Subtracting Integers
Integers are the whole numbers and their negatives: Adding and subtracting them trips people up for one reason — the signs. Is positive or negative? What does even mean? There are exactly two rules for addition and one move for subtraction, and once you have them, every problem looks the same.
This skill sits under almost everything that comes after it: solving equations, working with slope, combining like terms. A sign mistake here becomes a wrong answer three steps later, so it is worth getting automatic now.
Adding integers: two rules
Same signs: add the absolute values and keep the shared sign. So becomes , and both numbers were negative, so the answer is . Two debts stack into a bigger debt.
Different signs: subtract the smaller absolute value from the larger, and take the sign of the number with the larger absolute value. For , compute ; the is bigger in size, so the answer is . The two numbers fight, and the bigger one wins.
One more useful fact: opposites add to zero. , always. That cancellation is exactly what the different-signs rule is doing — pairing up positives with negatives until one side runs out.
Subtracting integers: add the opposite
Every subtraction problem can be rewritten as an addition problem: keep the first number, change subtraction to addition, and flip the sign of the second number. So becomes , and becomes .
That is the whole trick. Once you rewrite, you are back to the two addition rules from the last section — you never need a separate set of subtraction rules.
Why does it work? Subtracting a number and adding its opposite move you the same direction on the number line. Subtracting moves you to the left; adding also moves you to the left. Same trip, different wording.
The classic trap: subtracting a negative
The expression makes students freeze, but the rewrite handles it: change to , giving . Two negatives sitting side by side — a subtraction sign next to a negative sign — always turn into addition.
If the signs pile up and you lose track, slow down and rewrite the whole problem as addition first. It costs five seconds and prevents almost every sign error.
Worked examples
Example 1: same signs
Add .
Answer:
Example 2: different signs
Add .
Answer:
Example 3: subtraction becomes addition
Subtract .
Answer:
Example 4: subtracting a negative
Subtract .
Answer:
Try one yourself
Common questions
Why does subtracting a negative give a bigger answer?
Because subtracting a number and adding its opposite are the same move. The opposite of a negative is a positive, so is the same as . On a number line, subtracting sends you steps to the right.
How do I know whether a sum is positive or negative?
If the signs match, the sum keeps that sign. If they differ, the sum takes the sign of the number with the larger absolute value — compare sizes and the bigger one wins. So is positive and is negative.
Do these rules work for decimals and fractions too?
Yes. The sign rules care only about the signs, not what kind of number is attached. follows the exact same different-signs rule as .
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