The Number System
Multiplying & Dividing Rational Numbers
Multiplying and dividing positive and negative fractions and decimals.
Prerequisites
- Adding & Subtracting Rational Numbers
Why Negative Times Negative Is Positive
Before reading on, think this through: why should a negative number multiplied by a negative number give a positive result? It's a rule most people memorize without ever really seeing why it has to be true — so look for the pattern in this sequence first: 3 × (−2) = −6, 2 × (−2) = −4, 1 × (−2) = −2, 0 × (−2) = 0. What comes next?
Each step down by 1 in the first factor increases the product by 2. Continuing that exact pattern: (−1) × (−2) should equal 0 + 2 = 2, and (−2) × (−2) should equal 4. The pattern that already held for every earlier step demands that a negative times a negative comes out positive — it's not an arbitrary rule, it's the only outcome that keeps the pattern consistent.
Definition — Sign Rules for Multiplication and Division
Worked Example — Multiplying Rational Numbers
Worked Example — Dividing Rational Numbers
Worked Example — A Multi-Step Expression with Signed Fractions
Tip
Common Mistakes
Applying the addition sign rules (comparing distances from zero) to multiplication or division, which follow a completely different rule.
Multiplication and division sign rules only depend on whether the signs match, not on the size of either number — same signs give positive, different signs give negative, regardless of magnitude.
Losing track of the sign partway through a multi-step expression with several signed numbers.
Resolve the sign of each intermediate result immediately after each operation, rather than trying to track every sign mentally until the very end.
Key Takeaways
- Multiplying or dividing two numbers with the same sign gives a positive result; different signs give a negative result.
- This rule is consistent with the pattern of steadily increasing a product by a fixed amount as the pattern of factors continues into negative numbers.
- In a chain of multiplications and divisions, counting the total number of negative factors quickly predicts the final sign.
Summary
Sign rules for multiplying and dividing rational numbers follow logically from extending a numeric pattern, not from an arbitrary convention. The next unit turns from the number system to geometry, beginning with classifying angles by their measure.
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