Integers: Learn It 4

Adding Integers

Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with integers.

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more difficult.

Understanding the addition and subtraction of negative numbers can be challenging due to their abstract nature and the scarcity of everyday contexts where these operations are used. Math educators often employ various methods, including number lines, financial scenarios, and temperature changes, to illustrate these concepts with real world examples.

We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.

This figure has a blue circle labeled positive and a red circle labeled negative.

 

If we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in the figure below.

A blue counter represents [latex]+1[/latex]. A red counter represents [latex]-1[/latex]. Together they add to zero.

This figure has a blue circle over a red circle. Beside them is the statement 1 plus negative 1 equals 0.

 

Model the following expressions:

  1. [latex]-5+\left(-3\right)[/latex]
  2. [latex]-2+\left(-4\right)[/latex]
  3. [latex]-2+\left(-5\right)[/latex]

Now let’s see what happens when the signs are different.

Model: [latex]-5+3[/latex]

Notice that there were more negatives than positives, so the result is negative.

Model each addition.

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

Now that you have modeled adding small positive and negative integers, you can visualize the model in your mind to simplify expressions with any integers.

For example, if you want to add [latex]37+\left(-53\right)[/latex], you don’t have to count out [latex]37[/latex] blue counters and [latex]53[/latex] red counters.

Picture [latex]37[/latex] blue counters with [latex]53[/latex] red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because [latex]53 - 37=16[/latex], there are [latex]16[/latex] more negative counters.

[latex]37+\left(-53\right)=-16[/latex]

adding positive and negative integers

Operation Description Result
[latex]5 + 3[/latex] Both positive, sum positive Sum is positive
[latex]-5 + (-3)[/latex] Both negative, sum negative Sum is negative
[latex]5 + (-3)[/latex] Different signs, more positives Sum is positive
[latex]-5 + 3[/latex] Different signs, more negatives Sum is negative

When the signs are the same, add the numbers. When the signs are different, subtract the smaller absolute value from the larger absolute value to determine the sum’s sign based on which number is larger.

Simplify:

  1. [latex]19+\left(-47\right)[/latex]
  2. [latex]-32+40[/latex]

Think of positive numbers as money you have and negative numbers as money you owe. This will help you determine if your answer is positive or negative. 

[latex](-4) + 7[/latex] would be owing [latex]$4[/latex] and having [latex]$7[/latex], once you settle up, you still have [latex]$3[/latex]. So the answer would be positive [latex]3[/latex].

Another example is [latex](-3) + (-5)[/latex]. This means you owe [latex]$3[/latex] and you owe [latex]$5[/latex], so you owe [latex]$8[/latex], which would be represented by [latex]-8[/latex]. 

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers when evaluating expressions. In our first example we will evaluate a simple variable expression for a negative value.

Evaluate [latex]x+7\text{ when}[/latex]:

  1. [latex]x=-2[/latex]
  2. [latex]x=-11[/latex]

In the next example, we are give two expressions,[latex]n+1[/latex], and [latex]-n+1[/latex]. We will evaluate both for a negative number. This practice will help you learn how to keep track of multiple negative signs in one expression.

When [latex]n=-5[/latex], evaluate:

  1. [latex]n+1[/latex]
  2. [latex]-n+1[/latex]

Applications With Adding Integers

Recall that we were introduced to some situations in everyday life that use positive and negative numbers, such as temperatures, banking, and sports. For example, a debt of [latex]$5[/latex] could be represented as [latex]-$5[/latex]. Let’s practice translating and solving a few applications.

Solving applications is easy if we have a plan. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question.

The temperature in Buffalo, NY, one morning started at [latex]7[/latex] degrees below zero Fahrenheit. By noon, it had warmed up [latex]12[/latex] degrees. What was the temperature at noon?

In the next example, a football team gaining and losing yardage can be represented with positive and negative numbers.

A football team took possession of the football on their [latex]42[/latex]-yard line. In the next three plays, they lost [latex]\text{6 yards,}[/latex] gained [latex]4[/latex] yards and then lost [latex]8[/latex] yards. On what yard line was the ball at the end of those three plays?