Dividing a Negative Number by a Negative Number: Rules & Examples

Dividing a negative number by a negative number always gives a positive result. For example, −12 ÷ −3 = 4, not −4. The reason is that division follows the same sign rules as multiplication: two matching signs produce a positive answer, while two different signs produce a negative one. Since both numbers here are negative, the two minus signs cancel, and the quotient is positive. Once you know the rule, the arithmetic itself works exactly the same as dividing positive numbers.

The Basic Rule

A negative number is any value less than zero, written with a minus sign (for example, −6 or −25). Division asks how many times one number fits into another. When you divide, you treat the two signs separately from the digits: first divide the numbers as if they were positive, then decide the sign of the answer.

Because a negative divided by a negative has two identical signs, those signs cancel out. The rule is simple: same signs give a positive answer, different signs give a negative answer. So −20 ÷ −4 = 5, and −100 ÷ −5 = 20.

This mirrors what happens in multiplication, which is why the two operations share one set of sign rules. As Khan Academy explains, the sign of a product or quotient depends only on how many negative factors are involved — an even count gives a positive result, an odd count gives a negative one.

Why Two Negatives Cancel

Think of a negative sign as a “reverse.” Dividing by a negative reverses the direction of the result, and dividing by another negative reverses it again. Two reversals bring you back to the original direction, so the answer ends up positive.

Sign Rules for Division

The table below shows all four possible sign combinations, so you can see where the negative-by-negative case fits. The digit values divide normally; only the sign changes.

Notice that only two rows produce a positive answer: positive ÷ positive and negative ÷ negative. Whenever the two signs match, the quotient is positive. Whenever they differ, it is negative.

A Quick Way to Remember

• Same signs → positive answer.
• Different signs → negative answer.

This single rule covers every division problem with signed numbers, which Math Is Fun reinforces with the idea that “like signs” give a positive and “unlike signs” give a negative.

Step-by-Step Example

The method never changes: divide the digits, then apply the sign rule. Take −144 ÷ −12.

1. Divide the digit values: 144 ÷ 12 = 12.
2. Check the signs: negative ÷ negative → positive.
3. Answer: 12.

The two negatives cancel and the quotient is positive. If you ever get a negative answer for a negative-by-negative problem, it means you lost track of a sign — double-check before moving on.

The same approach works for fractions. Divide −3/4 by −1/2 by multiplying by the reciprocal: −3/4 × −2/1. Multiply the digits to get 6/4, or 3/2, then apply the sign rule: negative ÷ negative → positive. The answer is 3/2, or 1.5.

Common Mistakes to Avoid

Most errors with signed division come from mixing up operations or losing track of a sign. Watch for these pitfalls.

• Confusing division with subtraction. Subtracting a negative changes the sign, but dividing by a negative follows its own rule. −8 − (−2) = −6, while −8 ÷ (−2) = 4. These are completely different operations.
• Forgetting the second negative. If you divide −18 by 6, the answer is −3 because the signs differ. It is only when both numbers are negative that the result turns positive.
• Mishandling multiple negatives. In longer expressions, count the negative signs. An even number gives a positive result; an odd number gives a negative one.
• Dropping a sign in fractions. When a fraction like −3/−6 appears, the two negatives cancel, so it equals positive 1/2. Do not treat a negative in the numerator and a negative in the denominator as separate problems.
• Assuming the digits change. The sign rule only affects whether the answer is positive or negative. The numbers themselves divide exactly as they would if both were positive.

If a problem mixes several operations, simplify one step at a time and check the sign after each division. This keeps errors from piling up.

Checking Your Work Without a Calculator

Because division reverses multiplication, you can always confirm a signed division by multiplying the quotient by the divisor and seeing whether you recover the dividend. This check catches sign errors that a quick glance might miss.

• If you get −15 ÷ −3 = −5, multiply back: −3 × −5 = 15, not −15, so the sign is wrong. The correct answer is 5.
• If you get −20 ÷ −4 = 5, multiply back: −4 × 5 = −20, which matches, so the answer is correct.

Running this one-step check takes seconds and works for fractions and larger numbers too. It is one of the most dependable habits for anyone learning to work confidently with signed numbers.

Key Takeaways

• Dividing a negative number by a negative number always gives a positive answer.
• The rule follows the same sign logic as multiplication: same signs → positive, different signs → negative.
• To solve, divide the digit values first, then apply the sign rule.
• Only the negative ÷ negative and positive ÷ positive cases yield a positive quotient.
• Common errors involve confusing division with subtraction, missing a negative sign, or miscounting negatives in longer expressions.
• You can verify any division by multiplying the quotient by the divisor and checking that you get the dividend back.