SECTION 2.1 the Integers

2.1 The Integers

1. Read, write and say integers. 2. Compare integers. SWBAT

3. Find opposites. 4. Find absolute values.

Read, Write and Say Integers “Got VISA?” In 1970, college students were not eligible for credit cards. Today, special high-rate credit cards are available to people who may have trouble paying their bills. When we borrow or take on debt, we can record this amount using a negative number. We record our income as a positive number. Together these are called signed numbers.

In this section we begin our study of signed numbers by studying the set of numbers called integers.

Definition: The integers are the counting numbers, their opposites, and zero. Zero is neither positive nor negative. We can write the integers using braces { } to denote a set: }

The diagram at right gives the relationship of groups within the integers. The counting numbers { 1, 2, 3, 4, . . . } were the first numbers humans used, and these are still the numbers children learn first. The whole numbers are the set of counting numbers and zero { 0, 1, 2, 3, . . . } . The integers do not include fractions or decimals. The integers can all be put on a number line. An integer number line stretches to infinity in both directions and lets us see which numbers are smaller and which are larger:

Positive integers are all greater than zero, so they are on the right of zero. Negative integers are all less than zero, so they are on the left of zero. Positive numbers do not need a sign in front of them; for instance, we assume that. Sometimes we may choose to use the sign for clarity, but it is never necessary. Negative integers, like all negative numbers, always have a dash () in front of them to show that they are negative. Expression How to Say It “ subtract ” “take 7 from 3” “3take away ” “3 minus 7” “negative seven” or “the opposite of ” “the opposite of ” “the opposite of ” “the opposite of the difference of and ” “negative four minus negative three" Subtraction is also shown by a dash. We also use a dash to mean “opposite." This can be confusing. As you read this section and study this unit, you will want to refer to the table below. It is a good idea to mark this page in your book! Integers are used in business, technology, science, the trades, medicine, games, and household finances.

Example 1: Fill in the correct negative integers. a. Joshua pays $382 for car insurance. In his check book he writes ______. ANSWER:

b. The coldest natural temperature ever recorded on Earth was below zero Fahrenheit , ______, recorded in Antarctica on July 21, 1983. ANSWER: F c. When Oregon State University scientists test for oxygen in the water off Cape Perpetua, the scientists take samples at 150 feet below sea level, that is ______. ANSWER: ft

Check Point 1

Translate into negative integers. a. Jonathan lost 38 points in this round. Jonathan’s score is ______in this round. b. The shore of the Dead Sea has an elevation of 1,371 feet below sea level. The shore of the Dead Sea has an elevation of ______ft. c. The Seahawks lost 4 yards on their last play. The Seahawk's yardage was recorded as ______yards on their last play. Compare Integers We compare integers two at a time. The first integer is either greater than (>) the second integer, less than (<) the second integer, or equal to (=) the second integer. The number line shows how two numbers are related. The farther left on the number line, the smaller the number. The farther right on the number line, the larger the number.

Example 2: Compare integers using < or > and use the number line to check your results.

a. b. ANSWERS: a. b.

Check Point 2

Compare integers using < or > and use the number line to check your results here. a. b.

Find Opposites DEFINITION: Opposites are numbers that are the same distance from zero and are located in opposite directions from zero on the number line.

We can find opposites on the number line. Notice that the opposite of a positive number is negative, and the opposite of a negative is positive. Zero is its own opposite. On the number line we see that 6 is the same distance from zero as -6, but since they are located on opposite sides of zero, they are opposites. Similarly, -2 and 2 are opposites: Often the opposite of a negative number shows up in technical reading. For example, in working a medical calculation, a nursing student comes to a point where she has on her paper. She needs to know what this means. We say this number is “the opposite of ” We also simplify this number by thinking about the number line. The opposite of negative 8 is positive 8.

PROPERTY: The Op-Op Property: For any number,

We can take the opposite of the opposite of the opposite of a number, and this will simplify too. In Example 3 we have simplified each number and hope that you will find the pattern for taking repeated opposites. Use this pattern to complete the Check Point problems.

Example 3: Simplify. a. =______b. = ______c. = ______d. = ______e. = ______f. =______

ANSWERS: a. 19 b. c. 2 d. e. 55 f.

Check Point 3

Simplify. a. ______b. ______c. ______d. ______e. ______Write the pattern for finding repeated opposites below and be prepared to discuss in class: ______

Find Absolute Values When we add, subtract, multiply, and divide integers, the distance the integer is from zero is very important. When we travel, distance is never negative. If we travel 20 miles north, this is the same distance as traveling 20 miles south. Both numbers are positive even though the directions are opposite. We use absolute value to find the distance a number is from zero no matter which direction it is from zero.

DEFINITION: The absolute value of any number is its distance from zero. Distance is always positive. We use straight brackets to represent the absolute value of the number The absolute value of any number is positive, . The opposite of the absolute value of any number is negative, .

Example 4: Simplify. a. _____ b. = _____ c. = _____ d. = _____ e. _____ f. = _____ g. = _____ h. = _____

ANSWERS: a. 3 b. 54 c. 73 d. 4,563 e. -3 f. -54 g. -73 h. -4,563

Check Point 4

a. b. c. = _____ d. e. f. = _____ g. h.  Check Point 5 Carefully determine if parentheses or absolute value bars are being used, and then simplify accordingly.

a. b. c. = _____ d. e. f. = _____ g. h.

Read Integers on a Graph and Solve Applications

Example 5: By reading the graph below, record the approximate average monthly low temperature for the months of January, February, and March for Barrow, Alaska. January: ______February: ______March: ______

ANSWERS: January ≈ -18°F , February ≈ -23°F, March ≈ -21°F

Check Point 6

In the table provided, write the difference in stock price for the given days. The first difference has been placed in the table for you.

Date Transaction Amount 9/26 RCC bookstore textbooks 9/27 Groceries 9/30 VISA payment

Example 6: Cheryl recorded all of her VISA transactions in a ledger. She is up to date through the month of September.

In the first week of October, she kept these receipts to record: October 1: Gas $42, October 1: prescription $26, October 7: payment to VISA $100.

Record these payments in her ledger to the right.

Check Point 7

Transaction Date Amount

Jeffrey is keeping track of his transactions. On October 1 he deposits $1,890, writes a rent check for $695, and writes the $297 check for his car payment. On October 3, Jeffrey writes a $252 check for groceries. On October 6, Jeffrey deposits a $40 refund check and then writes a check for $122 for cable. Fill in his check ledger. 2.1 Exercise Set Name ______Skills

Find the absolute value. 1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

Write the opposite of the following: 11 12. 13. 14. 15 . . 16 17. 18. 19. 20 . .

Find the integer that makes the following equations true. 21. 22. 23. 24.

For each pair of numbers below, determine which number is larger. Place the appropriate < or > symbols in between the pair. 25. 26. 27. 28.

29. 30. 31. 32.

33. 34. 35. 36.

Find when given the following values for x: 37. 38. 39. 40. 41. 42.

43. a. Circle the following numbers that are integers: 0

b. Circle the following numbers that are NOT integers: 0

44 Place the following numbers on the number . line: Use one of the symbols, =, < or > to compare the two numbers with grouping symbols. Simplify first. 45 46 47 48 . . . .

49 50 51 52 . . . .

Applications

Translate the words in italics in the following sentences using inequality symbols, < or >. For example: Mary makes more money per hour than Alan. The answer is: >

53. The number of cactus plants is less than other plants. ______

54. Jonny complains that his dish of ice cream is smaller than his sister’s. ______

55. I was taller than my sister at age 4. ______

56. The studs used in the living room are shorter than those in the bedroom. ______

57. Anita's spelling test score always exceeds that of her brother. ______

58. The water level behind the dam is lower than it was last year. ______59. Simplify . (Remember that absolute value symbols are grouping symbols.)

60. a. Simplify –(–4).

b. Why is this result different from that for number 59?

61. You were over (spent more than) your 62. You were over your budget last month budget last month by $50. The month by $32. The month before you were before you were under your budget by under your budget by $41. Express each $27. Express each of these numbers as of these numbers as integers. integers.

63. The coldest temperature recorded in 64. The coldest temperature ever recorded Fairbanks, Alaska, was F degrees below on Earth on July 21, 1983, was reported zero. Express this temperature as an as F below zero. Express this integer. temperature as an integer.

65. Death Valley, California, has an elevation 66. The Salton Trough runs in the of 86 meters (283 feet) below sea level. southwestern U.S. and into Mexico, and Express this elevation, in both meters has an elevation of 69 meters below sea and feet, using integers. level. Express this elevation as an integer.

67. The Dead Sea Depression, located along 68. The San Julian's Great Depression, the borders of Israel, Syria, and Jordan, located in Argentina, has an elevation of has an elevation of about 413 meters 103 meters (344 feet) below sea level. (1,355 feet) below sea level. Express this Express this elevation, in both meters elevation, in both meters and feet, using and feet, using integers. integers.

69. Your credit card bill for one month was 70. Your credit card bill for one month was $223. Express this debt as an integer. $491. Express this debt as an integer.

71. Your golf score was 4 below par. Express 72. (See number 71) You are 2 above par in this score as an integer. golf. Is this a positive or negative score? Review

For problems 73 and 74, simplify using correct order of operations and show your work below. 73. 74.

75. Find the area of the following shape: 76. Find the area of the following shape:

77. What is the average of 76, 89, 80, and 75? 78. What is the average of 15, 27, 25, and 5? 79. Add 189 + 54 mentally and then explain 80. Subtract 189 – 94 mentally and then how you did this mentally. explain how you did this mentally.

81. Divide 365 by 5 mentally and explain how 82. Multiply 1,200 by 5 mentally and explain you did this mentally. how you did this mentally.