When it comes to adding and subtracting negative numbers, we like to imagine we’re digging and filling to help us keep track of when we need to go up and down the number line ↕️.

The sign or combination of signs a number comes after tells us whether we need to fill up or dig down:

Sign(s)

Action

Visual

$−$

Dig down

$−5$

$+−$

Dig down

(Add

a hole)

$+$$−$

$+$

Fill up

$5$

$−−$

Fill up

(Take away

a hole)

$−$$−$

Check out our calculator or continue the lesson to explore how we use our imagination to help us add and subtract negative numbers!

Adding & Subtracting Negative Numbers Calculator

Step 1. Imagine the first number.

Original Question: $+=?$

The first number is $0$, which means we don't need to fill or dig.

Step 2. Imagine the second number.

Original Question: $+=?$

The second number is $0$, which means we don't need to fill or dig.

Step 3. Combine the numbers.

When we combine, the top of the hole or pile is our answer.

This helps us see our answer: $+=$.

Recap 🧢 How to Add & Subtract Negative Numbers

Single Sign

Double Signs

Step 1. Imagine the first number.

If the first number is positive, we fill up to the value of the first number.

If the first number is negative, we dig down to the value of the first number.

Step 2. Imagine the second number.

If the second number comes after a plus sign, we fill up to the value of the second number.

If the second number comes after a minus sign, we dig down to the value of the second number.

Step 3. Combine the numbers.

Overlap the piles and/or holes. The top of the final pile lines up with the final answer.

Continue to see how we can visualize adding and subtracting negative numbers in more detail, starting with a single sign in between our numbers.

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Single Sign

When we’re adding and subtracting negative numbers, and there’s only one sign in between the numbers we’re adding/subtracting, we can imagine the following:

For numbers that come after a positive/plus sign, like $+5$, we need to fill up.

$5$

For numbers that come after a negative/minus sign, like $−5$, we need to dig down.

$−5$

Let’s take a look at a few examples to make this make sense!

Let’s start by looking at an example where we have a positive number minus a number 👀.$5−3=?$

Step 1. Imagine the first number.

Original Question: $5−3=?$

The first number ($5$) is positive, so we fill up to the number $5$.

$5$

Step 2. Imagine the second number.

Original Question: $5−3=?$

The second number ($3$) comes after a minus sign, so we dig down to the number $−3$.

$5$

$−3$

Step 3. Combine the numbers.

When we combine, the top of the hole or pile is our answer.

This helps us see our answer: $5−3=2$.

Now, let’s see what happens when we have a negative number plus a number 👀.

$−6+2=?$

Step 1. Imagine the first number.

Original Question: $−6+2=?$

The first number ($−6$) is negative, so we dig down to the number $−6$.

$−6$

Step 2. Imagine the second number.

Original Question: $−6+2=?$

The second number ($2$) comes after a plus sign, so we fill up to the number $2$.

$2$

$−6$

Step 3. Combine the numbers.

When we combine, the top of the hole or pile is our answer.

This helps us see our answer: $−6+2=−4$.

Now, let’s see what happens when we have a negative number minus a number 👀.

$−1−2=?$

Step 1. Imagine the first number.

Original Question: $−1−2=?$

The first number ($−1$) is negative, so we dig down to the number $−1$.

$−1$

Step 2. Imagine the second number.

Original Question: $−1−2=?$

The second number ($2$) comes after a minus sign, so we dig down to the number $−2$.

$−1$

$−2$

Step 3. Combine the numbers.

When we combine, the top of the hole or pile is our answer.

This helps us see our answer: $−1−2=−3$.

Awesome! Practice with some additional problems or keep scrolling to explore what we do when there are two signs in between our numbers.

Practice: Single Sign

Question 1 of 3: $3−7=?$

Step 1. Imagine the first number.

Is the first number positive or negative?

That's right! Since the first number is positive we need toto the number.

Question 1 of 3: $3−7=?$

Step 2. Imagine the second number.

Does the second number come after a plus sign or a minus sign?

The second number comes after a minus sign, so weto the number.

Question 1 of 3: $3−7=?$

Step 3. Combine the numbers. The top of the pile or hole lines up with the final answer.

Based on the combined graph, $3−7=$

$3$

Nice work! Our answer is $3−7=−4$

So, what do we do when we have double signs in a problem, like $3−(−5)=?$

Double Signs

When we’re adding and subtracting negative numbers, and there are two signs in between the numbers we’re adding/subtracting, we can imagine the following:

If a number comes after a plus sign and is negative, like $3+(−2)$, that means we need to add a hole (dig down).

$3$

$−2$

If a number comes after a minus sign and is negative, like $3−(−2)$, that means we need to take away a hole (fill up).

$3$

$2$

Let’s take a look at a few examples to make this make sense!

Let’s start by looking at an example where we have a positive number plus a negative number 👀.

Step 1. Imagine the first number.

Original Question: $6+(−4)=?$

The first number ($6$) is positive, so we fill up to the number $6$.

$6$

Step 2. Imagine the second number.

Original Question: $6+(−4)=?$

The second number ($−4$) comes after a plus sign and is negative, which means we need to add a hole (dig down) equal to $−4$.

$6$

$−4$

Step 3. Combine the numbers.

When we combine, the top of the hole or pile is our answer.

This helps us see our answer: $6+(−4)=2$.

Now, let’s see what happens when we have a positive number minus a negative number 👀.

Step 1. Imagine the first number.

Original Question: $5−(−3)=?$

The first number ($5$) is positive, so we fill up to the number $5$.

$5$

Step 2. Imagine the second number.

Original Question: $5−(−3)=?$

The second number ($−3$) comes after a minus sign and is negative, which means we need to take away a hole (fill up) equal to $3$.

$5$

$3$

Step 3. Combine the numbers.

When we combine, the top of the hole or pile is our answer.

This helps us see our answer: $5−(−3)=8$.

Now, let’s see what happens when we have a negative number plus a negative number 👀.

Step 1. Imagine the first number.

Original Question: $−5+(−3)=?$

The first number ($−5$) is negative, so we dig down to the number $−5$.

$−5$

Step 2. Imagine the second number.

Original Question: $−5+(−3)=?$

The second number ($−3$) comes after a plus sign and is negative, which means we need to add a hole (dig down) equal to $−3$.

$−5$

$−3$

Step 3. Combine the numbers.

When we combine, the top of the hole or pile is our answer.

This helps us see our answer: $−5+(−3)=−8$.

Now, let’s see what happens when we have a negative number minus a negative number 👀.

Step 1. Imagine the first number.

Original Question: $−5−(−3)=?$

The first number ($−5$) is negative, so we dig down to the number $−5$.

$−5$

Step 2. Imagine the second number.

Original Question: $−5−(−3)=?$

The second number ($−3$) comes after a minus sign and is negative, which means we need to take away a hole (fill up) equal to $3$.

$3$

$−5$

Step 3. Combine the numbers.

When we combine, the top of the hole or pile is our answer.

This helps us see our answer: $−5−(−3)=−2$.

Notice a Pattern? 💡

Adding a negative number is the same as subtracting a positive number: $3+(−2)=3−2$.

Subtracting a negative number is the same as adding a positive number: $3−(−2)=3+2$.

Awesome! If you’d like to practice additional problems with double operations, take a moment to complete the quick practice below before closing out this lesson! ⚡

Practice: Double Signs

Question 1 of 5: $3+(−5)=?$

Step 1. Imagine the first number.

Is the first number positive or negative?

That's right! Since the first number is positive we need toto the number.

Question 1 of 5: $3+(−5)=?$

Step 2. Imagine the second number.

Does the second number $−5$ come after a plus sign or a minus sign?

That's right! The second number comes after a plus sign and is negative, which means we need to equal to the number.

Question 1 of 5: $3+(−5)=?$

Step 3. Combine the numbers. The top of the pile or hole lines up with the final answer.

Based on the combined graph, $3+(−5)=$

$3$

Nice work! Our answer is $3+(−5)=−2$

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