Multiplying and Dividing Negative Numbers

For Elementary Students

The Sign Rules Made Easy

When you multiply or divide with negative numbers, there's a simple pattern to remember!

Think about it like this: Are the two signs the SAME or DIFFERENT?

Same Signs = Positive Answer

When both numbers have the same sign, the answer is POSITIVE!

Both positive:

3 × 5 = 15
Positive × Positive = Positive ✓

Both negative:

(−3) × (−5) = 15
Negative × Negative = Positive ✓

Memory trick: Two negatives make a positive! Like a double flip!

Different Signs = Negative Answer

When the numbers have different signs, the answer is NEGATIVE!

Positive and negative:

3 × (−5) = −15
Positive × Negative = Negative

Negative and positive:

(−3) × 5 = −15
Negative × Positive = Negative

Quick Reference Chart

Super simple rule:

• SAME signs → + (positive answer)
• DIFFERENT signs → − (negative answer)

Multiplication Examples

Example 1: 4 × 6 = ?

• Both positive (same signs)
• Answer: 24 (positive)

Example 2: (−4) × (−6) = ?

• Both negative (same signs)
• Answer: 24 (positive)

Example 3: 4 × (−6) = ?

• Different signs
• Answer: −24 (negative)

Example 4: (−4) × 6 = ?

• Different signs
• Answer: −24 (negative)

Division Works the Same Way!

The exact same rules work for division!

Same signs = Positive:

20 ÷ 4 = 5 (both positive)
(−20) ÷ (−4) = 5 (both negative)

Different signs = Negative:

20 ÷ (−4) = −5 (different signs)
(−20) ÷ 4 = −5 (different signs)

Why Does Negative × Negative = Positive?

Think about patterns:

3 × 2 = 6
3 × 1 = 3
3 × 0 = 0
3 × (−1) = −3   (we go down by 3 each time)
3 × (−2) = −6

Now with negatives:

(−3) × 2 = −6
(−3) × 1 = −3
(−3) × 0 = 0
(−3) × (−1) = 3   (pattern continues, we go UP by 3!)
(−3) × (−2) = 6

The pattern shows: negative times negative gives positive!

More Than Two Numbers

When you multiply lots of numbers, count the negative signs:

Even number of negatives → Positive answer Odd number of negatives → Negative answer

Example 1: (−2) × (−3) × (−4) = ?

• Count negatives: 1, 2, 3 → 3 negatives (odd!)
• Multiply the numbers: 2 × 3 × 4 = 24
• Odd negatives → negative answer
• Answer: −24

Example 2: (−2) × (−3) × (−4) × (−5) = ?

• Count negatives: 4 negatives (even!)
• Multiply: 2 × 3 × 4 × 5 = 120
• Even negatives → positive answer
• Answer: 120

For Junior High Students

Sign Rules for Multiplication and Division

When multiplying or dividing integers, the sign of the result depends on the signs of the operands.

Fundamental rule:

• Same signs → positive result
• Different signs → negative result

This applies to both multiplication and division.

Multiplication Sign Rules

Rule 1: (+) × (+) = (+)

• Positive times positive equals positive
• Example: 7 × 3 = 21

Rule 2: (−) × (−) = (+)

• Negative times negative equals positive
• Example: (−7) × (−3) = 21

Rule 3: (+) × (−) = (−)

• Positive times negative equals negative
• Example: 7 × (−3) = −21

Rule 4: (−) × (+) = (−)

• Negative times positive equals negative
• Example: (−7) × 3 = −21

Summary table:

Division Sign Rules

The exact same rules apply to division.

Rule 1: (+) ÷ (+) = (+)

• Example: 24 ÷ 6 = 4

Rule 2: (−) ÷ (−) = (+)

• Example: (−24) ÷ (−6) = 4

Rule 3: (+) ÷ (−) = (−)

• Example: 24 ÷ (−6) = −4

Rule 4: (−) ÷ (+) = (−)

• Example: (−24) ÷ 6 = −4

Why Negative × Negative = Positive

Pattern approach:

Consider the sequence:

3 × 3 = 9
3 × 2 = 6    (decrease by 3)
3 × 1 = 3    (decrease by 3)
3 × 0 = 0    (decrease by 3)
3 × (−1) = −3  (decrease by 3)
3 × (−2) = −6  (decrease by 3)

Now multiply by −3:

(−3) × 3 = −9
(−3) × 2 = −6   (increase by 3)
(−3) × 1 = −3   (increase by 3)
(−3) × 0 = 0    (increase by 3)
(−3) × (−1) = 3  (increase by 3: positive!)
(−3) × (−2) = 6  (increase by 3)

Logical approach: Multiplication by a negative "reverses direction"

• Multiplying by −1 reverses the sign
• Multiplying by −1 twice gets you back to positive
• (−1) × (−1) = 1

Distributive property proof:

0 = (−3) × 0
0 = (−3) × [3 + (−3)]
0 = (−3) × 3 + (−3) × (−3)
0 = −9 + (−3) × (−3)
9 = (−3) × (−3)

Therefore (−3) × (−3) = 9 (positive)

Multiplying Multiple Integers

When multiplying more than two integers, count the number of negative factors.

Rule:

• Even number of negative factors → positive result
• Odd number of negative factors → negative result

Example 1: (−2) × (−3) × (−5)

Count negatives: 3 (odd)

Calculate magnitude: 2 × 3 × 5 = 30
Apply sign: odd negatives → negative
Answer: −30

Example 2: (−2) × (−3) × (−4) × (−5)

Count negatives: 4 (even)

Calculate magnitude: 2 × 3 × 4 × 5 = 120
Apply sign: even negatives → positive
Answer: 120

Example 3: (−1) × (−1) × (−1) × (−1) × (−1)

Count negatives: 5 (odd)

(−1)⁵ = −1

Example 4: (−1) × (−1) × (−1) × (−1)

Count negatives: 4 (even)

(−1)⁴ = 1

Combining Operations

When expressions mix multiplication and division, process from left to right, tracking signs.

Example: (−12) × 3 ÷ (−6)

Method 1: Left to right

(−12) × 3 = −36  (different signs)
(−36) ÷ (−6) = 6  (same signs)
Answer: 6

Method 2: Track signs separately

Signs: (−) × (+) ÷ (−) → two negatives (even) → positive
Magnitude: 12 × 3 ÷ 6 = 36 ÷ 6 = 6
Answer: +6

Real-Life Applications

Temperature change: Temperature drops 5°F per hour for 3 hours

Change = (−5) × 3 = −15°F

Debt: Removing a $20 debt 3 times

(−20) × (−3) = 60
(Subtracting negative debt = gaining money)

Elevation: Descending 10 meters per minute for 4 minutes

(−10) × 4 = −40 meters (below starting point)

Banking: Withdrawing $50 three times

(−50) × 3 = −150 (account decreases by $150)

Physics: Velocity in opposite direction

Speed = −30 m/s (going backwards)
Time = 2 seconds
Distance = (−30) × 2 = −60 m (60 m in reverse direction)

Properties

Closure: Multiplying two integers always gives an integer

Commutative: a × b = b × a

(−3) × 5 = 5 × (−3) = −15

Associative: (a × b) × c = a × (b × c)

[(−2) × 3] × (−4) = (−2) × [3 × (−4)]
(−6) × (−4) = (−2) × (−12)
24 = 24 ✓

Identity: a × 1 = a

(−7) × 1 = −7

Zero property: a × 0 = 0

(−15) × 0 = 0

Multiplicative inverse: For non-zero a, there exists 1/a such that a × (1/a) = 1

Common Mistakes

Mistake 1: Forgetting the sign rules

❌ (−4) × (−5) = −20 ✓ (−4) × (−5) = 20 (same signs → positive)

Mistake 2: Miscounting negatives in multi-factor products

❌ (−2) × (−3) × (−4) = 24 (thinking even negatives) ✓ (−2) × (−3) × (−4) = −24 (3 negatives is odd)

Mistake 3: Confusing addition and multiplication rules

❌ Thinking (−3) × (−4) = −7 (adding instead of multiplying) ✓ (−3) × (−4) = 12

Mistake 4: Sign errors in division

❌ (−20) ÷ (−4) = −5 ✓ (−20) ÷ (−4) = 5 (same signs → positive)

Tips for Success

Tip 1: Memorize the basic rule: same signs = positive, different signs = negative

Tip 2: For multiple factors, count the negative signs first

Tip 3: Calculate the magnitude separately from the sign

Tip 4: Check your answer: does the sign make sense?

Tip 5: Practice with small numbers first (like −2, −3) before harder problems

Tip 6: Remember: multiplication and division follow the same sign rules

Advanced: Exponents with Negatives

Even exponents: Always produce positive results

(−3)² = (−3) × (−3) = 9
(−2)⁴ = 16

Odd exponents: Preserve the negative sign

(−3)³ = (−3) × (−3) × (−3) = −27
(−2)⁵ = −32

Important distinction:

−3² = −(3 × 3) = −9  (exponent applies to 3 only)
(−3)² = (−3) × (−3) = 9  (exponent applies to −3)

Practice