Four operations, a few sign rules, and one skill that unlocks all of it: knowing exactly when the answer is positive or negative, and why.
A night temperature of −3°C. A bank account that is Rs 500 overdrawn. A lift that goes down to the second basement. Each of these uses a negative number. Almost every mistake with integers comes from one thing: losing track of signs. This guide explains the four operations (add, subtract, multiply, divide) in plain words first, then gives the exact rule, then shows why the rule is true, so you can remember it instead of guessing.
Integers are whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, … There are no fractions or decimals. Zero is an integer and is neither positive nor negative. (For where integers sit among other kinds of number, see Number Systems.)
On a number line, numbers to the right of zero are positive and numbers to the left are negative. A number further right is always greater. That means −2 is greater than −7, even though 7 looks “bigger”. So −7 < −2 < 0 < 3.
Two numbers that are the same distance from zero on opposite sides are opposites: 5 and −5, or −12 and 12. The opposite of 0 is 0. The absolute value of a number is its distance from zero, ignoring the sign. It is written with bars: |7| = 7 and |−7| = 7. Absolute value is never negative. We will use it because every sign rule below compares distances from zero.
A minus sign can mean two different things. In 8 − 3 it is an operation (take away). In −3 it is part of the number (negative three). Putting a negative number in brackets, as in 8 − (−3), keeps the two jobs apart.
Think of adding as moving along the number line. Adding a positive number moves you right. Adding a negative number moves you left. The number you start at is where you stand, and the answer is where you land.
The number line works for small numbers. For bigger numbers, use the rule:
| Case | Rule |
|---|---|
| Same signs | Add the absolute values and 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. |
Picture a positive chip (+) and a negative chip (−). One of each cancels out and makes zero, like earning Rs 1 and then owing Rs 1. We call this a zero pair. To add (−5) + 3, lay out five negative chips and three positive chips. Three zero pairs cancel, and two negative chips are left. So (−5) + 3 = −2.
(−4) + (−7). Both are negative, so add the absolute values: 4 + 7 = 11. Keep the negative sign: −11. (Owe Rs 4, then owe Rs 7 more: you owe Rs 11.)
9 + (−15). The signs differ. The absolute values are 9 and 15. Subtract: 15 − 9 = 6. The larger absolute value belongs to −15, so the answer is negative: −6. Check on the number line: start at 9 and move 15 left to land on −6.
(−8) + 12. The absolute values are 8 and 12. Subtract: 12 − 8 = 4. The larger absolute value belongs to +12, so the answer is 4.
At night a hill station is at −3°C. By afternoon the temperature has risen by 11 degrees. The new temperature is (−3) + 11 = 8°C. Another check: (−47) + 38. The absolute values are 47 and 38, the difference is 9, and 47 is the larger and is negative, so the answer is −9.
This adds the absolute values as if both signs matched. They do not.
The fix: when the signs differ you subtract. The answer is 9 + (−15) = −6. A quick check: adding a negative number can never make the total bigger, so an answer of 24 (bigger than 9) cannot be right.
There is no separate subtraction rule to memorise. Subtracting a number is the same as adding its opposite. Rewrite the subtraction as an addition, then use the addition rules.
Why is this true? Look at this pattern, where each line takes away one less than the line before:
| Calculation | Answer |
|---|---|
| 5 − 3 | 2 |
| 5 − 2 | 3 |
| 5 − 1 | 4 |
| 5 − 0 | 5 |
| 5 − (−1) | 6 |
| 5 − (−2) | 7 |
The answers go up by 1 each time, so the pattern must continue: taking away −1 gives 6, which is the same as 5 + 1. Taking away a debt makes you richer.
7 − (−5) = 7 + 5 = 12.
(−10) − 6 = (−10) + (−6) = −16. Also 5 − 12 = 5 + (−12) = −7.
(−3) − (−9) = (−3) + 9 = 6. And (−15) − (−15) = (−15) + 15 = 0, because anything minus itself is zero.
The highest temperature in a week is 14°C and the lowest is −6°C. The range is 14 − (−6) = 14 + 6 = 20°C. In the same way, a bird at +30 m above sea level and a diver at −12 m are 30 − (−12) = 42 m apart. A difference is always “larger minus smaller”.
The two minus signs were treated as if they were one.
The fix: the first minus is the operation (take away) and the second is the sign of the number. “Take away negative 3” means “add the opposite”: 5 − (−3) = 5 + 3 = 8. Test it with a pattern: 5 − 3 = 2, 5 − 2 = 3, 5 − 1 = 4 and so on, so 5 − (−3) must be bigger than 5, not smaller.
Multiplication is repeated addition. 3 × (−2) means three lots of −2: (−2) + (−2) + (−2) = −6. So a positive times a negative is negative. What about a negative times a negative? Use a pattern again. Keep the second number as −2 and lower the first number by 1 each line:
| Calculation | Answer |
|---|---|
| 3 × (−2) | −6 |
| 2 × (−2) | −4 |
| 1 × (−2) | −2 |
| 0 × (−2) | 0 |
| (−1) × (−2) | 2 |
| (−2) × (−2) | 4 |
The answers rise by 2 each time, so (−1) × (−2) must be +2 and (−2) × (−2) must be +4. That is why two negatives make a positive. The rule depends only on whether the two signs match:
(a) (−6) × (−7): same signs, so positive: 42. (b) 8 × (−9): different signs, so negative: −72. First multiply the absolute values (6 × 7 and 8 × 9), then decide the sign.
Count the negative signs. An even number of negatives gives a positive answer. An odd number gives a negative answer. Each negative flips the sign once, like pressing a light switch: two presses return it to how it began, three presses leave it the other way.
(a) (−3) × (−2) × (−4) has three negatives (odd). 3 × 2 × 4 = 24, so the answer is −24.
(b) (−2) × (−2) × (−2) × (−2) has four negatives (even). 2 × 2 × 2 × 2 = 16, so the answer is 16.
(c) Multiplying −1 by itself five times gives −1 (five negatives, odd).
Negatives cancel in pairs. A leftover one stays.
The fix: work in pairs. (−3) × (−2) = 6 (a pair cancels). Then 6 × (−4) = −24 (one negative is left over). Three negatives is an odd number, so the final sign is negative.
Division is the opposite of multiplication, so it follows the same sign rule: same signs give a positive answer, different signs give a negative answer. A useful check is to multiply your answer by the divisor and see if you get back to the start.
(a) (−36) ÷ 6 = −6 (different signs). Check: (−6) × 6 = −36 ✓.
(b) (−48) ÷ (−8) = 6 (same signs). Check: 6 × (−8) = −48 ✓.
(c) 72 ÷ (−8) = −9. (d) 0 ÷ (−5) = 0.
Division by zero is not allowed. There is no number that, multiplied by 0, gives 7, so 7 ÷ 0 has no answer. Also, dividing integers does not always give an integer: (−7) ÷ 2 = −3.5, which is not an integer (it is still a correct answer).
| Operation | Same signs | Different signs |
|---|---|---|
| Addition | Add the absolute values, keep the sign | Subtract, keep the sign of the larger absolute value |
| Subtraction | Add the opposite, then use the addition rule | |
| Multiplication | Positive | Negative |
| Division | Positive | Negative |
When an expression has several operations, follow the order BODMAS: Brackets, Orders (powers and roots), Division and Multiplication (left to right), then Addition and Subtraction (left to right). With negative numbers one trap catches nearly everyone.
This is one of the most common traps in school mathematics.
The fix: without brackets, the power applies only to the 3. So −3² means −(3²) = −9. To square negative three itself you need brackets: (−3)² = (−3) × (−3) = 9. Brackets change the answer, so always notice them.
Powers of negative numbers follow the sign-counting rule: an even power gives a positive answer and an odd power gives a negative answer. (−2)³ = −8, (−2)⁴ = 16, but −2⁴ = −(2⁴) = −16. Also (−1)¹⁰⁰ = 1 and (−1)¹⁰¹ = −1.
A minus sign in front of a bracket changes every sign inside it, or, equivalently, it finds the opposite of the result: −(3 − 8) = −(−5) = 5.
Work out −8 + 3 × (−2) − (−6) ÷ 2.
Step 1 (× and ÷ 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.
Work out (−5 + 2) × (−3) − 4. Brackets: −5 + 2 = −3. Multiply: (−3) × (−3) = 9. Subtract: 9 − 4 = 5.
(a) 12 − 3 × (−4) + (−18) ÷ (−3). Multiplication and division first: 3 × (−4) = −12 and (−18) ÷ (−3) = 6. Then 12 − (−12) + 6 = 12 + 12 + 6 = 30.
(b) (−2)³ + 5 × (−4) − (−7). Power first: (−2)³ = −8. Then 5 × (−4) = −20. So −8 + (−20) − (−7) = −28 + 7 = −21.
(c) Compare 4 − (−3)² = 4 − 9 = −5 with (4 − (−3))² = 7² = 49. Same digits, different brackets, very different answers.
Calculator tip. To enter a negative number, use the (−) key, not the subtraction key, and put negative numbers in brackets when you raise them to a power. If you type −3² and the calculator shows −9, it is following the same BODMAS rule as above.
Tap any expression to see it solved step by step, with the rule used at each stage.
Tap an expression above to see the full breakdown.
Type two whole numbers (negative numbers are fine, for example −7), then tap an operation. You get the answer and the rule that was used.
Tap an operation above to see the answer and the rule.
Tap the correct answer for each question. Your score is tracked as you go, and your best score is saved on this device.
1. (a) The signs differ, 15 − 8 = 7, and the larger absolute value is negative: −7. (b) (−4) + 10 = 6.
2. Working left to right: (−5) × 6 = −30, then (−30) ÷ (−3) = 10.
3. −2² = −4 and (−2)² = 4, so the total is −4 + 4 = 0.
1. (a) −5 (b) 3 + 8 = 11.
2. (a) 42 (b) −7.
3. 9 − (−7) = 9 + 7 = 16°C.
4. Multiply and divide first: 3 × (−2) = −6 and (−12) ÷ 4 = −3. Then 5 − (−6) + (−3) = 5 + 6 − 3 = 8.
5. (a) 16 (b) −16 (c) −64.
6. (−2) × 3 = −6, so the third integer is −30 ÷ (−6) = 5. Check: (−2) × 3 × 5 = −30 ✓.
7. 2 − 7 + 4 − 6 = −7, the seventh basement.
Negative numbers feel abstract until they are attached to something physical. Start with a floor-marked number line (a strip of tape on the floor works well) and let learners walk the additions: face the positive direction, step forwards for a positive number and backwards for a negative one. Use the context of temperature, floors in a building or money owed, and move to chips for zero pairs when the numbers get bigger. For multiplication, patterns (like the tables above) give a reason for the rule, which sticks longer than a chant. Insist on brackets around negative numbers while learners are still building the habit, and encourage a quick estimate of the sign of the answer before any calculation.
This article is original EDUSAMBAM writing for students up to O-Level and similar courses. Exact examination requirements differ between boards, so compare your practice with the specification and past papers of your own board.
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