# Subtracting Integers

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Eric Clark

## DescriptionSubtracting Integers

Subtracting integers is not difficult, if you follow one easy rule. Integers are positive and negative whole numbers including zero, such as -5, -2, 0, 3, 9, and so on.

To find the difference of integers, or of positve and negative numbers, think of subtraction another way: instead of taking away a number, you can add the opposite. Opposite numbers have the same distance from zero, or also called 'absolute value.' On the number line, you can see that -2 and 2 have the same distance from zero. Therefore 2 and -2 are opposite numbers. Opposite numbers are always numbers that look similar apart from the positive or negative sign.

So if you have to subtract a negative number from a positive number, simply think of it as adding the opposite, positive number. This concept helps you figuring out the solution even without using the number line or your calculator. The same concept works for adding integers, especially for negative numbers: instead of adding a negative number to a positive number, just subtract the opposite, positive number.

This video shows you how to subtract integers involving something you possible know from your childhood: a see-saw. This will help you understand the concept behind the rule, so you will remember it for the rest of your math career. This video also includes a more complex example, but don't worry: you will be guided step by step through the example and you might discover that the rule makes it much easier than you thought.

Apply and extend previous understandings of numbers to the system of rational numbers. CCSS.MATH.CONTENT.6.NS.C.5

### TranscriptSubtracting Integers

Have you ever tried to balance a seesaw? I will explain to you what this has to do with subtracting integers. How can we bring this seesaw into balance?

We need to put a value here, on the right side, to get it in balance. In this case, a positive 3 balances the seesaw. We can write this as an addition: + 3. Let us go back to the first situation. Can you imagine another way to bring this seesaw in balance? Right!

### Subtracting a Negative Number – Example 1

Another way is to take something away on the left side. In this case, negative 3 can be taken away from the left side. We can express this mathematically as minus a negative 3. As you can see the seesaw is in balance again, which is the same as in the situation before. Taking negative 3 away equals adding positive 3. Remember: Subtracting a number equals the same as adding the opposite. So, subtracting a negative 3 is the same as adding a positive 3.

### Subtracting a Negative Number – Example 2

Now let's look at a different example. Find the value of negative 2 minus negative 4 minus 3. Instead of minus 3 you can also write minus positive 3. You have learned that subtracting a number equals the same as adding the opposite number. In our example minus negative four becomes plus positive four and instead of minus positive three you can write plus negative three. Ok, let's calculate from left to right: −2 + 4 = 2 and 2 + −3 = −1.

You have learned that subtracting a negative number is the same as adding the opposite number. But might this knowledge help our little friend in his situation? Maybe...

## Subtracting Integers Exercise

Would you like to practice what you’ve just learned? Practice problems for this video Subtracting Integers help you practice and recap your knowledge.
• ### Find the integer that brings the seesaw back into balance.

Hints

Which of the numbers on the left side of the seesaw do not have an opposite pair on the right side?

The absolute value of the left side is $16$. Remember, absolute value is the value of a number without the sign.

Solution

If we look at the left side of the equation, we see the integers $-7$ and $-9$. On the right-hand side, we have a $+7$.

The opposite of $-7$ is $+7$, since the absolute value of these numbers is equal.

Since there's also a $-9$ on the left-hand side, we can balance the seesaw by putting a number that is equal to the opposite of $-9$ on the right-hand side.

In this case, a $+9$ would balance the equation (seesaw).

• ### Describe how to solve the equation $-2 - (-4) -3$.

Hints

Subtracting a negative number is the same as adding a positive number.

Subtracting a positive number is the same as adding a negative number.

Remember to use the proper Order of Operations. (PEMDAS)

Solution

• $(-2)-(-4)-3$
First, we can simplify the signs. Knowing that subtracting a negative number is the same as adding a positive number, we can simplify this equation as follows:

• $(-2)+(+4)-3$
• $(-2)+4-3$
At this point, we can solve. Alternatively, if we need more support, we can change the next minus sign to a plus sign and rewrite the equation like so:

• $(-2)+4+(-3)$
Now that we have our equation, we just need to follow the order of operations to solve. Remember, we have to work left to right.

• $(-2)+4=2$
• $2+(-3)=-1$
The answer is $-1$!

• ### Calculate the temperature change by subtracting integers.

Hints

Subtracting a negative number is the same as adding a positive number.

To calculate the difference of two numbers they have to be subtracted.

The result is always a positive value.

Solution

The temperature on day one is $4$ degrees. The temperature on day two is $-3$ degrees. In order to find the difference in the temperatures, you should subtract $-3$ from $4$. We can write this equation as follows:

$4-(-3)$

Since we know that subtracting a negative number is the same as adding a positive number, we can rewrite the equation and solve:

$4+3=7$

So the temperature changed by $7$ degrees.

$\begin{array}{rcl} 4 - (-3) & = & 4 + 3\\ & = & 7 \end{array}$

• ### Evaluate John's final score by adding and subtracting integers.

Hints

Adding a negative number is the same as subtracting a positive number.

Subtracting a negative number is the same as adding a positive number.

To solve mathematical expressions, follow the rules of PEMDAS.

Solution

Let's look at John's throws:

John's first throw results in a positive $2$. On John's second throw, he hits $(+/-)$ and a $3$ on his third throw. The fourth throw is a $(-/-)$, and he hits a $6$ on his final throw.

That will leave us with $2+(-3)-(-6)$. Using the Order of Operations (PEMDAS), we can now solve from left to right.

$\begin{array}{rcl} 2 + (-3) -(-6) & = & ~~~2 + (-3) + 6\\ & = & -1 + 6 \\ & = & ~~~5 \end{array}$

John's final score is $5$.

• ### Identify the correct statements regarding subtracting integers.

Hints

Try comparing different combinations of adding and subtracting positive and negative numbers.

Solution

The statements are provided below:

• 'Subtracting a positive number is the same as adding the negative of the number.' This definition is true. $4-(-4)$ is the same as $4+4$. Try it out!
• 'Adding two numbers always results in a larger number.' This statement is false. If just one of the numbers is negative, this answer is not true. When we add $4+(-3)$, the answer, $1$, is smaller than $4$.
• 'Subtracting two numbers always results in a larger number.' This statement is also false. While this is true if just one of the numbers is negative, this answer is not true. When we add $6-2$, the answer, $4$, is smaller than $6$.
• 'Subtracting a negative number is the same as subtracting a positive number.' This answer is false. In the equation $8-(-1)$, subtracting a positive number will result in $7$, which is incorrect. When we add $8-(-1)$, $9$ is the correct answer.
• ### Identify the equivalent expressions.

Hints

Adding two positive integers always yields a positive sum while adding two negative integers always yields a negative sum.

However, subtracting a negative number is like adding the same number.

Solution

Let's simplify the signs to find an equivalent equation.

• If we have an equation $7 - (-1) + 3$, and we know that subtracting a negative number is the same as adding a positive number, we can simplify this equation to $7 + 1 + 3$.
• For the equation $-7 + (-1) + (-3)$, we know that adding a negative number is the same as subtracting a positive number. So now we can simplify this equation to $-7 - 1 - 3$.
• Given the equation $7 - 1 - (-3)$, if we remember that subtracting a negative number is the same as adding a positive number, we can simplify this equation to $-7 - 1 + 3$.
• Lastly, for the equation $-7 - (-1) + 3$, we know that subtracting a negative number is the same as adding a positive number, so we can simplify this equation to $-7 + 1 + 3$.