A number like −5/7 can be written with the minus sign in front, in the numerator, or in the denominator, and it always has the same value. The sign rule only decides products and quotients. In addition and subtraction, what matters is where the numbers sit on the number line, which is why adding and subtracting negatives does not work like multiplying.
| Situation | What to do with the sign | Example |
|---|---|---|
| Moving the sign | It is the same number in all three positions | 6/(−11) = −6/11 |
| Adding or subtracting | Compare absolute values and use a common denominator | −4/9 + 1/3 = −1/9 |
| Multiplying or dividing | Same signs give positive; different signs give negative | (−3/5) ÷ (3/10) = −2 |
| Negative mixed number | The sign applies to the whole mixed number | −1 1/5 = −6/5 |
Depending on the row, the sign is moved, compared, or counted, and only in the multiplying and dividing row does the “negative times negative” rule apply.
Subtracting and multiplying with the same digits
With the same digits and the same signs, a subtraction and a product give different results, because the operation is different. That contrast explains most of the mistakes on this topic.
Worked example: −4/7 − 3/7 versus (−4/7) × (−3/7)
Calculate −4/7 − 3/7 and (−4/7) × (−3/7).
- Subtraction. The denominators already match, so you combine the numerators with their signs: −4 − 3 = −7.
- Simplify the subtraction. −7/7 = −1.
- Product. Both factors are negative, so the result is positive. Multiply 4 × 3 = 12 on top and 7 × 7 = 49 on the bottom.
The subtraction gives −1 and the product gives 12/49.
To check, −1 + 3/7 = −4/7, the starting point of the subtraction. And 12/49 ÷ (−3/7) = 12/49 × (−7/3) = −84/147 = −4/7, the first factor of the product.
In the subtraction you start at a negative number and move another three sevenths to the left, ending at −1. In the product there is no movement: you multiply two quantities, and matching signs give a positive product. Turning the two signs of the subtraction into a plus, as in a product, would give −1/7, which is a mistake.
Where to put the sign
Moving the sign from the denominator to the numerator does not change the value. For example, 6/(−11) = −6/11 = −(6/11). To see why, multiply the numerator and denominator by −1. If both terms are negative, the signs cancel, so (−5)/(−7) = 5/7.
The usual choice is to keep the denominator positive and put the sign in front, because that makes comparing and simplifying easier. Zero has no sign: 0/(−7) equals 0. The denominator, with or without a sign, can never be zero.
The number line confirms it. −5/7 and 5/7 are the same distance from zero, one on each side, so they are opposites and add up to 0.
Adding and subtracting with negatives
Here you do not count signs; you think in positions. Adding a negative number moves you to the left, and adding a positive number moves you to the right. In −4/9 + 1/3 you switch to ninths, −4/9 + 3/9, and the result is −1/9. It is negative because 4/9 outweighs 1/3: the sign of the addend with the larger absolute value wins.
Subtracting is the same as adding the opposite. That is why 1/8 − (−3/4) = 1/8 + 6/8 = 7/8, a result greater than 1/8. The parentheses show that the second minus sign belongs to the number being subtracted. The add fractions and subtract fractions calculators accept negative addends and show the common denominator.
Multiplying and dividing with negatives
In products and quotients of nonzero values, the same signs give positive and different signs give negative. First decide the sign, then calculate with the absolute values. With (−7/9) × (3/14), the sign is negative and the calculation is 21/126. Reduced, the result is −1/6.
To divide, flip only the divisor and keep its sign. In (−3/5) ÷ (3/10) you switch to (−3/5) × (10/3) = −30/15 = −2. If you flip the first number instead of the divisor, the result is wrong. If the divisor is 0, there is no reciprocal and the division is undefined. For longer calculations, the multiply fractions calculator and the divide fractions calculator apply the same rule.
Negative mixed numbers
In −1 1/5, the sign applies to the whole mixed number: it means −(1 + 1/5), which is −6/5. It is not read as −1 + 1/5, which equals −4/5. The same happens with −2 1/4: −(8/4 + 1/4) = −9/4, while −2 + 1/4 = −7/4.
Convert the positive part first and apply the sign at the end. If you are unsure how a mixed number reads, write it as an improper fraction. The mixed number to fraction tool has separate fields for the whole number and the fraction, so the sign does not depend on a space.
Practice with signs
First decide the sign you expect, then calculate and check with the inverse operation or the number line. When you place each result on the line, see whether it lands on the side of zero you expected.
Practice
- −7/10 + 3/10
- 2/9 − (−4/9)
- (−3/4) × (8/9)
- (−5/8) ÷ (−5/6)
Show the answers
- −4/10 = −2/5. The two addends are on opposite sides, and the one with the larger absolute value wins.
- 2/9 + 4/9 = 6/9 = 2/3, because subtracting a negative adds its opposite.
- −24/36 = −2/3. One negative factor gives a negative product.
- (−5/8) × (−6/5) = 30/40 = 3/4. Both signs match, so the result is positive.
To estimate without calculating whether a result should be greater or less than 0, 1/2, or 1, see the guide to benchmark fractions.
Sources
The English Wikipedia article Negative number, in its Multiplication subsection, states the sign rule: the product of a positive and a negative number is negative, and the product of two negative numbers is positive (accessed October 17, 2026).