Multiplying & Dividing Integers
Two operations, one master rule for signs. Faster and cleaner than addition — once you see the pattern.
The Sign Rules — The Whole Lesson in Four Cells
Multiplication and division of integers follow the same sign rules. Unlike addition, you don’t need to compare absolute values or think about direction. There are only two outcomes: same signs → positive, different signs → negative.
Different signs → Negative (+ × − = −) (− × + = −)
The rule is identical for division: same signs → positive quotient, different signs → negative quotient.
The rule that two negatives multiply to a positive surprises most students. Here is how to see it naturally — through a pattern that makes the result unavoidable:
The pattern leaves no room for debate. Each time the second factor decreases by 1, the result increases by 3. Once we cross zero, the results must become positive. This is not a convention — it is a logical necessity for arithmetic to remain consistent.
Imagine you are accumulating debt at R50 per day. Going forward in time (positive direction) increases your debt (negative). Going backward in time (negative direction) means you had less debt — which is a positive outcome.
−50 × −3 days = +R150
Three days ago, you had R150 more than you do now. Negative × negative = positive.
For any multiplication or division of integers, follow two steps:
Step 2: Multiply or divide the absolute values Always handle the sign separately from the magnitude. This prevents errors.
When multiplying a chain of integers, the sign of the final result depends entirely on how many negative factors there are:
Odd number of negative factors → negative result
The absolute values are multiplied normally — only the sign is governed by this count.
A negative fraction can place its minus sign in three equivalent positions. All three mean exactly the same thing:
Apply BODMAS carefully with negative integers.
−4² — the power applies only to 4, then the negative is applied: −(4²) = −(16) = −16
This distinction is the source of one of the most common algebra errors. When in doubt, use brackets.
(−3)² = 9 | (−2)⁴ = 16 | (−5)⁶ = 15625
A negative number raised to an odd power is always negative:
(−3)³ = −27 | (−2)⁵ = −32 | (−1)⁷ = −1
−3² = −(3²) = −9
They are NOT the same. The brackets make all the difference.
(−2) × (−5) = 10 (same signs → positive)
10 + 3 = 13
(−1)⁹⁹ = −1 (odd power → negative)You don’t need to compute — the even/odd rule tells you everything about the sign.
The share price fell R24 in total.The −3 represents the daily fall (negative change). 8 is the number of days (positive). Different signs → negative result.
Multiply the equations: (a×b) × (a÷b) = (−24)(−6) → a² = 144 → a = 12 or −12
If a = 12: b = −24÷12 = −2. Check: 12÷(−2) = −6 ✓
If a = −12: b = 2. Check: (−12)÷2 = −6 ✓
Solutions: (a,b) = (12,−2) or (−12, 2)
- Same signs (+ × + or − × −) always produce a positive result
- Different signs (+ × − or − × +) always produce a negative result
- The same sign rules apply equally to division
- For a chain of factors: count the negatives — even count → positive, odd count → negative
- (−n)² is positive; −n² is negative — brackets change everything
- Even powers of negative numbers are always positive; odd powers are always negative
- Division by zero is undefined — never allowed