Adding and Subtracting Integers — Definition, Formula & Examples
Adding and subtracting integers means combining positive and negative whole numbers using sign rules that determine whether the result is positive, negative, or zero.
For any integers and , addition follows the rules of absolute value comparison: if the signs match, add the absolute values and keep the common sign; if the signs differ, subtract the smaller absolute value from the larger and take the sign of the integer with the greater absolute value. Subtraction is defined as , converting every subtraction into addition of the opposite.
Key Formula
Where:
- = The first integer
- = The integer being subtracted
- = The opposite (additive inverse) of b
How It Works
When you add two integers with the same sign, add their absolute values and keep that sign: and . When you add two integers with different signs, find the difference of their absolute values and use the sign of the number farther from zero: because and , so the answer is negative. For subtraction, change the problem to addition by flipping the sign of the second number. For example, becomes . This single rule — subtract means add the opposite — turns every subtraction problem into an addition problem you already know how to solve.
Worked Example
Problem: Compute .
Step 1: Rewrite the subtraction as addition of the opposite.
Step 2: Add the first two values. The signs differ, so subtract absolute values: . Because 9 > 4, the result takes the negative sign.
Step 3: Now add 6. The signs differ again: . Because 6 > 5, the result is positive.
Answer:
Why It Matters
Integer arithmetic is the foundation for solving equations in pre-algebra and algebra. It also appears constantly in real contexts like tracking temperature changes, bank account balances, and elevation differences.
Common Mistakes
Mistake: Treating as subtraction instead of addition, writing instead of .
Correction: Subtracting a negative means adding: . Always convert subtraction to adding the opposite first.