What are integers?
Integers are whole numbers with a sign: the positive numbers 1, 2, 3, …, the negative numbers −1, −2, −3, … and zero. Zero is neither positive nor negative.
Every integer has an opposite (also called its additive inverse). The opposite of 5 is −5. A number plus its opposite is always 0.
The absolute value |a| is the distance from 0. It is never negative: |−7| = 7 and |7| = 7.
Signed decimals
Numbers like −2.5 or −0.75 are not integers, but they follow exactly the same rules. Everything on this page works for signed decimals and fractions too.
Ordering integers on the number line
On a number line, numbers get bigger as you go right. So:
- Any positive number is bigger than any negative number: 1 > −100.
- Between two negatives, the one closer to 0 is bigger: −2 > −7.
Think of temperature: −2 °C is warmer than −7 °C.
Adding and subtracting integers
Adding
Start at the first number. Adding a positive number moves you right. Adding a negative number moves you left.
- Same signs: add the sizes, keep the sign. −3 + (−4) = −7.
- Different signs: subtract the smaller size from the bigger one, keep the sign of the bigger size. −8 + 5 = −3.
Subtracting
Subtracting means adding the opposite: a − b = a + (−b).
- 5 − 9 = 5 + (−9) = −4
- 2 − (−4) = 2 + 4 = 6
Addition and subtraction are inverse operations: they undo each other. If −3 + 8 = 5, then 5 − 8 = −3. Use this to check your answers.
Multiplying and dividing: the sign rules
First work out the size of the answer as normal. Then decide the sign:
- (+) × (+) = + and (−) × (−) = +
- (+) × (−) = − and (−) × (+) = −
Division follows the same rules: −12 ÷ (−4) = 3 and 12 ÷ (−4) = −3.
Why is negative × negative positive?
Look at the pattern: 3 × (−2) = −6, 2 × (−2) = −4, 1 × (−2) = −2, 0 × (−2) = 0. Each time the answer goes up by 2. Keep going: (−1) × (−2) = 2, (−2) × (−2) = 4. The pattern forces the answer to be positive.
Many signs
Count the negative signs. An even number of negatives gives +, an odd number gives −. (−1) × (−2) × (−3) = −6 because there are three negatives.
Signed decimals
−2.5 × 4 = −10 and −1.2 ÷ (−0.4) = 3. Same rules, decimal sizes.
Dividing by 0 is never allowed.
Properties and mental calculation
- Commutative: a + b = b + a and a × b = b × a. Not for subtraction or division.
- Associative: (a + b) + c = a + (b + c); the same for ×.
- Distributive: a × (b + c) = a × b + a × c.
- Identity: a + 0 = a and a × 1 = a.
- Inverse: a + (−a) = 0.
Mental tricks: group opposites first. −17 + 25 + 17 = 25. Use the distributive law: −6 × 99 = −6 × 100 + 6 = −594.
Order of operations stays the same: brackets, then powers, then × and ÷, then + and −. Careful: (−3)² = 9 but −3² = −9.
Key formulas and definitions
- a − b = a + (−b)
- a + (−a) = 0
- (+)(+) = +, (−)(−) = +
- (+)(−) = −, (−)(+) = −
- |a| = distance of a from 0
- a × (b + c) = a × b + a × c
Worked examples
1. Order from smallest to largest: 3, −8, 0, −2, −11.
On the number line from left to right: −11, −8, −2, 0, 3.
2. Find −7 + 12.
Different signs. Sizes 12 and 7; 12 − 7 = 5. The bigger size (12) is positive, so the answer is 5.
3. Find −4 − 9.
Add the opposite: −4 + (−9). Same signs, add sizes: 4 + 9 = 13, keep minus. Answer −13.
4. At 6 a.m. it was −5 °C. By noon the temperature rose 11 degrees. What was it at noon?
−5 + 11 = 6. It was 6 °C.
5. Find (−6) × (−7) and 48 ÷ (−8).
(−6) × (−7): same signs → +, 6 × 7 = 42, so 42. 48 ÷ (−8): different signs → −, 48 ÷ 8 = 6, so −6.
6. Find −2.5 × 4 ÷ (−0.5).
Left to right. −2.5 × 4 = −10. Then −10 ÷ (−0.5) = 20 (same signs → +, 10 ÷ 0.5 = 20). Answer 20.
7. Find (−2)³ − 3 × (−4) + (−10) ÷ 5.
Powers first: (−2)³ = −8. Then × and ÷: 3 × (−4) = −12; −10 ÷ 5 = −2. Now: −8 − (−12) + (−2) = −8 + 12 − 2 = 2.
Common mistakes
- Thinking −7 is bigger than −2 because 7 > 2. On the number line −2 is further right, so −2 > −7.
- Writing 3 − (−4) = −1. Two minus signs side by side become plus: 3 + 4 = 7.
- Using the × sign rule for addition: −3 + (−4) is −7, not +7. The 'two negatives make a positive' rule is only for × and ÷.
- Mixing up (−3)² = 9 with −3² = −9. Without brackets, only 3 is squared.