2.2: Subtracting Integers

In Section 1.2, we stated that “Subtraction is the opposite of addition.” Thus, to subtract 4 from 7, we walked seven units to the right on the number line, but then walked 4 units in the opposite direction (to the left), as shown in Figure \(\PageIndex{1}\).

Figure \(\PageIndex{1}\): Subtraction requires that we reverse direction.

Thus, 7 − 4 = 3. The key phrase is “add the opposite.” Thus, the subtraction 7 − 4 becomes the addition 7 + (−4), which we would picture on the number line as shown in Figure \(\PageIndex{2}\).

Figure \(\PageIndex{2}\): Subtraction means add the opposite.

Figure \(\PageIndex{1}\) and Figure \(\PageIndex{2}\) provide ample evidence that the subtraction 7−4 is identical to the addition 7+(−4). Again, subtraction means “add the opposite.” That is, 7 − 4=7+(−4).

Defining Subtraction

Subtraction means “add the opposite.” That is, if a and b are any integers, then

\[a − b = a + (−b).\nonumber \]

Thus, for example, −123−150 = −123+(−150) and −57−(−91) = −57+91. In each case, subtraction means “add the opposite.” In the first case, subtracting 150 is the same as adding −150. In the second case, subtracting−91 is the same as adding 91.

Example 1

Find the differences: (a) 4 − 8, (b) −15 − 13, and (c) −117 − (−115).

Solution

In each case, subtraction means “add the opposite.”

a) Change the subtraction to addition with the phrase “subtraction means add the opposite.” That is, 4−8 = 4+(−8). We can now perform this addition on the number line.

Thus, 4 − 8=4+(−8) = −4.

b) First change the subtraction into addition by “adding the opposite.” That is, −15 − 13 = −15 + (−13). We can now use physical intuition to perform the addition. Start at the origin (zero), walk 15 units to the left, then an additional 13 units to the left, arriving at the answer −28. That is,

\[ \begin{aligned} −15 − 13 & = −15 + (−13) \\ ~ & = −28. \end{aligned}\nonumber \]

c) First change the subtraction into addition by “adding the opposite.” That is, −117 − (−115) = −117 + 115. Using “Adding Two Integers with Unlike Signs” from Section 2.2, first subtract the smaller magnitude from the larger magnitude; that is, 117 − 115 = 2. Because −117 has the larger magnitude and its sign is negative, prefix a negative sign to the difference in magnitudes. Thus,

\[ \begin{aligned} −117 − (−115) & = −117 + 115 \\ & = −2. \end{aligned}\nonumber \]

Exercise

Use each of the techniques in parts (a), (b), and (c) of Example 1 to evaluate the difference −11 − (−9).

Answer

−2

Order of Operations

We will now apply the “Rules Guiding Order of Operations” from Section 1.5 to a number of example exercises.

Example 2

Simplify −5 − (−8) − 7.

Solution

We work from left to right, changing each subtraction by “adding the opposite.”

\[ \begin{aligned} -5-(-8) -7=-5+8+(-7) ~ & \textcolor{red}{ \text{ Add the opposite of } -8, \text{ which is 8.}} \\ ~ & \textcolor{red}{ \text{ Add the opposite of 7, which is } -7.} \\ =3 +(-7) & \textcolor{red}{ \text{ Working left to right, } -5+8=3.} \\ =-4 ~ & \textcolor{red}{3 +(-7) = -4.} \end{aligned}\nonumber \]

Exercise

Simplify: −3 − (−9) − 11.

Answer

−5

Grouping symbols say “do me first.”

Example \(\PageIndex{1}\)

Simplify −2 − (−2 − 4).

Solution

Parenthetical expressions must be evaluated first.

\[ \begin{aligned} -2(-2-4)=-2-(-2+(-4)) ~ & \textcolor{red}{ \text{ Simplify the parenthetical expression}} \\ ~ & \textcolor{red}{ \text{ first. Add the opposite of 4, which is } -4.} \\ = -2 -(-6) ~ & \textcolor{red}{ \text{ Inside the parentheses, } -2 + (-4) = -6.} \\ =-2 + 6 ~ & \textcolor{red}{ \text{ Subtracting a } -6 \text{ is the same as adding a 6.}} \\ =4 ~ & ~ \textcolor{red}{ \text{ Add: } -2 + 6 = 4.} \end{aligned}\nonumber \]

Exercise

Simplify: −3 − (−3 − 3).

Answer

3

Change as a Difference

Suppose that when I leave my house in the early morning, the temperature outside is 40^◦ Fahrenheit. Later in the day, the temperature measures 60◦ Fahrenheit. How do I measure the change in the temperature?

The Change in a Quantity

To measure the change in a quantity, always subtract the former measurement from the latter measurement. That is:

\[ \colorbox{cyan}{Change in a Quantity} = \colorbox{cyan}{Latter Measurement} - \colorbox{cyan}{Former Measurement}\nonumber \]

Thus, to measure the change in temperature, I perform a subtraction as follows:

\[ \begin{aligned} \colorbox{cyan}{Change in Temperature} & = \colorbox{cyan}{Latter Measurement} & - & \colorbox{cyan}{Former Measurement} \\ ~ & = 60^{ \circ} \text{F} & - & 40^{ \circ} \text{F} \\ ~ & = 20^{ \circ} \text{F} \end{aligned}\nonumber \]

Note that the positive answer is in accord with the fact that the temperature has increased.

Example 4

Suppose that in the afternoon, the temperature measures 65^◦Fahrenheit, then late evening the temperature drops to 44^◦ Fahrenheit. Find the change in temperature.

Solution

To measure the change in temperature, we must subtract the former measurement from the latter measurement.

\[ \begin{aligned} \colorbox{cyan}{Change in Temperature} & = \colorbox{cyan}{Latter Measurement} & - & \colorbox{cyan}{Former Measurement} \\ ~ & = 44^{ \circ} \text{F} & - & 65^{ \circ} \text{F} \\ ~ & = -11^{ \circ} \text{F} \end{aligned}\nonumber \]

Note that the negative answer is in accord with the fact that the temperature has decreased. There has been a “change” of −11^◦ Fahrenheit.

Exercise

Marianne awakes to a morning temperature of 54^◦ Fahrenheit. A storm hits, dropping the temperature to 43^◦ Fahrenheit. Find the change in temperature.

Answer

−11◦ Fahrenheit

Example 5

Sometimes a bar graph is not the most appropriate visualization for your data. For example, consider the bar graph in Figure \(\PageIndex{3}\) depicting the Dow Industrial Average for seven consecutive days in March of 2009. Because the bars are of almost equal height, it is difficult to detect fluctuation or change in the Dow Industrial Average.

Let’s determine the change in the Dow Industrial average on a day-to-day basis. Remember to subtract the latter measurement minus the former (current day minus former day). This gives us the following changes.

Consecutive Days	Change in Dow Industrial Average
Sun-Mon	6900 - 7000 = -100
Mon-Tues	6800 - 6900 = -100
Tues-Wed	6800 - 6800 = 0
Wed-Thu	7000 - 6800 = 200
Thu-Fri	7100 - 7000 = 100
Fri-Sat	7200 - 7100 = 100

Figure \(\PageIndex{3}\): Profit and loss bar graph.

We will use the data in the table to construct a line graph. On the horizontal axis, we place the pairs of consecutive days (see Figure \(\PageIndex{4}\)). On the vertical axis we place the Change in the Industrial Dow Average. At each pair of days we plot a point at a height equal to the change in Dow Industrial Average as calculated in our table.

Figure \(\PageIndex{4}\): Plotting change in Dow versus consecutive days.

Note that the data as displayed by Figure \(\PageIndex{4}\) more readily shows the changes in the Dow Industrial Average on a day-to-day basis. For example, it is now easy to pick the day that saw the greatest increase in the Dow (from Wednesday to Thursday, the Dow rose 200 points).

Exercises

In Exercises 1-24, find the difference.

1. 16 − 20

2. 17 − 2

3. 10 − 12

4. 16 − 8

5. 14 − 11

6. 5 − 8

7. 7 − (−16)

8. 20 − (−10)

9. −4 − (−9)

10. −13 − (−3)

11. 8 − (−3)

12. 14 − (−20)

13. 2 − 11

14. 16 − 2

15. −8 − (−10)

16. −14 − (−2)

17. 13 − (−1)

18. 12 − (−13)

19. −4 − (−2)

20. −6 − (−8)

21. 7 − (−8)

22. 13 − (−14)

23. −3 − (−10)

24. −13 − (−9)

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In Exercises 25-34, simplify the given expression.

25. 14 − 12 − 2

26. −19 − (−7) − 11

27. −20 − 11 − 18

28. 7 − (−13) − (−1)

29. 5 − (−10) − 20

30. −19 − 12 − (−8)

31. −14 − 12 − 19

32. −15 − 4 − (−6)

33. −11 − (−7) − (−6)

34. 5 − (−5) − (−14)

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In Exercises 35-50, simplify the given expression.

35. −2 − (−6 − (−5))

36. 6 − (−14 − 9)

37. (−5 − (−8)) − (−3 − (−2))

38. (−6 − (−8)) − (−9 − 3)

39. (6 − (−9)) − (3 − (−6))

40. (−2 − (−3)) − (3 − (−6))

41. −1 − (10 − (−9))

42. 7 − (14 − (−8))

43. 3 − (−8 − 17)

44. 1 − (−1 − 4)

45. 13 − (16 − (−1))

46. −7 − (−3 − (−8))

47. (7 − (−8)) − (5 − (−2))

48. (6 − 5) − (7 − 3)

49. (6 − 4) − (−8 − 2)

50. (2 − (−6)) − (−9 − (−3))

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51. The first recorded temperature is 42^◦F. Four hours later, the second temperature is 65^◦F. What is the change in temperature?

52. The first recorded temperature is 79^◦F. Four hours later, the second temperature is 46^◦F. What is the change in temperature?

53. The first recorded temperature is 30^◦F. Four hours later, the second temperature is 51^◦F. What is the change in temperature?

54. The first recorded temperature is 109^◦F. Four hours later, the second temperature is 58^◦F. What is the change in temperature?

55. Typical temperatures in Fairbanks, Alaska in January are −2 degrees Fahrenheit in the daytime and −19 degrees Fahrenheit at night. What is the change in temperature from day to night?

56. Typical summertime temperatures in Fairbanks, Alaska in July are 79 degrees Fahrenheit in the daytime and 53 degrees Fahrenheit at night. What is the change in temperature from day to night?

57. Communication. A submarine 1600 feet below sea level communicates with a pilot flying 22,500 feet in the air directly above the submarine. How far is the communique traveling?

58. Highest to Lowest. The highest spot on earth is on Mount Everest in Nepal-Tibet at 8,848 meters. The lowest point on the earth’s crust is the Mariana’s Trench in the North Pacific Ocean at 10,923 meters below sea level. What is the distance between the highest and the lowest points on earth? Wikipedia http://en.Wikipedia.org/wiki/Extremes_on_Earth

59. Lowest Elevation. The lowest point in North America is Death Valley, California at -282 feet. The lowest point on the entire earth’s landmass is on the shores of the Dead Sea along the Israel-Jordan border with an elevation of -1,371 feet. How much lower is the Dead Sea shore from Death Valley?

60. Exam Scores. Freida’s scores on her first seven mathematics exams are shown in the following bar chart. Calculate the differences between consecutive exams, then create a line graph of the differences on each pair of consecutive exams. Between which two pairs of consecutive exams did Freida show the most improvement?

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Answers

1. −4

3. −2

5. 3

7. 23

9. 5 11. 11 13. −9 15. 2 17. 14 19. −2 21. 15 23. 7 25. 0 27. −49 29. −5 31. −45 33. 2 35. −1 37. 4 39. 6 41. −20 43. 28 45. −4 47. 8 49. 12 51. 23◦ F 53. 21◦ F 55. −17 degrees Fahrenheit 57. 24,100 feet 59. 1,089 feet lower