left to right. You Try It! EXAMPLE 7. Given x = 2/3, y = −3/5, and z = 10/9, evaluate xy + yz. Solution: Following Tips for Evaluating Algebraic Expressions, first replace all occurrences of variables in the expression xy + yz with open parentheses. Next, substitute the given values of variables (2/3 for x, −3/5 for y, and 10/9 for z) in the open parentheses. xy + yz = = + Replace variables with parentheses. 2 10 3 3 − + − Substitute: 2/3 for x, −3/5 3 5 5 9 for y, and 10/9 for z. Given a = −1/2, b = 2/3, and c = −3/4, evaluate the expression a + bc and simplify the result. 38 CHAPTER 1. THE ARITHMETIC OF NUMBERS Use the Rules Guiding Order of Operations to simplify. 6 30 + − 15 45 2 2 =− + − 5 3 2 5 2 3 =− · + − · 5 3 3 5 6 10 =− + − 15 15 16 =− 15 =− Answer: −1 Multiply. Reduce. Make equivalent fractions with a least common denominator. Add. Thus, if x = 2/3, y = −3/5, and z = 10/9, then xy + yz = −16/15 You Try It! Simplify: (−1/3)4 EXAMPLE 8. Given x = −3/5, evaluate −x3 . Solution: First, replace each occurrence of the variable x with open parentheses, then substitute −3/5 for x. 3 −x3 = − ( ) 3 3 =− − 5 3 3 3 =− − − − 5 5 5 27 =− − 125 = Answer: 1/81 27 125 Replace x with open parentheses. Substitute −3/5 for x. Write −3/5 as a factor three times. The product of three negative fractions is negative. Multiply numerators and denominators. The opposite of −27/125 is 27/125. Hence, −x3 = 27/125, given x = −3/5. You Try It! Given x = −3/4 and y = −4/5, evaluate x2 − y 2 . EXAMPLE 9. Given a = −4/3 and b = −3/2, evaluate a2 + 2ab − 3b2 . Solution: Following Tips for Evaluating Algebraic Expressions, first replace all occurrences of variables in the expression a2 + 2ab − 3b2 with open parentheses. 39 1.3. THE RATIONAL NUMBERS Next, substitute the given values of variables (−4/3 for a and −3/2 for b) in the open parentheses. 2 2 a + 2ab − 3b = = 2 +2 −3 2 2 2 3 3 4 4 − +2 − − −3 − 3 3 2 2 Next, evaluate the exponents: (−4/3)2 = 16/9 and (−3/2)2 = 9/4. 16 2 + = 9 1 4 3 3 9 − − − 3 2 1 4 Next, perform the multiplications and reduce. 16 24 27 + − 9 6 4 16 27 = +4− 9 4 = Make equivalent fractions with a common denominator, then add. 16 4 36 27 9 − · +4· · 9 4 36 4 9 64 144 243 = + − 36 36 36 35 =− 36 = Thus, if a = −4/3 and b = −3/2, then a2 + 2ab − 3b2 = −35/36 Answer: −31/400 Fractions on the Graphing Calculator We must always remember that the graphing calculator is an “approximating machine.” In a small number of situations, it is capable of giving an exact answer, but for most calculations, the best we can hope for is an approximate answer. However, the calculator gives accurate results for operations involving fractions, as long as we don’t use fractions with denominators that are too large for the calculator to respond with an exact answer. 40 CHAPTER 1. THE ARITHMETIC OF NUMBERS You Try It! Simplify using the graphing calculator: 4 8 − + 5 3 EXAMPLE 10. Use the graphing calculator to simplify each of the following expressions: (a) 2 1 + 3 2 (b) 2 5 · 3 7 (c) 3 1 ÷ 5 3 Solution: We enter each expression in turn. a) The Rules Guiding Order of Operations tell us that we must perform divisions before additions. Thus, the expression 2/3 + 1/2 is equivalent to: 2 1 + 3 2 4 3 = + 6 6 7 = 6 2/3 + 1/2 = Divide first. Equivalent fractions with LCD. Add. Enter the expression 2/3+1/2 on your calculator, then press the ENTER key. The result is shown in the first image in Figure 1.12. Next, press the MATH button, then select 1:IFrac (see the second image in Figure 1.12) and press the ENTER key again. Note that the result shown in the third image in Figure 1.12 matches the correct answer of 7/6 found above. Figure 1.12: Calculating 2/3 + 1/2. b) The Rules Guiding Order of Operations tell us that there is no preference for division over multiplication, or vice-versa. We must perform divisions and multiplications as they occur, moving from left to right. Hence: 2 × 5/7 3 10 = /7 3 10 1 = × 3 7 10 = 21 2/3 × 5/7 = Divide: 2/3 = Multiply: 2 3 2 10 ×5= 3 3 Invert and multiply. Multiply: 10 1 10 × = 3 7 21 41 1.3. THE RATIONAL NUMBERS This is precisely the same result we get when we perform the following calculation. 2 5 10 × = Multiply numerators and denominators. 3 7 21 Hence: 2 5 2/3 × 5/7 is equivalent to × 3 7 Enter the expression 2/3×5/7 on your calculator, then press the ENTER key. The result is shown in the first image in Figure 1.13. Next, press the MATH button, then select 1:IFrac (see the second image in Figure 1.13) and press the ENTER key again. Note that the result shown in the third image in Figure 1.13 matches the correct answer of 10/21 found above. Figure 1.13: Calculating 2/3 × 1/2. c) This example demonstrates that we need a constant reminder of the Rules Guiding Order of Operations. We know we need to invert and multiply in this situation. 3 1 3 3 ÷ = × Invert and multiply. 5 3 5 1 9 = Multiply numerators and denominators. 5 So, the correct answer is 9/5. Enter the expression 3/5/1/3 on your calculator, then press the ENTER key. Select 1:IFrac from the MATH menu and press the ENTER key again. Note that the result in the first image in Figure 1.14 does not match the correct answer of 9/5 found above. What have we done wrong? If we follow the Rules Guiding Order of Operations exactly, then: 3 /1/3 5 3 = /3 5 3 1 = × 5 3 1 = 5 3/5/1/3 = 3 5 3 3 Divide: /1 = 5 5 Divide: 3/5 = Invert and multiply. Multiply: 3 1 1 × = 5 3 5 42 CHAPTER 1. THE ARITHMETIC OF NUMBERS This explains the answer found in the first image in Figure 1.14. However, it also show that: 3/5/1/3 is not equivalent to 3 1 ÷ 5 3 We can cure the problem by using grouping symbols. 3 1 / 5 3 3 1 = ÷ 5 3 (3/5)/(1/3) = Parentheses first. / is equivalent to ÷. Hence: 3 1 ÷ 5 3 Enter the expression (3/5)/(1/3) on your calculator, then press the ENTER key. Select 1:IFrac from the MATH menu and press the ENTER key again. Note that the result in the second image in Figure 1.14 matches the correct answer of 9/5. (3/5)/(1/3) is equivalent to Figure 1.14: Calculating (3/5)/(1/3). Answer: 28/15 43 1.3. THE RATIONAL NUMBERS ❧ ❧ ❧ Exercises ❧ ❧ ❧ In Exercises 1-6, reduce the given fraction to lowest terms by dividing numerator and denominator by the their greatest common divisor. 20 50 36 2. 38 10 3. 48 1. 36 14 24 5. 45 21 6. 36 4. In Exercises 7-12, reduce the given fraction to lowest terms by prime factoring both numerator and denominator and cenceling common factors. 153 170 198 8. 144 188 9. 141 7. 171 144 159 11. 106 140 12. 133 10. In Exercises 13-18, for each of the following problems, multiply numerators and denominators, then prime factor and cancel to reduce your answer to lowest terms. 18 14 20 3 13. · − 16. − · − 8 13 2 6 2 18 16 19 14. · − 17. − · 16 5 8 6 19 18 14 7 15. − · − 18. − · 4 13 4 17 In Exercises 19-24, for each of the following problems, first prime factor all numerators and denominators, then cancel. After canceling, multiply numerators and denominators. 5 12 21 36 19. − · − 20. − · − 6 49 17 46 44 21. − 21 12 · 10 55 22. − 49 52 · 13 51 CHAPTER 1. THE ARITHMETIC OF NUMBERS 55 54 · − 29 11 55 7 · − 24. 13 49 23. In Exercises 25-30, divide. Be sure your answer is reduced to lowest terms. 27 45 50 5 28. − ÷ ÷ − 25. 28 23 39 58 13 7 31 4 ÷ − 29. − 26. ÷ − 10 28 25 5 4 48 60 34 30. − ÷ − 27. − ÷ 13 35 17 31 In Exercises 31-38, add or subtract the fractions, as indicated, and simplify your result. 5 1 2 1 31. − + 35. − − − 6 4 4 9 1 5 1 1 32. − + 36. − − − 7 8 2 8 8 1 8 4 33. − + − 37. − − 9 3 9 5 1 1 4 1 34. − + − 38. − − 3 2 7 3 In Exercises 39-52, simplify the expression. 8 5 2 39. − − 9 2 5 8 7 1 40. − − 5 6 2 2 5 7 1 + − − 41. − 6 2 3 2 3 1 5 42. + − 2 2 8 9 8 9 1 43. − − + − 5 7 5 2 44. 45. 46. 47. 48. 2 5 6 1 − + − − 3 7 3 7 9 5 7 − + − 8 2 2 1 3 9 + − 2 2 4 2 7 9 2 − − − 5 2 5 2 3 2 1 − 4 3 4 45 1.3. THE RATIONAL NUMBERS 4 − 9 3 5 1 50. − − 2 6 3 49. 8 9 2 4 − − − 3 7 7 8 3 1 5 1 52. − − − 2 3 8 8 6 2 − 5 5 51. In Exercises 53-70, evaluate the expression at the given values. 53. xy − z 2 at x = −1/2, y = −1/3, and z = 5/2 62. ab+bc at a = −8/5, b = 7/2, and c = −9/7 63. x3 at x = −1/2 2 54. xy −z at x = −1/3, y = 5/6, and z = 1/3 2 64. x2 at x = −3/2 2 55. −5x + 2y at x = 3/4 and y = −1/2. 56. −2x2 + 4y 2 at x = 4/3 and y = −3/2. 57. 2x2 − 2xy − 3y 2 at x = 3/2 and y = −3/4. 2 65. x−yz at x = −8/5, y = 1/3, and z = −8/5 66. x − yz at x = 2/3, y = 2/9, and z = −3/5 67. −x2 at x = −8/3 2 58. 5x − 4xy − 3y at x = 1/5 and y = −4/3. 59. x + yz at x = −1/3, y = 1/6, and z = 2/5. 68. −x4 at x = −9/7 60. x + yz at x = 1/2, y = 7/4, and z = 2/3. 69. x2 + yz at x = 7/2, y = −5/4, and z = −5/3 61. ab+bc at a = −4/7, b = 7/5, and c = −5/2 70. x2 +yz at x = 1/2, y = 7/8, and z = −5/9 71. a + b/c + d is equivalent to which of the following mathematical expressions? 73. a + b/(c + d) is equivalent to which of the following mathematical expressions? b +d c a+b (c) +d c (a) a + (b) a+b c+d (d) a + b c+d 72. (a + b)/c + d is equivalent to which of the following mathematical expressions? b +d c a+b (c) +d c (a) a + (b) a+b c+d (d) a + b c+d 75. Use the graphing calculator to reduce 4125/1155 to lowest terms. b +d c a+b (c) +d c (a) a + (b) a+b c+d (d) a + b c+d 74. (a + b)/(c + d) is equivalent to which of the following mathematical expressions? b +d c a+b (c) +d c (a) a + (b) a+b c+d (d) a + b c+d 76. Use the graphing calculator to reduce 2100/945 to lowest terms. 46 CHAPTER 1. THE ARITHMETIC OF NUMBERS 77. Use the graphing calculator to simplify 45 70 · . 84 33 78. Use the graphing calculator to simplify 34 13 + . 55 77 ❧ ❧ ❧ 79. Use the graphing calculator to simplify 35 28 − ÷ − . 33 44 80. Use the graphing calculator to simplify 11 11 . − − − 84 36 Answers ❧ ❧ ❧ 1. 2 5 25. − 580 39 3. 5 24 27. − 930 289 5. 8 15 29. 7. 9 10 31. − 7 12 9. 4 3 33. − 11 9 3 11. 2 35. − 1 36 45 13. − 13 37. − 76 45 39. − 109 90 171 15. 26 19 17. − 3 19. 10 49 21. − 23. − 126 275 270 29 98 65 41. 79 36 43. 53 35 45. − 131 8 47. − 323 50 49. 62 45