Subtracting Integers: Definition, Method, and Examples

What does subtracting integers mean?

Subtracting integers represents the process of taking away a positive or negative amount from a starting value. Because integers include negative values, subtraction can result in an increase or a decrease in value.

Every subtraction expression consists of a minuend (the starting number), a subtrahend (the number being subtracted), and the difference (the result). For integers, the most reliable method is to rewrite the subtraction using opposite numbers and additive inverses.

Change subtraction to adding the opposite

To subtract any integer, change the subtraction operation to addition, and change the subtrahend to its opposite sign.

1. Keep the first integer (the minuend) exactly the same.
2. Change the subtraction sign to an addition sign.
3. Change the sign of the second integer (the subtrahend) to its opposite.
4. Calculate the result using the rules of integer addition.

To subtract an integer, add its opposite.

Subtracting positive integers

When you subtract a positive integer, you are moving backward or decreasing the value of the starting number.

If the starting number is positive and greater than the number being subtracted, the result remains positive. If the starting number is positive but less than the number being subtracted, the result crosses zero and becomes negative. If the starting number is already negative, subtracting a positive number makes it even more negative.

Subtracting negative integers

When you subtract a negative integer, you are removing a deficit or taking away a negative amount, which has the exact same effect as adding a positive amount.

This is why subtracting a negative number increases the overall value. When evaluating $7-(-3)$, changing the subtraction to addition and taking the opposite of $-3$ results in $7+3=10$.

Use a number line

Using number lines helps to visualize subtraction as finding a distance and a direction.

When subtracting on a number line, you can rewrite the expression as an addition problem to determine which way to move. To add a positive integer, move to the right. To add a negative integer, move to the left.

Worked examples

Review these examples to understand how to apply the rules of integer subtraction to different scenarios.

Example 1: Subtracting two negative integers

Question: Evaluate $-15-(-24)$.

Method:

1. Keep the first integer exactly the same: $-15$.
2. Change the subtraction operation to addition and change the subtrahend to its opposite: $-15+24$.
3. Because the signs are different, find the difference of their absolute values: $24-15=9$.
4. The positive integer has the larger absolute value, so the final result is positive.

Answer: $9$

Check: Add the difference back to the subtrahend: $9+(-24)=-15$. This matches the original starting number.

Example 2: Subtracting a larger positive number

Question: What is $42-68$?

Method:

1. Rewrite the expression as addition of the opposite: $42+(-68)$.
2. Subtract the smaller absolute value from the larger absolute value: $68-42=26$.
3. Because the negative number has the larger absolute value, the result is negative.

Answer: $-26$

Check: Start with the answer and add the subtrahend: $-26+68=42$, which returns the original minuend.

Example 3: Applying subtraction to temperature changes

Question: The temperature at sunset was $5^{\circ }\text{C}$. By midnight, the temperature was $-4^{\circ }\text{C}$. Find the change in temperature using integer word problems principles.

Method:

1. The change is the final temperature minus the initial temperature.
2. Write the subtraction expression: $-4-5$.
3. Rewrite the subtraction as adding the opposite: $-4+(-5)$.
4. Because both integers are negative, add their absolute values and keep the negative sign.
5. $4+5=9$, so the result is $-9$.

Answer: The change in temperature is $-9^{\circ }\text{C}$.

Check: Start at $5^{\circ }\text{C}$ and apply the change: $5+(-9)=-4^{\circ }\text{C}$, which matches the final midnight temperature.

Common mistakes

When subtracting integers, learners often make sign errors. Being aware of these common pitfalls will help you calculate accurately.

• Subtracting instead of adding the opposite: You must change the operation to addition and change the sign of the subtrahend. Forgetting the second step leads to errors. For example, $8-(-3)$ is $8+3$, not $8+(-3)$.
• Changing the sign of the first number: Only the number being subtracted changes its sign. The starting number always stays the same. In $-6-4$, keep the $-6$ as it is to get $-6+(-4)$.
• Assuming subtraction always makes a number smaller: Subtracting a negative integer increases the overall value. For example, $10-(-5)=15$.

Frequently asked questions

Is subtracting integers the same as adding integers?

Subtracting integers relies heavily on adding and subtracting positive and negative numbers. Every subtraction expression can be written as an addition expression by adding the opposite of the subtrahend. Once rewritten, all the normal rules of integer addition apply.

What happens if you subtract zero from an integer?

Subtracting zero from any integer leaves the integer unchanged. For example, $-14-0=-14$. The difference is simply the starting value.

What happens if you subtract an integer from zero?

Subtracting an integer from zero results in the opposite of that integer. For example, $0-7=-7$, and $0-(-9)=9$.

Can subtracting integers result in a larger number?

Yes. Subtracting a negative integer is mathematically identical to adding a positive integer. For example, $12-(-4)$ is the same as $12+4$, which equals $16$.