Every addition and subtraction problem, however tangled the signs, comes down to one picture: start at a point, then move. Learn why each rule is true, not just what it says.
This is the topic where even confident students, and sometimes teachers, pause and double-check themselves. The good news: every addition and subtraction problem, however tangled the signs look, comes down to one simple picture. Start at a point, then move.
This guide goes slowly and explains why each rule is true, not just what the rule says. If a sign ever feels confusing again, come back to the number line. It never lies. It also ties in with Number Lines and Operations of Integers.
Adding a positive number always means moving to the right on the number line. The number you start at is where you put your finger, and the number you add is how many steps you take to the right.
2 + 3 means: start at 2, then take 3 steps to the right. You land on 5. Nothing more mysterious than that.
Subtracting a positive number always means moving to the left. It is the same idea in the opposite direction.
6 − 4 means: start at 6, then take 4 steps to the left. You land on 2.
This is the part most students already know. The confusion almost always starts once negative numbers get involved, so the next sections slow down exactly there.
Many students learn subtraction only as “taking away objects”. That works for 6 − 4 but breaks down for 3 − 8. You cannot take 8 apples away from a group of 3. But you can always move 8 steps to the left of 3 on a number line, and you land on −5.
3 − 8: start at 3 and move 8 steps left. You pass through 2, 1, 0, −1, −2, −3, −4 and land on −5. The “take away objects” model has no answer here, but the number line always does.
This belief comes from the take-away model and stops working the moment negative numbers exist.
The fix: you can. The answer is just negative: 3 − 8 = −5. On the number line there is no wall at zero. The line keeps going left for ever.
Adding a negative number moves you left, exactly like subtracting does. This is because +(−3) and −3 mean the same thing: a step of size 3 taken in the negative direction.
−1 + (−3) means: start at −1, then move 3 steps further left, because the number being added is negative. You land on −4. In practice “+ (−3)” behaves just like “− 3”.
The word “addition” tricks people into assuming rightward movement no matter what.
The fix: the direction depends on the sign of the number being added, not on the word “addition”. Adding a negative number always moves left. Rewrite +(−3) as −3 if that helps it click.
This is the most common breaking point in the whole topic, so read this section twice if you need to.
Subtraction normally means “move left”. But when the number you are subtracting is itself negative, you get two direction-flips stacked together, and two flips cancel out. That sends you right instead.
2 − (−4) means: start at 2. Subtraction says “move left”, but the number being subtracted is negative, which flips that direction. Left, flipped, becomes right. You move 4 steps right and land on 6, which is the same answer as 2 + 4.
This is the double-flip mistake: treating “subtract a negative” as if the two minus signs simply merge into one, instead of cancelling into a plus.
The fix: two minus signs next to each other (subtracting a negative) always become a plus: 3 − (−4) = 3 + 4 = 7. A simple check: rewrite “− (−4)” as “+ 4” before doing anything else, every single time.
Nothing special happens at zero. It is just another point on the line. It is worth practising, though, because that is where sign confusion is most likely to creep in.
−3 + 5 means: start at −3 and move 5 steps right. You pass through −2, −1, 0, 1, and land on 2. The line does not reset or change behaviour at zero. You simply keep counting through it.
Every number has an opposite: the number the same distance from zero on the other side. The opposite of 6 is −6. Adding a number and its opposite always gives 0, because you walk out and walk straight back.
Start at −6. Then −6 + 6 means move 6 steps right: you land on 0. In the same way, 9 + (−9) = 0, and −2.5 + 2.5 = 0. These “zero pairs” are useful: in a long sum, spot a number and its opposite and cancel them.
A longer expression is just a chain of moves. Work from left to right, and treat every sign as the direction of the next move. After each move, write down where you landed.
Work out 5 − 8 + 4 − (−2).
5 − 8 = −3. Then −3 + 4 = 1. Then 1 − (−2) = 1 + 2 = 3.
Rewrite every subtraction as adding a negative. Then add all the positives together, add all the negatives together, and combine the two totals.
(a) −7 + 3 − 5 = −7 + 3 + (−5). Positives: 3. Negatives: −7 and −5, which total −12. Then 3 + (−12) = −9.
(b) 4 − 9 + 6 − 11 + 2. Positives: 4 + 6 + 2 = 12. Negatives: −9 − 11 = −20. Then 12 − 20 = −8.
The number line also measures change. Change = final value − starting value. A positive change means the value rose, and a negative change means it fell.
(a) The temperature goes from −3 °C to 4 °C. Change = 4 − (−3) = 4 + 3 = +7. It rose by 7 degrees.
(b) The temperature goes from 6 °C to −5 °C. Change = −5 − 6 = −11. It fell by 11 degrees.
The distance between two numbers ignores direction, so it is always positive. The distance between −3 and 4 is 7, the same as the distance from 4 to −3.
| Expression Pattern | Direction of Movement | Why |
|---|---|---|
| start + positive | Right | Standard addition |
| start − positive | Left | Standard subtraction |
| start + negative | Left | Adding a negative behaves like subtracting |
| start − negative | Right | Two flips (subtract, then negative) cancel out |
Count the minus signs sitting next to each other right before the number. An even count means move right for that step, and an odd count means move left. “− (−4)” has two minus signs, so move right. “− 4” has one, so move left. Even three in a row works: 5 − (−(−2)) has three minus signs, which is odd, so it moves left: 5 − 2 = 3.
A submarine is at −40 metres (40 metres below sea level). It rises 25 metres: −40 + 25 = −15 metres, still below sea level but closer to the surface. If it then rises another 15 metres, −15 + 15 = 0, and it is at sea level. The sign tells you which side of sea level it is on, and the number tells you how far.
An account has a balance of −Rs 1,200 (overdrawn). A deposit of Rs 3,500 is made. Then Rs 2,000 is withdrawn. Finally the bank cancels (subtracts) a charge of −Rs 150 that was wrongly added.
−1,200 + 3,500 − 2,000 − (−150). Step by step: −1,200 + 3,500 = 2,300. Then 2,300 − 2,000 = 300. Then 300 − (−150) = 300 + 150 = Rs 450.
A football team has scored 12 goals and conceded 15. Goal difference = 12 − 15 = −3. In its next match it wins 3 to 0, which adds +3 to the goal difference: −3 + 3 = 0. The team is now level.
A stall makes +Rs 500 on Monday, −Rs 800 on Tuesday and +Rs 200 on Wednesday. Total = 500 − 800 + 200 = −Rs 100. After three days the stall is Rs 100 behind.
A lift starts at floor 4 and goes down 7 floors. 4 − 7 = −3, which is basement level 3. Then it rises 5 floors: −3 + 5 = floor 2.
Negative numbers were once very controversial. The Indian mathematician Brahmagupta wrote rules for arithmetic with debts and fortunes in the year 628. But in Europe, some mathematicians were still refusing to accept negative numbers in the late 1700s, even though merchants were already using them to track debts.
A gold ring marks your starting point. Read the expression, work out which way to move and how far, then tap where you land. Your best score is saved on this device.
1. Start at −4 and move 9 right: 5.
2. −2 + 7 = 5.
3. 6 − 11 = −5, then −5 + 3 = −2.
1. −7 + 3 = −4, then −4 − 5 = −9.
2. 4 + 6 = 10.
3. 5 − 8 = −3, then −3 + 4 = 1, then 1 + 2 = 3.
4. Change = final − start = −5 − 6 = −11, a fall of 11 °C.
5. −1,200 + 3,500 − 2,000 − (−150) = 2,300 − 2,000 + 150 = Rs 450.
Draw a number line on the floor with chalk or tape and let learners walk the moves. Ask them to say aloud which way they will go before they step. Use the pattern in Section 5 to show that subtracting a negative is not an arbitrary rule. Real contexts such as temperature, lifts and bank balances give the signs meaning. Whenever a mistake occurs, ask the learner to draw the move, rather than correcting the sign for them.
Signed numbers appear in temperature, finance, altitude, sport and every branch of science. The skill of tracking direction and distance with positive and negative numbers is also what makes algebra, coordinates and graphs possible later on.
10 questions. Select an answer for each, then submit to see your score instantly.