Positive and Negative Integers


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1 0.4 Positive and Negative Integers 0.4 OBJECTIVES 1. Represent integers on a number line 2. Order signed numbers 3. Evaluate numerical expressions involving absolute value When numbers are used to represent physical quantities (altitudes, temperatures, and amounts of money are examples), it may be necessary to distinguish between positive and negative quantities. It is convenient to represent these quantities with plus ( ) or minus ( ) signs. For instance, The altitude of Mount Whitney is 14,495 feet (ft) above sea level ( 14,495). 14,495 ft Mount Whitney The altitude of Death Valley is 282 ft below sea level ( 282). 282 ft Death Valley The temperature in Chicago is 10 below zero ( 10 )
2 36 CHAPTER 0 AN ARITHMETIC REVIEW An account could show a gain of $100 ( 100), or a loss of $100 ( 100). These numbers suggest the need to extend the whole numbers to include both positive numbers (like 100) and negative numbers (like 282). To represent the negative numbers, we extend the number line to the left of zero and name equally spaced points. Numbers used to name points to the right of zero are positive numbers. They are written with a positive ( ) sign or with no sign at all. +6 and 9 are positive numbers Numbers used to name points to the left of zero are negative numbers. They are always written with a negative ( ) sign. 3 and 20 are negative numbers Read negative 3. Positive and negative numbers considered together are signed numbers. Here is the number line extended to include both positive and negative numbers. NOTE 0 is not considered a signed number. Zero is neither positive nor negative Negative numbers Positive numbers NOTE The dots are called ellipses and indicate that the pattern continues. The numbers used to name the points shown on the number line above are called the integers. The integers consist of the natural numbers, their negatives, and the number 0. We can represent the set of integers by {..., 3, 2, 1, 0, 1, 2, 3,...}
3 POSITIVE AND NEGATIVE INTEGERS SECTION Example 1 Representing Integers on the Number Line Represent the following integers on the number line shown. 3, 12, 8, 15, CHECK YOURSELF 1 Represent the following integers on a number line. 1, 9, 4, 11, 8, The set of numbers on the number line is ordered. The numbers get smaller moving to the left on the number line and larger moving to the right When a set of numbers is written from smallest to largest, the numbers are said to be in ascending order. Example 2 Ordering Signed Numbers Place each set of numbers in ascending order. (a) 9, 5, 8, 3, 7 From smallest to largest, the numbers are 8, 5, 3, 7, 9 Note that this is the order in which the numbers appear on a number line as we move from left to right. (b) 3, 2, 18, 20, 13 From smallest to largest, the numbers are 20, 13, 2, 3, 18 CHECK YOURSELF 2 Place each set of numbers in ascending order. (a) 12, 13, 15, 2, 8, 3 (b) 3, 6, 9, 3, 8
4 38 CHAPTER 0 AN ARITHMETIC REVIEW The least and greatest numbers in a set are called the extreme values. The least element is called the minimum and the greatest element is called the maximum. Example 3 Labeling Extreme Values For each set of numbers, determine the minimum and maximum values. (a) 9, 5, 8, 3, 7 From our previous ordering of these numbers, we see that 8, the least element, is the minimum, and 9, the greatest element, is the maximum. (b) 3, 2, 18, 20, is the minimum and 18 is the maximum. CHECK YOURSELF 3 For each set of numbers, determine the minimum and maximum values. (a) 12, 13, 15, 2, 8, 3 (b) 3, 6, 9, 3, 8 Integers are not the only kind of signed numbers. Decimals and fractions can also be thought of as signed numbers. Example 4 Identifying Signed Numbers that are Integers Which of the following signed numbers are also integers? (a) 145 is an integer. (b) 28 is an integer. (c) 0.35 is not an integer. (d) 2 3 is not an integer. CHECK YOURSELF 4 Which of the following signed numbers are also integers? Sometimes we refer to the negative of a number as its opposite. But what is the opposite of the opposite of a number? It is the number itself. The next example illustrates.
5 POSITIVE AND NEGATIVE INTEGERS SECTION Example 5 Find the Opposite for Each Number (a) 5 The opposite of 5 is 5. (b) 9 The opposite of 9 is 9. CHECK YOURSELF 5 Find the opposite for each number. (a) 17 (b) 12 An important idea for our work in this chapter is the absolute value of a number. This represents the distance of the point named by the number from the origin on the number line. 5 units 5 units The absolute value of 5 is 5. The absolute value of 5 is also 5. The absolute value of a positive number or zero is itself. The absolute value of a negative number is its opposite. In symbols we write 5 5 and 5 5 Read the absolute Read the absolute value of 5. value of negative 5. The absolute value of a number does not depend on whether the number is to the right or to the left of the origin, but on its distance from the origin. Example 6 Simplifying Absolute Value Expressions (a) 7 7 (b) 7 7 (c) 7 7 This is the negative, or opposite, of the absolute value of negative 7. (d) Absolute value bars serve as another set of grouping symbols, so do the operation inside first. (e) (f) Here, evaluate the absolute values, then subtract.
6 40 CHAPTER 0 AN ARITHMETIC REVIEW CHECK YOURSELF 6 Evaluate. (a) 8 (b) 8 (c) 8 (d) 9 4 (e) 9 4 (f) 9 4 CHECK YOURSELF ANSWERS (a) 13, 8, 3, 2, 12, 15 (b) 9, 3, 3, 6, (a) minimum is 13; maximum is 15 (b) minimum is 9; maximum is , 1054, 0, and (a) 17; (b) (a) 8; (b) 8; (c) 8; (d) 13; (e) 5; (f ) 5
7 Name 0.4 Exercises Section Date Represent each quantity with a signed number. 1. An altitude of 400 feet (ft) above sea level 2. An altitude of 80 ft below sea level 3. A loss of $ A profit of $ A decrease in population of 25, An increase in population of 12,500 Represent the integers on the number lines shown. ANSWERS , 15, 18, 8, , 4, 5, 13, 9 Which numbers in the following sets are integers? , 2, 175, 234, , 0.35, 3, 700, 26 5 Place each of the following sets in ascending order , 5, 2, 0, 7, 1, , 7, 1, 8, 6, 1, , 2, 11, 4, 6, 1, , 18, 5, 11, 15, 14, , 7, 7, 6, 3, , 13, 14, 14, 15, For each set, determine the maximum and minimum values , 6, 0, 10, 3, 15, 1, , 1, 3, 11, 4, 2, 5, , 15, 0, 7, 9, 16, 3, , 0, 22, 31, 18, 5, , 0, 1, 2, 5, 4, , 7, 3, 5, 10, 5 Find the opposite of each number
8 ANSWERS Evaluate
9 ANSWERS Label each statement as true or false. 53. All whole numbers are integers. 54. All nonzero integers are signed numbers. 55. All integers are whole numbers. 56. All signed numbers are integers. 57. All negative integers are whole numbers. 58. Zero is neither positive nor negative. Place absolute value bars in the proper location on the left side of the expression so that the equation is true ( 2) ( 3) ( 2) ( 3) 11 Represent each quantity with a signed number. 63. Soil erosion. The erosion of 5 centimeters (cm) of topsoil from an Iowa corn field. 64. Soil formation. The formation of 2.5 cm of new topsoil on the African savanna. 65. Checking accounts. The withdrawal of $50 from a checking account. 66. Saving accounts. The deposit of $200 in a savings account. 67. Temperature. The temperature change pictured F F 1:00 P.M. 2:00 P.M. 68. Stocks. An increase of 75 points in the DowJones average. JUNE JULY 43
10 ANSWERS Baseball. An eightgame losing streak by the local baseball team. 70. Population. An increase of 25,000 in the population of the city. 71. Positive trade balance. A country exported $90,000,000 more than it imported, creating a positive trade balance. 72. Negative trade balance. A country exported $60,000,000 less than it imported, creating a negative trade balance. For each collection of numbers given in exercises 73 to 76, answer the following: (a) Which number is smallest? (b) Which number lies farthest from the origin? (c) Which number has the largest absolute value? (d) Which number has the smallest absolute value? 73. 6, 3, 8, 7, , 3, 5, 4, , 6, 1, 0, 2, , 0, 2, 3, Simplify each of the following: ( 7) ( ( 7)) ( ( ( 7))) Based on your answers, generalize your results. Answers or ( 400) , , 175, , 5, 1, 0, 2, 3, , 6, 2, 1, 4, 5, , 6, 3, 3, 6, Max: 15; Min: Max: 21, Min: Max: 5; Min: True 55. False 57. False ( 2) F ,000, ; 8; 8; ; 6; 6;
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