Number and Operations · 3 min read
Operations with Integers
Adding Integers
When adding integers with the same sign, add the absolute values and keep the common sign. 5 + 8 = 13. (-5) + (-8) = -13.
When adding integers with different signs, subtract the smaller absolute value from the larger absolute value, then use the sign of the number with the larger absolute value. 5 + (-8) = -3 because |-8| > |5| and the answer takes the negative sign. (-5) + 8 = 3 because |8| > |-5| and the answer takes the positive sign.
Adding zero to any integer gives that integer. 7 + 0 = 7. (-7) + 0 = -7.
The commutative property applies: a + b = b + a. The associative property applies: (a + b) + c = a + (b + c).
Subtracting Integers
To subtract an integer, add its opposite. This rule transforms all subtraction problems into addition problems.
5 - 8 = 5 + (-8) = -3. 5 - (-8) = 5 + 8 = 13. (-5) - 8 = (-5) + (-8) = -13. (-5) - (-8) = (-5) + 8 = 3.
The pattern "subtracting a negative is adding a positive" is critical. Double negatives become positive.
Multiplying Integers
Dividing Integers
Division follows the same sign rules as multiplication. Positive ÷ positive = positive: 24 ÷ 6 = 4. Negative ÷ negative = positive: (-24) ÷ (-6) = 4. Positive ÷ negative = negative: 24 ÷ (-6) = -4. Negative ÷ positive = negative: (-24) ÷ 6 = -4.
Division by zero is undefined. You cannot divide any number by zero.
Zero divided by any nonzero integer equals zero: 0 ÷ 5 = 0. 0 ÷ (-5) = 0.
| Operation | Same signs | Different signs |
|---|---|---|
| Addition | Add the absolute values; keep the common sign: (−4) + (−7) = −11 | Subtract the absolute values; use the sign of the larger absolute value: (−9) + 4 = −5 |
| Subtraction | Add the opposite, then follow the addition rules: 3 − 8 = 3 + (−8) = −5 | Add the opposite: (−6) − (−10) = (−6) + 10 = 4 |
| Multiplication and division | Result is positive: (−3)(−5) = 15 | Result is negative: (−24) ÷ 6 = −4 |
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