Negative numbers
Add, subtract, multiply, and divide with signs.
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the idea
A negative number sits below zero: −5 is five steps down. On a number line, subtracting 5 moves you 5 steps left, so subtracting −5 moves you 5 steps right. It is the same as adding 5.
For × and ÷, work the numbers as if both were positive, then fix the sign: same signs give a positive, different signs give a negative.
Absolute value, written with bars like |−7|, is a number's distance from zero. A distance is never negative, so |−7| is 7.
techniques
Turn subtraction into addition
- Rewrite subtracting as adding the opposite: − (−12) becomes + 12, and − 5 becomes + (−5).
- Same signs: add the sizes and keep the sign.
- Different signs: take the smaller size from the bigger. The answer gets the sign of the number farther from zero.
- Equal sizes with different signs cancel, so the answer is 0.
worked example
−8 − (−15)
- Subtracting −15 is adding 15: −8 + 15.
- Different signs: 15 − 8 = 7.
- 15 is farther from zero and positive, so the answer is positive.
Answer: 7
Sign rule for × and ÷
- Work out the size, ignoring the signs.
- Same signs give a positive answer. Different signs give a negative one.
worked example
(−56) ÷ 7
- 56 ÷ 7 = 8.
- One negative and one positive: the signs differ, so the answer is negative.
Answer: −8
Absolute value is distance
- Bars around a calculation: work it out inside first.
- Then drop the sign. A distance from zero is never negative.
- Bars on each number separately: take each absolute value first, then do the calculation, which can come out negative.
worked example
|−3 − 8|
- Inside first: −3 − 8 = −11.
- Its distance from zero is 11.
Answer: 11
watch out for
- Reading − (−12) as − 12. Two minus signs in a row make a plus.
- Dropping the sign when dividing: 72 ÷ 8 is 9, but (−72) ÷ 8 is −9.
- Using the wrong order for the bars. |−4 − 9| works inside first, giving 13. |−4| − |9| takes each value first: 4 − 9, which is −5.
- Giving the answer the sign of the first number instead of the number farther from zero: 6 + (−15) is −9.
practice
Add & subtract with signs
worked example
−10 − (−17)
Answer: 7
- Subtracting a negative is adding: −10 − (−17) = −10 + 17.
- The signs differ: 17 − 10 = 7, and 17 is farther from zero, so the answer is 7.
Multiply & divide with signs
worked example
(−6) × 8
Answer: -48
- 6 × 8 = 48.
- The signs differ, so the result is negative: −48.
Absolute value
worked example
|−13| − |−12|
Answer: 1
- Each number has its own bars, so take each absolute value first: |−13| = 13 and |−12| = 12.
- Then 13 − 12 = 1.