Multiplying and Dividing Negatives

"A minus times a minus makes a plus." You have probably heard the chant. But why? It sounds like a rule someone made up to torture students. Why should (-6)\times(-4) come out to a cheerful, positive +24?

Here is a way to feel it before we prove it. Think of a negative number as a debt — money you owe. Multiplying by a negative means taking that thing away. So "take away a debt" makes you richer: a minus (removing) of a minus (debt) is a plus. Once you see it that way, the chant stops being magic and starts being obvious.

The one rule

Multiplying and dividing with negatives comes down to a single rule about signs. Work out the sign first; then find the size exactly as you always have.

Same signs give a positive answer. Different signs give a negative answer. This holds for multiplication and division alike:

(+)(+) = +\qquad (-)(-) = +

(+)(-) = -\qquad (-)(+) = -

So (-3) \times 4 = -12 (different signs), while (-3) \times (-4) = 12 (same signs). Division behaves identically — (-12) \div (-4) = 3 and (-12) \div 4 = -3.

When you multiply or divide:

same signs give a positive answer;
different signs give a negative answer;
two negatives cancel — they make a positive;
counting the negative factors: an even number of them gives a positive result, an odd number gives a negative result.

Worked examples

Say the sign out loud, then the size:

(-6)\times(-4) — same signs → positive; 6\times4=24, so the answer is +24.
3\times(-7) — different signs → negative; 3\times7=21, so -21.
(-20)\div 5 — different signs → negative; 20\div5=4, so -4.
(-45)\div(-9) — same signs → positive; 45\div9=5, so +5.

A real story: the plunging temperature

A cold front rolls in and the temperature drops by 3^\circ every hour. A "drop of 3" is a change of -3 per hour. Where will the temperature be 4 hours from now compared with now?

(-3) \times 4 = -12,

so 12^\circ colder. Different signs, negative answer — that matches the real cold. Now ask a sneakier question: what was the temperature 4 hours ago? "Hours ago" is negative time, -4:

(-3) \times (-4) = +12,

12^\circ warmer in the past — which is exactly right, because if it has been falling steadily, earlier it was higher. Minus times minus is plus, and the weather agrees.

Chains of negatives — just count them

When several negative numbers are multiplied together, you don't need to track the sign at every step. Just count the minus signs. An even count is positive; an odd count is negative (each pair of negatives cancels to a positive).

(-2)\times(-3)\times(-1) — three negatives (odd) → negative. Size 2\times3\times1=6, so -6.
(-1)\times(-2)\times(-1)\times(-5) — four negatives (even) → positive. Size 10, so +10.

Seeing why: follow the pattern

Here is a proof you can almost watch happen. Multiply -2 by a first number that steps down one at a time, and look at the answers:

3\times(-2) = -6

2\times(-2) = -4

1\times(-2) = -2

0\times(-2) = \phantom{-}0

Each time the first number drops by 1, the answer climbs by +2: -6, -4, -2, 0, \ldots To keep that steady pattern going, the next lines are forced to be:

(-1)\times(-2) = +2

(-2)\times(-2) = +4

The pattern demands that a negative times a negative be positive — any other answer would break the nice, even staircase. That is exactly the argument that convinced mathematicians to accept the rule.

Division is just multiplication in disguise

You never have to learn a separate sign rule for division, because dividing is the inverse of multiplying. Check it with the multiplication it reverses:

(-12)\div(-4)=3, because 3\times(-4)=-12. ✓ (Same signs → positive.)
(-12)\div 4=-3, because (-3)\times 4=-12. ✓ (Different signs → negative.)

Same "same-signs / different-signs" rule, no extra memorising. One idea, two operations.

The sign grid

Read off the sign of either operand along an edge; the cell where they meet is the sign of the answer. The diagonal of matching signs is positive; the off-diagonal of mixed signs is negative.

This is the single biggest trap in the whole topic. "Two negatives make a positive" is a rule for multiplying and dividing only. It does not apply to adding or subtracting.

Adding two negatives makes it MORE negative: -3 + (-5) = -8, not +8. Two debts stacked together is a bigger debt.
But multiplying two negatives flips to positive: (-3)\times(-5) = +15.

Same two numbers, same two minus signs, completely different answers — because the operation is different. Before you reach for "minus and minus makes plus", check you are actually multiplying or dividing. And in a long product, count the negatives carefully: miscounting by one flips the whole sign.

Do not feel bad if "minus times minus is plus" felt strange — you are in excellent company. For hundreds of years some of the sharpest mathematicians in Europe flatly distrusted negative numbers, calling them "false" or "absurd" and refusing to let them into their equations. The rule for multiplying them was argued over well into the 1700s.

What finally settled it was a beautifully stubborn idea: insist that the normal laws of arithmetic keep working. If the distributive law is to hold, then (-1)\times(-1) is forced to be +1 — there is no other value that keeps the maths consistent. And the debt picture makes it human: repeatedly removing a debt (a negative amount, taken away a negative number of times) genuinely leaves you richer. So minus-times-minus really is plus — not by decree, but because the whole system would fall apart otherwise.