What are integers?
Integers are all the whole numbers and their opposites:
…, −3, −2, −1, 0, 1, 2, 3, …
Integers don’t include fractions or decimals. So −4 is an integer, but −4.5 is not.
Absolute value
The absolute value of a number is its distance from zero on the number line. It’s written with two vertical bars:
| Expression | Read as | Value |
|---|---|---|
| |7| | absolute value of 7 | 7 |
| |−7| | absolute value of −7 | 7 |
| |0| | absolute value of 0 | 0 |
Distance is never negative, so absolute value is never negative.
Opposite integers have the same absolute value: 7 and −7 are both 7 units from zero.
Adding integers
Same signs: add the absolute values, keep the sign.
- 5 + 3 = 8
- (−5) + (−3) = −8
Different signs: subtract the absolute values; keep the sign of the number with the bigger absolute value.
- (−9) + 4 → 9 − 4 = 5, and −9 has the bigger absolute value → −5
- 9 + (−4) → 5
Zero pairs: Using coloured counters, one positive and one negative cancel out to make a zero pair. In (−9) + 4, four zero pairs disappear, leaving 5 negatives.
Subtracting integers
Subtracting is the same as adding the opposite:
- 6 − 10 = 6 + (−10) = −4
- 3 − (−5) = 3 + 5 = 8
- (−2) − (−7) = −2 + 7 = 5
Multiplying and dividing integers
| Signs | Result |
|---|---|
| positive × positive | positive |
| negative × negative | positive |
| positive × negative | negative |
| negative × positive | negative |
The same rules work for division.
- (−4) × 6 = −24
- (−3) × (−5) = 15
- (−20) ÷ (−4) = 5
- 18 ÷ (−3) = −6
Real-life integers
- Change in temperature from −4 °C to 9 °C: 9 − (−4) = 13 degrees.
- A diver at −30 m rises 12 m: −30 + 12 = −18 m.
Worked examples
Example 1: What is the absolute value of −9?
Its distance from zero: 9.
Example 2: (−8) + (−6)
Same signs: 8 + 6 = 14, keep the negative → −14.
Example 3: 4 − 11
4 + (−11) → different signs: 11 − 4 = 7, negative → −7.
Common mistakes
- Giving a negative answer for an absolute value.
- Forgetting to change the sign of the second number when subtracting a negative.
- Thinking two negatives always make a positive, even when adding: (−3) + (−4) = −7, not 7.
Try it yourself
- The absolute value of −15
- (−7) + 10
- (−3) − 8
- 5 − (−6)
- (−6) × (−7)
Answers: 1) 15 2) 3 3) −11 4) 11 5) 42
Quick recap
- Integers are whole numbers and their opposites.
- Absolute value is distance from zero, never negative.
- Add, subtract by adding the opposite, and follow the sign rules for multiplying.