5.3 The Rational Numbers 249 Directions 1. Boil water or milk and salt (if desired). 2. Stir in oats. 3. Stirring occasionally, cook over medium heat for 5 minutes. Servings 1 2 Water (or milk) 1 cup 13 4 cup Oats 1 2 cup 1 cup Salt (optional) dash 1 8 tsp b) If the top of the frame is to be 26 in. 1 4 from the ceiling, how far from the ceiling should the nail be placed? (Assume the wire will not stretch.) 33 in. c) Repeat part (b) if the wire will stretch in. 1 4 when the picture is hung. 32 3 4 in. 18 "3 8 6 "3 4 24 "1 8 106. Cooking Oatmeal Following are the instructions given on a box of oatmeal. Determine the amount of water (or milk) and oats needed to make 11 2 servings by: a) Adding the amount of each ingredient needed for 1 serving to the amount needed for 2 servings and dividing by 2. 1 3 8 cup water (or milk) and 3 4 cup oatmeal b) Adding the amount of each ingredient needed for 1 serving to half the amount needed for 1 serving. 1 1 2 cup water (or milk) and 3 4 cup oatmeal integers. The set of rational numbers is dense because between any two distinct rational numbers there exists a third distinct rational number. For example, we can determine a rational number between 0.243 and 0.244. The number 0.243 can be written as 0.2430, and 0.244 can be written as 0.2440. There are many numbers between these two numbers. Some of them are 0.2431, 0.2435, and 0.243912. In Exercises 107–110, determine a rational number between the two numbers in each pair. Many answers are possible. 107. 0.21 and 0.22 0.215 108. 4.005 and 4.05 4.0275 109. −2.176 and −2.175 −2.1755 110. 1.3457 and 1.34571 1.345705 Halfway Between Two Numbers To determine a rational number halfway between any two rational numbers given in fraction form, add the two numbers together and divide their sum by 2. In Exercises 111–114, determine a rational number halfway between the two fractions in each pair. 111. 1 4 and 3 4 1 2 112. 1 5 and 2 5 3 10 113. 1 100 and 1 10 11 200 114. 7 13 and 8 13 15 26 115. Consider the rational number 0.9. a) Use the method from Example 8 to convert 0.9 to a quotient of integers. 1 1 or 1 b) Determine a number halfway between 0.9 and 1 by adding the two numbers and dividing by 2. 0.9 c) Determine + 1 3 2 3 by adding the fractions. Now express 1 3 and 2 3 as repeating decimals and determine the sum using the decimal representation of 1 3 and . 2 3 + + = + = 1 3 2 3 1, 0.3 0.6 0.9 d) What conclusion can you draw from parts (a), (b), and (c)? = 0.9 1 Research Activity 116. Ancient Greek Mathematicians The ancient Greeks were early contributors to mathematics. Write a report summarizing the ancient Greeks’ contributions to rational numbers. Include in your report what they learned and believed about the rational numbers. Dense Set of Numbers A set of numbers is said to be a dense set if between any two distinct members of the set there exists a third distinct member of the set. The set of integers is not dense, since between any two consecutive integers there is not another integer. For example, between 1 and 2 there are no other
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