Why a negative times a negative is positive — and how division rides along for free.
Point 4 of 5 in this lesson: 5.5.4 Reciprocals: partners that multiply to one
Picture standing at zero and taking a jump of −2 — two steps to the left. Multiplying by a positive number just repeats and stretches that jump in the same direction. But multiplying by a negative number does something extra: it grabs the jump and flips it to the other side of zero. Flip once and a leftward jump points right; flip a second time and it swings back to the left. That single picture — flipping direction — explains every sign rule in this lesson.
By the end you will be able to multiply any two rational numbers, predict the sign of a long product just by counting its negative factors, use the distributive law in both directions, find the reciprocal of a number, and divide by turning it into a multiplication. We keep the steady color habit: positive numbers are teal, negative numbers are red, zero is slate, and a distance or size is amber.
Multiplication started life as repeated addition, and that meaning still works when the numbers are negative. To compute 3 × (−2), add −2 to itself three times:
3 × (−2) = (−2) + (−2) + (−2) = −6
On the number line that is three jumps of two, all heading left — you land on −6. So far the only new thing is that the repeated step itself points left.
Now the heart of the matter. What does multiplying by a negative do? Multiplying by −1 gives the opposite of a number — it flips it across zero. So (−1) × (−2) flips −2 over to +2. To work out (−3) × (−2), take the three leftward jumps of 3 × (−2) and flip the whole thing across zero. Three jumps that landed on −6 now land on +6.
That flip gives us a rule you never have to memorize blindly, because you can always picture it. First decide the sign, then multiply the sizes (the absolute values):
| × | + | − |
|---|---|---|
| + | + | − |
| − | − | + |
SAME signs → positive. DIFFERENT signs → negative. Then multiply the absolute values to get the size. So (−4)×(+5) = −20 (different signs), and (−4)×(−5) = +20 (same signs).
Set the two factors. Watch the jumps stack up — and watch them flip to the other side of zero the moment a factor turns negative. The product is where you land.
Once you trust the flip, a long product holds no fear. Each negative factor flips the direction once. Flip an even number of times and you end up facing forward — the product is positive. Flip an odd number of times and you end up facing backward — the product is negative. The size is just the product of all the absolute values.
So to find the sign of a whole product, ignore the digits for a moment and count the minus signs. For example, (−2)(−3)(−1) has three negatives — an odd count — so the answer is negative, and the size is 2·3·1 = 6, giving −6.
The count-the-negatives rule decides the sign, but a single 0 among the factors makes the entire product 0 — there is nothing left to be positive or negative about. Always check for a zero factor first.
Flip each factor between + and −. The counter tallies the negatives and predicts the sign before it shows you the value. Even count → positive; odd → negative.
The friendly properties you met for whole numbers survive the move to negatives untouched. Multiplication is commutative — you may swap the order, (−4)×7 = 7×(−4) — and associative — you may regroup, so a long product can be reordered to make the arithmetic easy.
The most powerful is the distributive law, which links multiplication and addition:
a × (b + c) = a × b + a × c
It works both ways. Read left to right, you open the brackets — handy even when there are negatives inside:
−2(3 + (−5)) = −2×3 + −2×(−5) = −6 + 10 = 4.
Check the short way: 3 + (−5) = −2 first, then −2×−2 = 4. Same answer. ✓
Read right to left, you pull out a common factor — the secret behind fast mental arithmetic:
7×8 + 7×2 = 7×(8 + 2) = 7×10 = 70.
Both 56 and 14 share the factor 7, so collect it once and add the easy 8 + 2 = 10.
Set a (it may be negative), b, and c. See a(b+c) split into ab + ac, and check that both routes reach the same total.
Two numbers are reciprocals when their product is exactly 1. The reciprocal of a number a is 1a, and the reciprocal of a fraction pq is found by turning it upside down to qp. For example, the reciprocal of 5 is 15 (and 5 × 15 = 1), while the reciprocal of 34 is 43.
The sign comes along for the ride: the reciprocal of a negative number is negative, because a number and its reciprocal must multiply to a positive 1, and that needs same signs. So the reciprocal of −2 is −12, and indeed −2 × (−12) = 1.
There is no number you can multiply by 0 to get 1 — every product with 0 is 0. So 0 is the one number with no reciprocal. (This is the same reason you can never divide by zero, which you will see in the next section.)
Step through some numbers — whole numbers, fractions, and negatives. Watch the reciprocal flip over, keep its sign, and multiply back to exactly 1 — except for 0, which has none.
Here is the payoff for all the work on reciprocals. Dividing by a number is the same as multiplying by its reciprocal:
a ÷ b = a × 1b
Since division is just multiplication wearing a disguise, it must obey the very same sign rule: same signs give a positive quotient, different signs give a negative one. Nothing new to learn.
(−12) ÷ 3: different signs → negative; 12 ÷ 3 = 4; answer −4.
(−12) ÷ (−4): same signs → positive; 12 ÷ 4 = 3; answer 3.
One more handy fact about where the minus sign lives in a fraction. A single negative on top, on the bottom, or out front all mean the same value:
−pq = −pq = p−q
The reason is the sign rule once more: a negative divided by a positive, or a positive divided by a negative, both land on negative. Two minuses on top and bottom would cancel back to positive.
Set the two numbers. The widget rewrites a ÷ b as a × (1/b), decides the sign from the same rule, and shows the quotient.
Multiplying by a positive stretches a jump; multiplying by a negative stretches it and flips its direction — so two negatives flip you back to forward, and same signs make a positive, different signs make a negative. For a long product, just count the negatives: even → positive, odd → negative, and any single 0 makes the whole thing 0. The distributive law a(b+c)=ab+ac opens brackets and pulls out common factors. Two numbers are reciprocals when they multiply to 1 (flip the fraction, keep the sign; 0 has none), and dividing is multiplying by the reciprocal — so division obeys the exact same sign rule.
With all four operations on rational numbers in hand, the next lesson stacks them into powers (repeated multiplication, where the sign rule explains why (−2) squared is positive) and pins down the order of operations for mixed expressions.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in. Tap the answer you think is right.
This lesson develops multiplication and division of rational numbers, aligned to U.S. Common Core 7.NS.A.2: building signed products from the distributive property and the fact that (−1)(−1) = 1 (7.NS.A.2a), establishing signed quotients and the equivalent forms −(p/q) = (−p)/q = p/(−q) with a nonzero divisor (7.NS.A.2b), and applying the commutative, associative, and distributive properties to compute fluently (7.NS.A.2c). The single most common misconception is that "negative × negative = negative." The antidote runs through the whole lesson: the flip-twice picture on the number line (multiplying by a negative reverses direction, and reversing twice faces you forward again) backed up by the count-the-negatives rule (an even number of negative factors gives a positive product).