Integers: Learn It 5

Subtracting Integers

Building on the fundamental principles of addition, we now turn our attention to the operation of subtraction. Subtraction often requires us to think in reverse.

Just as with addition, we can use counters to help us track the difference between positive and negative values in this process. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.

We will model four subtraction scenarios using the numbers [latex]5[/latex] and [latex]3[/latex].

  1. [latex]5 - 3[/latex]
  2. [latex]- 5-\left(-3\right)[/latex]
  3. [latex]-5 - 3[/latex]
  4. [latex]5-\left(-3\right)[/latex]
  1. Interpret the expression. [latex]5 - 3[/latex] means [latex]5[/latex] take away [latex]3[/latex].
    Model the first number. Start with [latex]5[/latex] positives. This figure has 5 blue circles with the number 5 underneath.
    Take away the second number. So take away [latex]3[/latex] positives. This figure has 5 blue circles. Under the last two circles, it say 2 positives.
    Find the counters that are left. This figure has two blue circles.
      [latex]5 - 3=2[/latex]

    The difference between [latex]5[/latex] and [latex]3[/latex] is [latex]2[/latex].
  2. Interpret the expression. [latex]-5-\left(-3\right)[/latex] means [latex]-5[/latex] take away [latex]-3[/latex].
    Model the first number. Start with [latex]5[/latex] negatives. This figure has 5 red circles labeled with a negative 5
    Take away the second number. So take away [latex]3[/latex] negatives. This figure has 5 red circles. Three are circled. Underneath the two that are not it is labeled 2 negatives.
    Find the number of counters that are left. This figure has two red circles.
      [latex]-5-\left(-3\right)=-2[/latex]

    The difference between [latex]-5[/latex] and [latex]-3[/latex] is [latex]-2[/latex].
  3. Interpret the expression. [latex]-5 - 3[/latex] means [latex]-5[/latex] take away [latex]3[/latex].
    Model the first number. Start with [latex]5[/latex] negatives. This figure has 5 red circles labeled negative 5.

    Take away the second number.

    So we need to take away [latex]3[/latex] positives.

    		  

    But there are no positives to take away.

    Add neutral pairs until you have [latex]3[/latex] positives.

    		 This figure has 8 red circles with a space between the first 5 and last three. Underneath the last three red circles, there are 3 blue circles.
    Now take away [latex]3[/latex] positives. This figure has 8 red circles with a space between the first 5 and last three. Underneath the last three red circles, there are 3 blue circles. The 3 blue circles are circled together.
    Count the number of counters that are left. This figure has 8 red circles labeled 8 negatives.
      [latex]-5 - 3=-8[/latex]

    The difference of [latex]-5[/latex] and [latex]3[/latex] is [latex]-8[/latex].
  4. Interpret the expression. [latex]5-\left(-3\right)[/latex] means [latex]5[/latex] take away [latex]-3[/latex].
    Model the first number. Start with [latex]5[/latex] positives. This figure has 5 blue circles with the number 5 labeled underneath.
    Take away the second number, so take away [latex]3[/latex] negatives.  

    But there are no negatives to take away.

    Add neutral pairs until you have [latex]3[/latex] negatives.

    		 This figure has 8 blue circles with a space between the first 5 and last 3. Underneath the last 3 blue circles there are 3 red circles.
    Then take away [latex]3[/latex] negatives. This figure has 8 blue circles with a space between the first 5 and last 3. Underneath the last 3 blue circles there are 3 red circles that are circled together.
    Count the number of counters that are left. 8 blue circles labeled 8 positives.
     

    The difference of [latex]5[/latex] and [latex]-3[/latex] is [latex]8[/latex].

    [latex]5-\left(-3\right)=8[/latex]

Now you can try an example that summarizes the situations above, with different numbers. Recall the different scenarios:

  • subtracting a positive number from a positive number
  • subtracting a positive number from a negative number
  • subtracting a negative number from a positive number
  • subtracting a negative number from a negative number
Model each subtraction.

  1. [latex]8 − 2[/latex]
  2. [latex]−5 − 4[/latex]
  3. [latex]6 − (−6)[/latex]
  4. [latex]−8 − (−3)[/latex]

Each of the examples so far have been carefully constructed so that the sign of the answer matched the sign of the first number in the expression.  For example, in  [latex]−5 − 4[/latex], the result is [latex]-9[/latex], which is the same sign as [latex]-5[/latex]. Now we will see subtraction where the sign of the result is different from the starting number.

Model each subtraction expression:

  1. [latex]2 - 8[/latex]
  2. [latex]-3-\left(-8\right)[/latex]

Now that you have seen subtraction modeled with color counters, we can move on to performing subtraction of integers without the models.

  • Subtract [latex]-23 - 7[/latex].
    • Think: We start with [latex]23[/latex] negative counters. We have to subtract [latex]7[/latex] positives, but there are no positives to take away. So we add [latex]7[/latex] neutral pairs to get the [latex]7[/latex] positives. Now we take away the [latex]7[/latex] positives. So what’s left? We have the original [latex]23[/latex] negatives plus [latex]7[/latex] more negatives from the neutral pair. The result is [latex]30[/latex] negatives.
      [latex]-23 - 7=-30[/latex]

      Notice, that to subtract [latex]\text{7,}[/latex] we added [latex]7[/latex] negatives.

  • Subtract [latex]30-\left(-12\right)[/latex].
    • Think: We start with [latex]30[/latex] positives. We have to subtract [latex]12[/latex] negatives, but there are no negatives to take away. So we add [latex]12[/latex] neutral pairs to the [latex]30[/latex] positives. Now we take away the [latex]12[/latex] negatives. What’s left? We have the original [latex]30[/latex] positives plus [latex]12[/latex] more positives from the neutral pairs. The result is [latex]42[/latex] positives.
      [latex]30-\left(-12\right)=42[/latex]

      Notice that to subtract [latex]-12[/latex], we added [latex]12[/latex].

While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters.

Have you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the Subtraction Property, written as follows: Subtracting a number is the same as adding its opposite.

[latex]a-b=a+(-b)[/latex]

Now let’s increase the complexity of the examples a little bit. We will use the order of operations to simplify terms in parentheses before we subtract from left to right.

Simplify: [latex]7-\left(-4 - 3\right)-9[/latex]

Applications With Subtracting Integers

It’s hard to find something if we don’t know what we’re looking for or what to call it. So when we solve an application problem, we first need to determine what we are asked to find. Then we can write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

Geography provides an application of negative numbers with the elevations of places below sea level.

Dinesh hiked from Mt. Whitney, the highest point in California, to Death Valley, the lowest point. The elevation of Mt. Whitney is [latex]14,497[/latex] feet above sea level and the elevation of Death Valley is [latex]282[/latex] feet below sea level. What is the difference in elevation between Mt. Whitney and Death Valley?

Managing your money can involve both positive and negative numbers.

Leslie has [latex]$25[/latex] in her checking account and she has a transaction for [latex]$8[/latex].

    1. What is the balance after the transaction?
    2. She has a second transaction for [latex]$20[/latex]. What is the new balance after this transaction?