Four operations, a handful of sign rules, and one skill that unlocks all of it: knowing exactly when the answer flips positive or negative — and why.
Every mistake with integers comes down to one thing: losing track of signs. This guide covers all four operations in full — addition, subtraction, multiplication, and division — with the exact rule for each, worked examples for every case, and the traps that catch even confident students.
Integers are whole numbers and their negatives: ..., −3, −2, −1, 0, 1, 2, 3, ... — no fractions, no decimals. For a full breakdown of where integers fit among other number types, see our Number Systems article.
Before the operation rules make sense, one idea needs to be solid: absolute value — a number's distance from zero, ignoring its sign. Absolute value is written with two vertical bars and is always zero or positive, never negative.
|7| = 7. |−7| = 7. Both 7 and −7 sit exactly 7 steps from zero — distance doesn't care about direction.
Absolute value matters here because every sign rule below is really a rule about comparing distances from zero, then deciding the final sign separately.
Adding integers splits into two cases, depending on whether the signs match.
| Case | Rule |
|---|---|
| Same signs | Add the absolute values, keep the shared sign |
| Different signs | Subtract the smaller absolute value from the larger, keep the sign of the number with the larger absolute value |
(−4) + (−7): both negative. Add absolute values: 4 + 7 = 11. Keep the negative sign: −11.
9 + (−15): signs differ. Absolute values are 9 and 15. Subtract: 15 − 9 = 6. Since 15 has the larger absolute value and is negative, the answer is −6.
(−8) + 12: signs differ. Absolute values are 8 and 12. Subtract: 12 − 8 = 4. Since 12 has the larger absolute value and is positive, the answer is 4.
This adds the absolute values as if the signs matched — but they don't.
The fix: when signs differ, you subtract, not add. 9 + (−15) = −6 — check with a number line: starting at 9 and moving 15 steps left lands on −6.
Every subtraction problem can be rewritten as addition: subtracting a number is the same as adding its opposite. Once rewritten, the addition rules above take over completely — there's no separate subtraction rule to memorize.
7 − (−5) = 7 + 5 = 12. Subtracting −5 is the same as adding 5.
(−10) − 6 = (−10) + (−6) = −16. Subtracting 6 is the same as adding −6.
(−3) − (−9) = (−3) + 9 = 6.
For a deeper, step-by-step visual walkthrough of exactly why this "add the opposite" trick works, see our Addition & Subtraction on the Number Line article.
Multiplication has its own, simpler pair of rules, based purely on whether the two signs match.
| Signs | Result |
|---|---|
| Same signs (+ × + or − × −) | Positive |
| Different signs (+ × − or − × +) | Negative |
(−6) × (−7) = 42. Two negatives multiplied together give a positive.
8 × (−9) = −72. One positive, one negative — the result is negative.
When three or more integers multiply together, count the total number of negative signs. An even count of negatives gives a positive result; an odd count gives a negative result.
(−3) × (−2) × (−4): three negative signs (odd). Multiply the absolute values: 3 × 2 × 4 = 24. Odd count of negatives means the result is negative: −24.
(−2) × (−2) × (−2) × (−2): four negative signs (even). Multiply the absolute values: 2×2×2×2 = 16. Even count of negatives means the result is positive: 16.
It's tempting to assume any group of negatives cancels out, but the actual rule depends on counting them.
The fix: each negative sign flips the result once. Two flips cancel back to positive, but three flips leave it negative, just like flipping a light switch three times leaves it in the opposite state from where it started. Always count the negatives: even = positive, odd = negative.
Division follows the exact same sign rule as multiplication — same signs give a positive result, different signs give a negative result.
| Signs | Result |
|---|---|
| Same signs (+ ÷ + or − ÷ −) | Positive |
| Different signs (+ ÷ − or − ÷ +) | Negative |
(−36) ÷ 6 = −6.
(−48) ÷ (−8) = 6.
| Operation | Same Signs | Different Signs |
|---|---|---|
| Addition | Add, keep the shared sign | Subtract, keep sign of the larger absolute value |
| Subtraction | Rewrite as addition of the opposite, then use the addition rule | |
| Multiplication | Positive | Negative |
| Division | Positive | Negative |
When an expression combines several operations, they aren't done left to right — they follow a fixed order: brackets first, then exponents, then multiplication and division (left to right), then addition and subtraction (left to right). This order matters even more with negative numbers, because of one very sneaky trap.
This is one of the most common order-of-operations traps in all of mathematics — and it looks completely reasonable at first glance.
The fix: without parentheses, the exponent applies only to the 3, not to the negative sign. So −3² means −(3²) = −9. To actually square −3 itself, parentheses are required: (−3)² = (−3) × (−3) = 9. The presence or absence of parentheses completely changes the answer.
−8 + 3 × (−2) − (−6) ÷ 2
Step 1 — multiplication and division first: 3 × (−2) = −6, and (−6) ÷ 2 = −3
Step 2 — rewrite: −8 + (−6) − (−3)
Step 3 — left to right: −8 + (−6) = −14, then −14 − (−3) = −14 + 3 = −11
(−5 + 2) × (−3) − 4
Step 1 — brackets first: −5 + 2 = −3
Step 2 — multiplication: −3 × (−3) = 9
Step 3 — subtraction: 9 − 4 = 5
Tap any expression below to see it solved step by step, with the exact rule applied at each stage.
Tap an expression above to see the full breakdown.
A submarine at −40 metres rises 25 metres, then dives another 18 metres: −40 + 25 − 18 = −33 metres. A bank account with a −$200 balance receives another charge of −$150: −200 + (−150) = −$350 owed. Both situations use exactly the sign rules covered above — no new math required, just careful sign-tracking.
Negative numbers were treated with suspicion by many European mathematicians well into the 1700s — some called them "absurd" or "fictitious" — even while merchants and astronomers were already relying on them daily for tracking debts and celestial calculations.
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Integer operations aren't just a hurdle to clear before "real" math starts — they're the toolkit behind an enormous range of everyday and professional situations, anywhere a quantity can go above or below a starting point.
| Field | Where Integer Operations Show Up |
|---|---|
| Finance | Tracking bank balances, debts, profits and losses, credit card statements |
| Science | Temperature changes, chemical charge (ions), altitude and depth measurements |
| Sports | Golf scores relative to par, point differentials, win/loss goal differences |
| Computer Science | Memory offsets, error codes, coordinate systems in graphics and games |
| Everyday Life | Elevators below ground level, time zones ahead/behind, weight gained or lost |
More importantly, this topic builds a habit that carries into every later math course: staying disciplined about signs instead of guessing. Algebra, physics, and engineering all lean on exactly the same sign rules covered here — just applied to variables and formulas instead of plain numbers. Master it now, and every one of those later topics gets noticeably easier.
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