A pattern of subtractions
What is 5 − (−1)? Start with subtractions you already know, and take away 1 less each time: 5 − 2 = 3, 5 − 1 = 4, 5 − 0 = 5.
Each time the number taken away goes down by 1, the answer goes up by 1.
Down the table, the number taken away drops by 1 and the answer climbs by 1.
Carry the pattern below zero
The next number to take away is one less than 0, which is −1. The pattern has no reason to stop at zero, so the answer climbs by 1 once more, from 5 to 6. So 5 − (−1) = 6.
One more row gives 5 − (−2) = 7. Taking away −2 gives the same answer as adding 2: 5 − (−2) = 5 + 2 = 7.
Below zero the answer keeps climbing by 1: 5 − (−1) = 6 and 5 − (−2) = 7.
Taking away a debt
A debt gives the same answer. In a card game, a card worth −2 takes 2 points off your score. Your score is 5, and one of your cards is that −2 card. If the card is taken away from you, the 2 points it was costing you come back, and your score goes up to 7.
Taking away the −2 card is 5 − (−2), and it leaves you 2 points better off: 5 − (−2) = 5 + 2 = 7. Taking away a debt is a gain.
Taking away −2 moves 2 steps to the right, the same move as adding 2.
The rule, from any start
Subtracting a negative number gives the same answer as adding the positive number with the same digits: 7 − (−3) = 7 + 3 = 10.
It works when the start is below zero too. −3 − (−5) = −3 + 5, which climbs 3 steps to zero and 2 more beyond it, so −3 − (−5) = 2.
From −3, taking away −5 moves up 5 and stops at 2.
The usual mistakes
Subtracting the number. 7 − (−3) is not 7 − 3 = 4. The question takes away −3, and taking away a debt adds.
Making everything negative. 7 − (−3) is not −10. Taking away a negative number moves up the scale, not down.
Worked example: A Game Score with a Penalty Taken Away
Question In a card game each card either adds points to a player's score or takes points away. Mei has four cards: +15, −20, +8 and −12. (a) Find Mei's score. (b) The −12 card was dealt to her by mistake, so it is taken away from her. Find her new score.
1.Add the two gains first: 15 + 8 = 23.
The two gains add up to 15 + 8 = 23. 2.Add the two penalties: (−20) + (−12) = −32.
The two penalties add up to (−20) + (−12) = −32. 3.(a) Mei's score is 23 + (−32) = 23 − 32 = −9 points.
(a) Her score is 23 − 32 = −9 points. 4.Taking the −12 card away subtracts −12 from her score, so her new score is −9 − (−12).
Taking the −12 card away subtracts −12 from her score: −9 − (−12). 5.Subtracting a negative number is the same as adding its opposite: −9 − (−12) = −9 + 12 = 3. (b) Her new score is 3 points. Check: the three cards that remain give 15 − 20 + 8 = 3.
(b) −9 − (−12) = −9 + 12 = 3 points.
Answer: (a) −9 points; (b) 3 points
Common mistakes
- Working out −9 − 12 = −21 for part (b). Losing a penalty makes the score better, not worse. The card is worth −12, so taking it away is −9 − (−12) = −9 + 12.
- Writing 23 − 32 = 9 in part (a). The penalties are larger than the gains, so the score is below zero: 23 − 32 = −9.
More negative numbers and linear equations problems, worked step by step →