234 CHAPTER 5 Number Theory and the Real Number System 65. Time Zone Calculations Part of a World Standard Time Zones chart used by airlines and the United States Navy is shown. The scale along the bottom is just like a number line with the integers 12, 11, , 11, 12 − − … on it. Note that only part of the scale from −12 to 12 is shown. a) Determine the difference in time between Nairobi (zone 3) + and Los Angeles (zone 8). − 11 hours b) Determine the difference in time between Casablanca (zone 1) − and Rio de Janeiro (zone 3). − 2 hours c) Determine the difference in time between Boston (zone 5) − and Tokyo (zone 9). + 14 hours d) Determine the difference in time between Puerto Vallarta (zone 7) − and Seoul (zone 8). + 15 hours 74. The quotient of any two negative integers is a negative integer. False; the quotient of any two negative integers is a positive integer. 75. The sum of a positive integer and a negative integer is always a positive integer. False; the sum of a positive integer and a negative integer may be a positive integer, a negative integer, or zero. 76. The product of a positive integer and a negative integer is always a positive integer. 77. Explain why the quotient of a nonzero number divided by 0 is undefined. Division by zero is undefined because = a x 0 always leads to a false statement. 78. Explain why = − − a b a b . The quotient of two positive numbers is a positive number, and the quotient of two negative numbers is a positive number. Challenge Problems/Group Activities 79. Determine the quotient: − + − + − + − + − + − + − + − 1 2 3 4 5 99 100 1 2 3 4 5 99 100 −1 80. Pentagonal Numbers Triangular numbers and square numbers were introduced in the Section 1.1 Exercises. There are also pentagonal numbers, which were also studied by the Greeks. Four pentagonal numbers are 1, 5, 12, and 22. 1 12 22 5 a) Determine the next three pentagonal numbers. 35, 51, and 70 b) Describe a procedure to determine the next five pentagonal numbers without drawing the figures. * c) Is 72 a pentagonal number? Explain how you determined your answer. Since 70 and 92 are consecutive pentagonal numbers, 72 is not a pentagonal number. 81. Place the appropriate plus or minus signs between each digit so that the total will equal 1. = 0123456789 1 +−++−+−−+= 01234567891 Recreational Mathematics 82. Four 4’s Refer to the Recreational Math box on page 228. a) Use the rules given on page 228 to represent the following whole numbers: 12, 15, 16, 17, 20. * b) We will now change our rules to allow the number 44 to count as two of the four 4’s. Use the number 44 and two other fours to represent the whole number 10. * Research Activity 83. History of the Number 0 Write a report on the history of the number 0 in the Hindu–Arabic numeration system. False; the product of a positive integer and a negative integer is always a negative integer. 8 pm 9 pm 10 pm 1 pm 2 pm 3 pm 11 pm 1 am 2 am 3 am 4 am 5 am 6 am 7 am 8 am 9 am 10 am +8 +10 +12 +2 -10 -12 -8 -6 -4 -2 0 11 am Midnight Noon Hours INTERNATIONAL DATE LINE MONDAY SUNDAY PRIME MERIDIAN Rio de Janeiro Munich Calgary Tokyo Sydney Seoul Casablanca Moscow Bogota Nairobi Baghdad Paris London Boston Amsterdam Los Angeles Puerto Vallarta 66. Bus Stops A city bus currently has 23 people on it. At the next stop, 10 people get off the bus, and 8 people get on the bus. At the next stop, 15 people get off the bus, and 6 people get on the bus. At the next stop, 2 people get off the bus, and 17 people get on the bus. At this point, how many people are on the bus? 27 people Concept/Writing Exercises In Exercises 67–76, determine whether the statement is true or false. Modify each false statement to make it a true statement. 67. Every whole number is an integer. True 68. Every integer is a whole number. False; every integer is not a whole number. 69. The difference of any two negative integers is a negative integer. False; the difference of two negative integers may be a positive integer, a negative integer, or zero. 70. The sum of any two negative integers is a negative integer. True 71. The product of any two positive integers is a positive integer. True 72. The difference of a positive integer and a negative integer is always a negative integer. False; the difference of a positive integer and a negative integer will always be positive. For example, − − = 5 ( 4) 9. 73. The quotient of a negative integer and a positive integer is always a negative integer. True *See Instructor Answer Appendix
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