We can easily add and subtract integers using the number line. Notice that the number line has a plus sign on the right and a minus sign on the left. If you have a problem such as 5 – 7, you can begin at 5 on the number line, then move 7 units to the left, or in the direction of the minus sign. After you have moved 7 units to the left, look at the number just under the spot on the number line that you are pointing to. It is a -2.

Similarly, you can have a problem such as -9 + 4. Begin by finding -9 on the number line, then move 4 units towards the plus sign and look at the number under the unit you are pointing to. You will find that it is -5.

Notice that you are moving towards the sign that is in front of the second number.

You can even use the number line to add and subtract several numbers, such as 5 – 7 + 3 + 6 – 2

Begin at 5 on the number line, move towards the minus sign 7 units, then towards the plus sign 3 units, then towards the plus sign another 6 units and finally towards the minus sign 2 units. You should end just above the 5 on the number line.

As you get skilled using the number line, you will see that when you are going towards the positive sign, you are going to the right, and when you are going towards the negative sign, you are going to the left. You should quickly get accustomed to this and be able to use the number line correctly for adding and subtracting even if the plus and minus signs are omitted.

Another good way to add or subtract integers is to understand the concept of absolute value. Knowing this idea, you can add and subtract integers easily.
 
 
Absolute Value
http://www.purplemath.com/modules/absolute.htm
 
Absolute value of an integer is its distance from zero. Opposite numbers have the same absolute value. For example, the absolute values of –3 and 3 or –5 and 5 are the same (Figure 2.7). If you count from zero, you will find that you are moving 5 units to get to both the number 5 and the number -5.

Figure 2.7

The absolute value of any number is positive. Absolute value of a number usually shown by placing the number between segments | |. These are some samples of showing absolute values of numbers.

1. |–12| = 12
    
2. |+39| = 39
    
3. |20| = 20
    
4. |–31| = 31
    
5. |0| = 0

Practice 4. Which number is at the furthest point from zero on a number line?
–2, 23, –34, 8, 29, 32, –22
 
Answer
 
 
How to Add and Subtract Integers?
http://www.iit.edu/~smart/dvorber/lesson4.htm
 
In adding and subtracting integers, we always deal with two cases.
 

1. All the integers are of like signs. Examples below are of this case.
    
   1. + 32 + 121 + 45 + 10 + 38
       
   2.  –21 – 45 – 67 – 76 – 101 – 3
       

   In all such cases, add the absolute values of the numbers. Then choose the sign of the numbers as the sign of the result.
   For example, in (a) the sign of the result is +. In (b), the sign of the result is –.
    

2. Two integers are of unlike signs. (–12 + 45) or (43 – 121) are samples of such case. To find the result, subtract the smaller absolute value from the larger one. Determine the sign of the number with larger absolute value. Use this sign as the sign of the result. (note to parents and students: You will not always have a calculator with you. Learning this concept will help you to add and subtract positive and negative numbers even when you do not have a calculator with you.)
   For example, in (–12 + 45) or (43 – 121), we find the results as below.
   
       –12 + 45 = +33
       43 – 121 = –78.

Practice 5. Calculate.

(a) 32 + 121 + 22 + 56 + 11
 
(b) –11 – 32 – 43 – 11 – 54
 
(c) 21 + 11 – 5 + 23 – 7 – 23

 
Answers
 
Practice 6. Place the sum of each row in the right column. Then add up the numbers in this column. This should be 63. Try doing this without a calculator.

Table 2.8
 

To check your addition and subtraction, also do the indicated operations down each column, then add across. The answer in the bottom right-hand rectangle should be the same, whether you are adding the far right hand column or adding across the bottom row.

Answer

 
Real Life Application 3. In Table 2.10, the operations of a company are shown during five days. Using integers, find the final profit of the company.

Table 2.10

Answer

 
General Rules for Combining Integers
http://www.mathgoodies.com/lessons/vol5/addition.html

If a and b are integers, then

a + (–b) = a – b
a – (+b) = a – b
a – (–b) = a + b
a + (+b) = a + b

Notice that when you have two negatives, or a negative and a minus sign on the same number, that number becomes positive. This works in the English language, too. The statement: “I refuse to talk to you.” Is a negative statement. The statement: “I will not refuse to talk to you.” Is a statement with two negatives in it – ‘refuse’ and ‘not’, but it is a positive statement whose meaning is that I will talk to you.

Practice 7. Compute

(a) 32 – (+45)
 
(b) –49 + (–23)
 
(c) 31 + (+28)
 
(d) –46 – (–26)

Answer
 
 
Practice 8. Calculate

(a) 12 + (–32) – 25
 
(b) –21 – (+43) + 19
 
(c) 15 – (–32) + 17
 
Answer
 
 
Real Life Application 4. The altitudes of Valley A and Mountain B are
–84 ft and 63 ft. What is the difference between these altitudes?

Answer


Practical Exercise 1 Compute
 
(a) –11 + 23 – (+31) – (–87)
 
(b) 34 – 43 – (–11) + (–34)
 
(c) –24 – (–11) – (–29)
 

Answer
 

Now let's do Practice Exercise 2-2 (top).