Before this page: be able to lay negative numbers out as arrows on the line from adding and subtracting negative numbers, and divide by a one-digit number using short division.
One negative number makes the product negative
Multiplying by 3 lays the same arrow down three times:
(−4) × 3
One arrow of −4
−4 runs four units to the left of zero.
Swapping the two numbers leaves the product unchanged, so 3 × (−4) = −12 as well: one negative number multiplied by a positive one gives a negative product.
Example: work out 6 × (−7)
Multiply the sizes, then fix the sign.
- 6 × 7 = 42 The two distances from zero multiply to give the length of the answer.
- One of the numbers is negative. Six arrows of −7 all run to the left.
- 6 × (−7) = −42 An arrow 42 units long, pointing left.
Multiplying by a negative number turns the arrow round
Arrows can be laid out three times or twice, never −2 times. (−3) × (−2) is reached instead by stepping the second number down from 3, one at a time.
Lowering the second number by one takes away one −3, a move of 3 to the right, and the same move carries on as the second number passes zero:
(−3) × 3 down to (−3) × (−2)
Three lots of −3
The arrow reaches −9, nine units to the left of zero.
Multiplying by a negative number turns the arrow round and stretches it by the size of that number. Turning −3 round gives 3: (−3) × (−1) = 3. Stretching that arrow to twice its length gives (−3) × (−2) = 6.
(−3) × (−2) = 6
- Two numbers with the same sign multiply to a positive answer, and two with different signs to a negative one.
- A zero gives an arrow of no length, so the product is zero and has no sign.
A decimal product takes its sign the same way. 0.8 × (−5) = −4 has one negative number in it, and (−0.6) × (−7) = 4.2 has two. The digits there multiply as 6 × 7 = 42, and the point is placed by the method on multiplying decimals.
Example: work out (−5) × (−6)
Multiply the sizes, then count the negatives.
- 5 × 6 = 30 The length of the answer, before its sign.
- Both numbers are negative. Two negative numbers are two turns, so an arrow of 30 turned twice points right.
- (−5) × (−6) = 30 A right-pointing arrow 30 units long.
A quotient takes its sign the way a product does
Dividing cuts an arrow into equal parts, or counts how many of one arrow fit inside another:
−12 ÷ 3 and −12 ÷ (−3)
Twelve units to the left
−12 is one arrow, twelve units long, running left from zero.
One negative number makes a quotient negative, and two make it positive.
Arrows of −3 laid from zero never reach 12, however many of them there are. Dividing is the inverse of multiplying, so 12 ÷ (−3) is the number that multiplies −3 to give 12. That number is −4, because (−3) × (−4) = 12.
Example: work out −4.2 ÷ 6
Divide −4.2 into six equal parts.
- 4.2 ÷ 6 = 0.7 Sharing 42 tenths between 6 gives 7 tenths.
- One of the numbers is negative. Six equal parts of a left-pointing arrow all point left.
- −4.2 ÷ 6 = −0.7 Each part is an arrow 0.7 long, pointing left.
A decimal divisor is turned into a whole number first, by the method on dividing by a decimal. One negative number then makes the quotient negative: −7.2 ÷ 0.9 = −8.
Each negative number in a product turns the arrow once more
In (−2) × 3 × (−5), the two negative numbers turn the arrow twice, which leaves it facing the way it started; a third would turn it round again.
Counting the negatives
The size of the answer comes from multiplying the numbers without their signs: 2 × 3 × 5 = 30.
An even count of negative numbers leaves the answer positive, and an odd count leaves it negative.
(−2) × 3 × (−4) = 24
Example: work out (−1) × (−2) × (−3)
Multiply the sizes, then count the negatives.
- 1 × 2 × 3 = 6 The length first.
- All three numbers are negative. Three turns leave the arrow pointing left, as one turn does.
- (−1) × (−2) × (−3) = −6 An arrow 6 units long, pointing left.
A negative divisor turns the arrow round in the same way, so ((−6) × (−4)) ÷ (−3) = 24 ÷ (−3) = −8 has three negative numbers and a negative answer.
Common mistakes
- Writing a minus because both numbers had one −20 ÷ (−5) is not −4. Four arrows of −5 reach −20, and the answer counts those arrows: it is 4.
- Turning two negatives into a positive in a subtraction Nothing is being multiplied in −4 − 3. Starting at −4 and moving a further 3 to the left reaches −7, not 7.
- Pairing the negatives off and dropping the odd one (−2) × (−3) × (−4) is not 24. The third minus turns the arrow round once more, and it finishes pointing left, at −24.
Summary
- Three arrows of −4 laid tip to tail reach −12, so (−4) × 3 = −12.
- Multiplying by a negative number turns the arrow round, so (−3) × (−2) = 6.
- One negative number makes a product or a quotient negative, and two make it positive: 0.8 × (−5) = −4.
- A division answers a multiplication, so 12 ÷ (−3) = −4, since (−3) × (−4) = 12.
- In a longer product, an even count of negative numbers leaves the answer positive: (−2) × 3 × (−5) = 30.