Subtracting Integers Lesson

In our last lesson, we learned how to perform addition with integers. In this lesson, we will learn how to perform subtraction with integers. The subtraction process will only take on two additional steps from the addition process. When we subtract one integer from another:

• Note: Leave the Minuend or leftmost number unchanged
• Change the subtraction operation into addition
• Change the subtrahend or number being subtracted away into its opposite (change the sign)
• Find the sum of the two integers

Let's take a look at a few examples.
Example 1: Subtract each.
-11 - 5

• Change the subtraction operation into addition
• -11 - 5 » -11 + 5
• Change the subtrahend or number being subtracted away into its opposite (change the sign)
• -11 + 5 » -11 + (-5)
• Find the sum of the two integers
• -11 + (-5) = -16

-11 - 5 = -16
Example 2: Subtract each.
-8 - (-12)

• Change the subtraction operation into addition
• -8 - (-12) » -8 + (-12)
• Change the subtrahend or number being subtracted away into its opposite (change the sign)
• -8 + (-12) » -8 + 12
• Find the sum of the two integers
• -8 + 12 = 4

-8 - (-12) = 4
Example 3: Subtract each.
-29 - (-15)

• Change the subtraction operation into addition
• -29 - (-15) » -29 + (-15)
• Change the subtrahend or number being subtracted away into its opposite (change the sign)
• -29 + (-15) » -29 + 15
• Find the sum of the two integers
• -29 + 15 = -14

-29 - (-15) = -14

Subtracting More than two Integers

We previously learned that addition is commutative. This means we can add any number of addends in any order and not change the result. The same is not true for subtraction. Subtraction is not commutative; when we work with the subtraction operation, the order matters. If we are faced with a problem with multiple subtraction operations, we can change the subtraction into addition of the opposite. Once this is done, we can add in any order and obtain the correct result. To subtract more than two integers:

• Note: Leave the leftmost number unchanged
• Change each subtraction operation into addition
• Change each number being subtracted away into its opposite (change the sign)
• Find the sum of the integers

Let's take a look at a few problems:
Example 4: Subtract each.
-4 - (-8) - 5 - (-6)

• Change each subtraction operation into addition
• -4 - (-8) - 5 - (-6) » -4 + (-8) + 5 + (-6)
• Change each number being subtracted away into its opposite (change the sign)
• -4 + (-8) + 5 + (-6) » -4 + 8 + (-5) + 6
• Find the sum of the integers
• -4 + 8 + (-5) + 6 = -4 + (-5) + 8 + 6 = -9 + 14 = 5

-4 - (-8) - 5 - (-6) = 5
Example 5: Subtract each.
7 - 9 - (-24) - 19 - (-2)

• Change each subtraction operation into addition
• 7 - 9 - (-24) - 19 - (-2) » 7 + 9 + (-24) + 19 + (-2)
• Change each number being subtracted away into its opposite (change the sign)
• 7 + 9 + (-24) + 19 + (-2) » 7 + (-9) + 24 + (-19) + 2
• Find the sum of the integers
• 7 + (-9) + 24 + (-19) + 2 = 7 + 24 + 2 + (-9) + (-19) = 33 + (-28) = 5

7 - 9 - (-24) - 19 - (-2) = 5

Why Does Subtracting Away a Negative Turn Into Adding a Positive?

You may be a bit confused as to why subtracting away a negative turns into adding a positive. Let's first think about a simple subtraction pattern.

Using a Subtraction Pattern:

• 6 - 3 = 3
• 6 - 2 = 4
• 6 - 1 = 5
• 6 - 0 = 6
• 6 - (-1) = 7
• 6 - (-2) = 8

In our pattern above, the minuend or starting amount is always 6. As we work our way down the pattern, the subtrahend or amount being taken away is being decreased by 1, which causes the difference to increase by 1. Using this pattern, we can see that subtracting away a negative is the same as adding a positive.
6 - (-1) = 6 + 1 = 7
6 - (-2) = 6 + 2 = 8

Using The Number Line:

Let's begin with a simple question. What is the distance between 1 and 6 on the number line? To find the distance, we can count the number of units to travel from 1 to 6 or from 6 to 1. From our image above, we can see that the distance between 1 and 6 on the number line is 5. Notice that we can obtain the same result with a simple subtraction. To keep the formula simple, let's just subtract the smaller number from the larger.
6 - 1 = 5
Now let's ask a second question. What is the distance between -2 and 6 on the number line? To find the distance, we can count the number of units to travel from -2 to 6 or from 6 to -2. From our image above, we can see that the distance between -2 and 6 on the number line is 8. As we did before, let's subtract the smaller number from the larger.
6 - (-2) = ?
Since we already know the answer is 8, this suggests the following:
6 - (-2) = 6 + 2 = 8

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