Multiplying by a whole number is repeated adding: $3 \times (-4)$ means three groups of $-4$, so $(-4) + (-4) + (-4) = -12$. But what could $(-3) \times (-4)$ mean? Nobody can make "negative three groups." A pattern answers the question.
Key idea
Multiply the absolute values. The product is positive when the signs are the same and negative when they are different. With more factors, an even number of negative signs gives a positive product; an odd number gives a negative one.
Why it works
Look at the pattern: $3 \times (-4) = -12$, $2 \times (-4) = -8$, $1 \times (-4) = -4$, $0 \times (-4) = 0$. Each time the first factor goes down by 1, we take away one group of $-4$, and the product goes up by 4. Keep the pattern going: $(-1) \times (-4) = 4$ and $(-2) \times (-4) = 8$.
The distributive property says the same thing: $$0 = (-1) \times (4 + (-4)) = (-1)(4) + (-1)(-4) = -4 + (-1)(-4)$$ The only number that makes this true is $(-1)(-4) = 4$.
The distributive property says the same thing: $$0 = (-1) \times (4 + (-4)) = (-1)(4) + (-1)(-4) = -4 + (-1)(-4)$$ The only number that makes this true is $(-1)(-4) = 4$.
Worked examples
Worked example: Different signs
Find $(-6) \times 7$.
- Absolute values: $6 \times 7 = 42$.
- The signs are different, so the product is negative.
Answer: $-42$
Worked example: Same signs
Find $(-6) \times (-7)$.
- Absolute values: $6 \times 7 = 42$.
- Both factors are negative, so the product is positive.
Answer: $42$
Worked example: Three factors
Find $(-2) \times (-3) \times (-5)$.
- Absolute values: $2 \times 3 \times 5 = 30$.
- Three negative signs: the first two make a positive, and the third makes it negative again.
Answer: $-30$
Common mistake
Using the multiplying rule when adding: "two negatives make a positive," so $-3 + (-4) = 7$. Not so: adding two negatives is two moves to the left, $-3 + (-4) = -7$. The rule "same signs give a positive" is only for multiplying and dividing. Before you answer, name the operation.
A story where a negative times a negative happens: the temperature falls 2 degrees every hour, a rate of $-2$. Three hours ago is time $-3$. The change from now to then is $(-3) \times (-2) = 6$: it was 6 degrees warmer.