Integers Multiplication & Division Calculator
Master the rules of multiplying and dividing positive and negative integers with our comprehensive guide and interactive calculator
Calculate Integer Operations
Enter two integers to multiply or divide
What Are Integers?
Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals. The set of integers extends infinitely in both directions: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Understanding how to multiply and divide integers is fundamental to algebra, real-world problem-solving, and advanced mathematics.
Integer Classifications:
- Positive Integers: Numbers greater than zero (1, 2, 3, 4, 5, ...)
- Negative Integers: Numbers less than zero (-1, -2, -3, -4, -5, ...)
- Zero: Neither positive nor negative (0)
The Golden Rules: Sign Patterns
🔑 The Master Rule
SAME SIGNS = POSITIVE (+)
DIFFERENT SIGNS = NEGATIVE (−)
This simple rule applies to BOTH multiplication AND division!
💡 Memory Tip:
Think of signs as directions: When both numbers "agree" (same sign), you get a positive result. When they "disagree" (different signs), you get a negative result. This works for both multiplication and division!
Integer Multiplication: Detailed Rules
Rule 1: Positive × Positive = Positive
When multiplying two positive integers, the result is always positive. This is the most intuitive rule since it matches basic multiplication.
Examples:
• 5 × 3 = 15
• 7 × 8 = 56
• 12 × 4 = 48
Rule 2: Negative × Negative = Positive
When multiplying two negative integers, the negatives "cancel out" and the result is positive. Think of it as: a negative times a negative reverses the direction twice, bringing you back to positive.
Examples:
• −5 × −3 = 15
• −7 × −8 = 56
• −12 × −4 = 48
Why does this work?
Consider temperature: If it drops 3 degrees per hour (−3) for 4 hours backward in time (−4), the net change is +12 degrees. Two negatives reverse each other!
Rule 3: Positive × Negative = Negative
When multiplying a positive integer by a negative integer (or vice versa), the result is always negative. The order doesn't matter—different signs always produce a negative result.
Examples:
• 5 × −3 = −15
• −7 × 8 = −56
• 12 × −4 = −48
• −9 × 6 = −54
Integer Division: Detailed Rules
The sign rules for division are identical to multiplication! Division is the inverse operation of multiplication, so the same sign patterns apply.
Rule 1: Positive ÷ Positive = Positive
When dividing two positive integers, the quotient is positive (when evenly divisible).
Examples:
• 15 ÷ 3 = 5
• 56 ÷ 8 = 7
• 48 ÷ 4 = 12
Rule 2: Negative ÷ Negative = Positive
When dividing two negative integers, the negatives cancel out and the quotient is positive. Same signs = positive result!
Examples:
• −15 ÷ −3 = 5
• −56 ÷ −8 = 7
• −48 ÷ −4 = 12
Verification Trick:
Check division by multiplication: If −15 ÷ −3 = 5, then 5 × (−3) must equal −15. But wait! We know 5 × −3 = −15 ✓. The math checks out!
Rule 3: Different Signs = Negative
When dividing integers with different signs (one positive, one negative), the quotient is always negative. It doesn't matter which number is positive and which is negative—different signs always yield a negative result.
Examples:
• 15 ÷ −3 = −5
• −56 ÷ 8 = −7
• 48 ÷ −4 = −12
• −54 ÷ 6 = −9
Step-by-Step Process for Integer Operations
Universal 3-Step Method
Step 1
Ignore the Signs
Multiply or divide the absolute values (positive versions) of the numbers.
Step 2
Apply Sign Rule
Same signs → Positive
Different signs → Negative
Step 3
Write Final Answer
Attach the correct sign to your result from Step 1.
Worked Example: −12 × 7
Step 1: Ignore signs → 12 × 7 = 84
Step 2: Check signs → One negative (−12), one positive (7) = Different signs
Step 3: Different signs = Negative result → Answer: −84
Common Mistakes to Avoid
❌ Mistake #1: Forgetting Negative × Negative = Positive
Wrong: −5 × −3 = −15
Right: −5 × −3 = +15
Remember: Two negatives make a positive! Same signs always give positive results.
❌ Mistake #2: Treating Division Differently from Multiplication
Wrong: Thinking −20 ÷ −4 = −5 (applying different rules)
Right: −20 ÷ −4 = +5 (same sign rules as multiplication)
Remember: Division follows THE EXACT SAME sign rules as multiplication!
❌ Mistake #3: Confusing Addition/Subtraction Rules with Multiplication/Division
Wrong: Applying "take the sign of the bigger number" to −5 × 3
Right: Different signs = negative result → −5 × 3 = −15
Remember: Multiplication and division have simpler rules than addition/subtraction!
❌ Mistake #4: Sign Errors in Multi-Step Problems
Wrong: −3 × −2 × −4 = +24 (forgetting to count all signs)
Right: −3 × −2 = +6, then +6 × −4 = −24
Remember: Work left to right, applying sign rules at each step. Odd number of negatives = negative result; even number = positive.
Real-World Applications
🌡️ Temperature Changes
If temperature drops 3°C per hour (−3) for 5 hours, total change: −3 × 5 = −15°C
Application: Negative × Positive = Negative (temperature decreases)
💰 Banking & Finance
Withdrawing $50 each day (−50) for 6 days: −50 × 6 = −$300 (account decreases by $300)
Application: Debits (negative) × frequency = Total change
🏔️ Elevation Changes
A submarine descends 200 meters (−200). Dividing by 4 dives: −200 ÷ 4 = −50 meters per dive
Application: Negative ÷ Positive = Negative (below sea level)
📊 Stock Market
Stock drops $5 per day (−5) for 8 days: −5 × 8 = −$40 total loss. Reversing the loss period (−8 days = going back in time): −5 × −8 = +$40
Application: Understanding losses and gains over time
⚡ Physics: Velocity
Object moves left at 10 m/s (−10) for 3 seconds: −10 × 3 = −30 meters (30 meters left of starting point)
Application: Direction (sign) × time = Displacement
🎮 Game Scoring
Lose 10 points (−10) on each of 5 wrong answers: −10 × 5 = −50 points. Removing those 5 wrong answers (−5): −10 × −5 = +50 points restored
Application: Penalties and score adjustments
Practice Problems
Test Your Understanding
Problem 1: 9 × (−7)
Show Solution
Step 1: Ignore signs → 9 × 7 = 63
Step 2: Different signs (+ and −) → Negative result
Step 3: Answer: −63
Problem 2: (−12) × (−5)
Show Solution
Step 1: Ignore signs → 12 × 5 = 60
Step 2: Same signs (both −) → Positive result
Step 3: Answer: +60 or 60
Problem 3: 56 ÷ (−8)
Show Solution
Step 1: Ignore signs → 56 ÷ 8 = 7
Step 2: Different signs (+ and −) → Negative result
Step 3: Answer: −7
Problem 4: (−45) ÷ (−9)
Show Solution
Step 1: Ignore signs → 45 ÷ 9 = 5
Step 2: Same signs (both −) → Positive result
Step 3: Answer: +5 or 5
Problem 5: (−6) × (−3) × (−2)
Show Solution
Step 1: (−6) × (−3) = +18 (same signs → positive)
Step 2: (+18) × (−2) = −36 (different signs → negative)
Alternative: 3 negative numbers = odd number of negatives → negative result
Answer: −36
Quick Reference Card
INTEGER MULTIPLICATION & DIVISION RULES
(+) × (+) = (+)
Positive × Positive
= Positive
(−) × (−) = (+)
Negative × Negative
= Positive
(+) × (−) = (−)
Positive × Negative
= Negative
(−) × (+) = (−)
Negative × Positive
= Negative
🔑 MASTER RULE
SAME SIGNS = POSITIVE | DIFFERENT SIGNS = NEGATIVE
📝 Division follows the SAME rules as multiplication!
(+) ÷ (+) = (+) | (−) ÷ (−) = (+) | (+) ÷ (−) = (−) | (−) ÷ (+) = (−)
About the Author
Adam
Co-Founder at RevisionTown
Math Expert specializing in various international curricula including IB (International Baccalaureate), AP (Advanced Placement), GCSE, IGCSE, and standardized test preparation. Dedicated to making mathematics accessible and understandable for students worldwide through clear explanations, practical examples, and interactive learning tools.
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