Sign Rules for Multiplication and Division
Multiply and divide with positive and negative integers and rational numbers, understanding the rules for the sign of the product or quotient
What a learner can do afterwards
- Apply the sign rules when multiplying two integers (positive × negative, negative × negative)
- Apply the sign rules when dividing two integers
- Solve multi-step real-world problems involving all four operations with positive and negative rational numbers
1 · Read
When you multiply two numbers, the signs tell you the sign of the answer. If both numbers have the same sign, positive times positive or negative times negative, the answer is positive. If the numbers have different signs, one positive and one negative, the answer is negative.
Look at −3 × −4. Both numbers are negative, so they have the same sign. The answer is positive: −3 × −4 = 12.
Now look at division: −20 ÷ 5. The numbers have different signs, one negative and one positive, so the answer is negative: −20 ÷ 5 = −4. Multiplication and division follow the exact same sign rule.
A fast way to check bigger problems: count how many negative numbers you are multiplying or dividing. An even count of negatives gives a positive answer, and an odd count gives a negative answer. This works with fractions and decimals too, not just whole numbers.
Same signs make a positive answer, different signs make a negative answer, whether you multiply or divide.
More to read
Negative numbers: multiplication and division FAQ
A minus sign tells a number to face the other way on the number line. Multiplying by a negative flips the direction. Flip twice and you face forward again: that is why two negatives make a positive. The full rules: positive x positive is positive; negative x negative is positive; mixed signs give negative. Division follows the same pattern because it undoes multiplication. (-5) x (-6) = 30 since both flips cancel. (-20) / 4 = -5 since one flip survives. Learn same-plus, mixed-minus and signs stop being scary.
Why is a Negative Times a Negative Always Positive?
Why must negative times negative be positive? Because the familiar rules should keep working everywhere. Count down: 3 x (-4) = -12, 2 x (-4) = -8, 1 x (-4) = -4, 0 x (-4) = 0. Each step adds 4. The next step must add 4 again: (-1) x (-4) = 4. The pattern forces it. If two negatives gave a negative, the steady +4 staircase would break. Maths keeps its patterns unbroken, so negative times negative is positive by necessity, not by decree.
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2 · Watch
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Where it sits
16 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.