Multiplication is repeated jumping and division counts how many jumps fit. Seen this way, the sign rules, including why two negatives make a positive, stop being rules to memorise.
Multiplication and division are the operations most people remember as “times tables and long division”. On the number line they become something much more visual: equal jumps. Seeing them as jumps makes the sign rules (why a negative times a negative is positive) a matter of direction instead of memorisation. It also explains why you can never divide by zero.
This guide builds in order: multiplication as repeated jumps, multiplying by negatives, why two negatives make a positive (explained three different ways), division as fitting jumps, and finally mixed expressions and real problems. It follows on from Addition & Subtraction on the Number Line, and every worked example here has been checked twice.
Multiplication is repeated addition, and repeated addition is repeated jumping. Start at 0. In 3 × 4, the first number is how many jumps, and the second number is how big each jump is. So 3 × 4 means: three jumps of size 4, starting from zero.
Three jumps of 4 land on 4, then 8, then 12. The last landing point is the product. Every time you read “×”, translate it into “this many jumps of this size”.
5 × 3 means five jumps of 3: 3, 6, 9, 12, 15. The same product comes from three jumps of 5: 5, 10, 15. Both routes land on 15, which is why 5 × 3 = 3 × 5. The order of the factors does not change the product.
This works for fractions and decimals too. 3 × 0.5 means three jumps of one half: 0.5, 1, 1.5.
It is easy to mix up the size of a jump with the number of jumps, or to add the two numbers directly.
The fix: 3 + 4 would mean one jump of 3 then one jump of 4. But 3 × 4 means three jumps, each of size 4. That lands on 12, not 7. Ask yourself two questions: “how many jumps?” (the first number) and “how big is each?” (the second).
That last idea, reflection, is the key to understanding every sign rule that follows.
When the jump size is negative, each jump goes left. So a positive number of jumps of a negative size takes you into the negatives.
3 × (−4) means three jumps, each of size −4: you land on −4, then −8, then −12.
(a) 2 × (−5): two jumps of −5 land on −5, then −10.
(b) 4 × (−2): −2, −4, −6, −8.
(c) A positive times a negative is always negative: the jumps are all in the leftward direction.
Multiplication makes numbers bigger only for positive factors greater than 1.
The fix: each of the three jumps goes left, because the jump size is −4. After the jumps you are at −12, which is smaller than where you started. “Bigger” in size (12 away from zero) does not mean a bigger number.
This is the rule that surprises almost everyone: (−3) × (−4) = +12. It is not an arbitrary rule. There are three independent ways to see why it must be true.
Look at what happens to (−3) × n as n goes down by 1 each time. The answers climb by 3 each time. The pattern must continue past zero.
Multiplying by −1 reflects a number across zero. So (−3) × (−4) means: first take three jumps of −4, which lands on −12, then flip that result across zero to land on +12. Two reversals bring you back to the positive side.
Imagine a company that loses Rs 4 million every year, a rate of −4 (million per year). What was its value 3 years ago? Three years ago is a time of −3. The change in value is rate × time = (−4) × (−3) = +12. Three years ago the company was Rs 12 million richer than today, which makes sense, because it has lost Rs 12 million since then.
The Indian mathematician Brahmagupta stated the sign rules in the year 628, including that a debt multiplied by a debt is a fortune. In Europe, some mathematicians were still refusing to accept negative numbers in the late 1700s. The rule that two negatives make a positive is not a convenient trick: it is the only choice that keeps the rest of arithmetic consistent.
| Multiplication | Result | Example |
|---|---|---|
| positive × positive | positive | 3 × 4 = 12 |
| positive × negative | negative | 3 × (−4) = −12 |
| negative × positive | negative | (−3) × 4 = −12 |
| negative × negative | positive | (−3) × (−4) = 12 |
Count the negative signs. An even number of negatives (0, 2, 4…) gives a positive answer, and an odd number (1, 3, 5…) gives a negative answer. This works for any number of factors: (−2) × (−3) × (−4) has three negatives, which is odd, so the answer is negative: 6 × (−4) = −24. It is also why (−2)³ = −8 but (−2)² = +4 (see Exponents & Powers).
The signs do not simply “carry over”. They combine.
The fix: work out the size first: 3 × 4 = 12. Then decide the sign: two negatives (an even count) give a positive. So (−3) × (−4) = +12. Compare with (−3) + (−4) = −7, where the signs do carry over. Addition and multiplication follow different sign rules.
Division asks the reverse question: how many jumps of a given size fit between 0 and a number? For 12 ÷ 4, count how many jumps of size 4 take you from 0 to 12.
Three jumps of 4 reach 12 exactly, so 12 ÷ 4 = 3. Division and multiplication are inverses: 3 × 4 = 12 and 12 ÷ 4 = 3 say the same thing from two directions.
(a) 20 ÷ 5 = 4, because 4 jumps of 5 reach 20.
(b) 14 ÷ 4: three jumps of 4 reach 12, which leaves 2. Two is half of a jump, so 14 ÷ 4 = 3 remainder 2, or 3½.
(c) 2 ÷ 1/2: how many jumps of one half fit into 2? Four of them (0.5, 1, 1.5, 2), so 2 ÷ 1/2 = 4. Dividing by a fraction less than 1 gives a bigger answer. See Multiply & Divide Fractions.
If the jumps fit exactly, the jump size is a factor of the number, and the number is a multiple of the jump size. This is the same idea as in Factors & Multiples.
The sign rules for division are exactly the same as for multiplication. You can always check a division by multiplying back: if a ÷ b = c, then c × b must equal a.
| Division | Result | Example | Check by multiplying |
|---|---|---|---|
| positive ÷ positive | positive | 12 ÷ 4 = 3 | 3 × 4 = 12 |
| negative ÷ positive | negative | (−12) ÷ 4 = −3 | (−3) × 4 = −12 |
| positive ÷ negative | negative | 12 ÷ (−4) = −3 | (−3) × (−4) = 12 |
| negative ÷ negative | positive | (−12) ÷ (−4) = 3 | 3 × (−4) = −12 |
How many jumps of size −4 reach +12? Jumps of −4 go left, so you would have to go backwards three times (3 jumps in the reverse direction) to get to +12. A backwards count is a negative count: −3. Check: (−3) × (−4) = +12. ✓
Counting one minus sign as the whole answer ignores that there are two of them.
The fix: count the negative signs: there are two, which is an even number, so the answer is positive. 12 ÷ 4 = 3, so (−12) ÷ (−4) = +3. Check by multiplying back: 3 × (−4) = −12. ✓
Each multiplication fact gives a whole family of facts. Knowing one gives you the rest, including with negatives.
From 3 × 4 = 12: 4 × 3 = 12, 12 ÷ 3 = 4 and 12 ÷ 4 = 3.
From (−3) × 4 = −12: 4 × (−3) = −12, (−12) ÷ 4 = −3 and (−12) ÷ (−3) = 4.
When multiplication or division sits inside a longer expression, use BODMAS (see BODMAS & DMAS Rules): multiply and divide first, left to right, then add and subtract.
(a) −2 × 3 + 5: multiply first, −2 × 3 = −6. Then −6 + 5 = −1.
(b) 12 ÷ (−3) − 2 × (−4): divide, 12 ÷ (−3) = −4. Multiply, 2 × (−4) = −8. Then −4 − (−8) = −4 + 8 = 4.
(c) The average of −4, 6, −2 and 0 is (−4 + 6 − 2 + 0) ÷ 4 = 0 ÷ 4 = 0.
(d) Powers with negative bases: (−3)² = (−3) × (−3) = 9, but −3² = −(3 × 3) = −9.
| Situation | Best Strategy | Why It Works |
|---|---|---|
| Multiplying by a positive | Repeated jumps from zero | Multiplication is repeated addition |
| Multiplying by a negative | Jump left | A negative jump size reverses direction |
| Negative × negative | Do the jumps, then flip across zero (or count negatives) | Two reversals cancel out |
| Dividing | Count how many jumps fit | Division asks “how many of these fit in that?” |
| Checking a division | Multiply the answer by the divisor | Division and multiplication are inverses |
| Not sure of the sign | Count the negative signs: even is positive, odd is negative | Each negative flips the direction once |
A shopkeeper’s account loses Rs 15 each week for 6 weeks: 6 × (−15) = −Rs 90. That is the same as six jumps of −15 on the number line. If the shopkeeper wants to know how many weeks of that loss add up to −Rs 90, the question is (−90) ÷ (−15) = 6 weeks.
The temperature is falling by 2 °C every hour. In 5 hours the change is 5 × (−2) = −10 °C. And 3 hours ago? Time is −3, so the change relative to now is (−3) × (−2) = +6 °C: it was 6 degrees warmer 3 hours ago. ✓
A diver descends 3 metres every minute. After 12 minutes the depth is 12 × (−3) = −36 m. To find how long it takes to reach −36 m, divide: (−36) ÷ (−3) = 12 minutes.
A business loses Rs 4,500 and the loss is shared equally between 3 partners: (−4,500) ÷ 3 = −Rs 1,500 each. Each partner is Rs 1,500 worse off.
Each round shows a multiplication expression. Work out the direction and size of the jumps, then tap where you land, starting from zero. Your best score is saved on this device.
1. One negative sign, so the answer is negative: 7 × 3 = 21, so −21.
2. Two negative signs, so the answer is positive: 36 ÷ 9 = 4.
3. Three negative signs, so the answer is negative: 2 × 3 × 4 = 24, so −24.
1. −42.
2. Two negatives give a positive: 40.
3. Two negatives give a positive: 56 ÷ 7 = 8.
4. 12 ÷ (−3) = −4 and 2 × (−4) = −8. Then −4 − (−8) = −4 + 8 = 4.
5. After 5 hours: 5 × (−2) = −10 °C. Three hours ago: (−3) × (−2) = +6 °C, meaning it was 6 °C warmer then.
Let learners physically walk the jumps on a floor number line, saying aloud “three jumps of minus four” before they step. Use the pattern in Section 4 to show that “negative times negative is positive” is forced by the pattern, not decided by a rule-maker. Keep reminding learners that addition and multiplication have different sign rules, since (−3) + (−4) and (−3) × (−4) behave differently. Real contexts such as temperature change, diving and shared losses give the signs meaning.
Signed multiplication and division are the foundation for algebra (where letters can stand for negative values), for gradient and graph work, and for any calculation involving rates of change. Understanding why the sign rules work, instead of just memorising them, is what lets you trust your answers in all of those later topics.
10 questions. Select an answer for each, then submit to see your score instantly.