Why does a negative times a negative equal a positive?

Think of the Town Matrix: A "Good Guy" is positive (+), a "Bad Guy" is negative (-), "Entering town" is positive (+), and "Leaving town" is negative (-). When a Bad Guy (-) Leaves Town (-), that is a GREAT thing for the town (+)! Mathematically, taking away a debt or negative quantity increases your overall value: (-4) × (-3) = +12.

What is the sign rule for multiplying and dividing integers?

The rule for multiplication and division is identical: If both numbers have the SAME sign (both positive or both negative), the result is ALWAYS POSITIVE (+). If the numbers have DIFFERENT signs (one positive and one negative), the result is ALWAYS NEGATIVE (-).

How do you determine the sign when multiplying more than two negative numbers?

Count the total number of negative signs in the multiplication chain: If the count of negative signs is EVEN (2, 4, 6...), the final answer is POSITIVE (+). If the count of negative signs is ODD (1, 3, 5...), the final answer is NEGATIVE (-). For example, (-2) × (-3) × (-4) has 3 negatives (odd), so the answer is -24.

What is the difference between -5^2 and (-5)^2?

In (-5)^2, the negative is inside the parentheses, meaning (-5) × (-5) = +25. In -5^2, order of operations applies exponents before the negative sign: -(5 × 5) = -25. This is one of the most common mistakes on algebra exams!

What happens when you divide an integer by zero vs dividing zero by an integer?

0 divided by any non-zero integer is always 0 (e.g. 0 ÷ -7 = 0). However, dividing ANY number by zero is UNDEFINED in mathematics (e.g. -7 ÷ 0 = undefined) because no number multiplied by 0 can equal -7.

arithmetic

Multiplying and Dividing Integers (Signed Numbers)

January 20, 2026

6 min read

PanMaths Team

⚡ THE 2 UNIVERSAL SIGN RULES

Same Signs ➔ Positive (+) 

 Different Signs ➔ Negative (-)

1. (+)(+) = + and (-)(-) = +2. (+)(-) = - and (-)(+) = -3. Multi-factor chain: Count minus signs (Even = +, Odd = -)

✖️➗ Pre-Algebra Masterclass • Middle School (Grades 6–8)

Multiplying & Dividing Signed Integers: The Town Matrix & Sign Rules

🔬 Try Sign Simulator 👇

💡 STEP 1: THE INTUITION

Why Negative × Negative = Positive (The Town Matrix)

🤠 THE SHERIFF'S TOWN MATRIX

Mental Model

1. (+) × (+) = (+)

Good Guy Enters Town

(+4) × (+3) = +12 (Good for town!)

2. (-) × (+) = (-)

Bad Guy Enters Town

(-4) × (+3) = -12 (Bad for town!)

3. (+) × (-) = (-)

Good Guy Leaves Town

(+4) × (-3) = -12 (Bad for town!)

4. (-) × (-) = (+)

Bad Guy Leaves Town

(-4) × (-3) = +12 (GREAT for town! 🎉)

📹 The Video Rewind Analogy: If a person walks backwards in a video ($-$) and you press reverse play ($-$), what do you see? They are moving forward into the future ($+$)!

📜 STEP 2: THE 2 GOLDEN SIGN RULES

The Universal Multiplication & Division Rules

1. SAME SIGNS = POSITIVE (+)

(+) × (+) = (+)  |  (-) × (-) = (+)

If both signs match, the answer is always POSITIVE. Examples: 6 × 7 = 42 and (-6) × (-7) = +42.

2. DIFFERENT SIGNS = NEGATIVE (-)

(+) × (-) = (-)  |  (-) × (+) = (-)

If the signs are opposites, the answer is always NEGATIVE. Examples: (-6) × 7 = -42 and 48 ÷ (-6) = -8.

⚡ Pro-Tip: The Negative Sign Counter Rule

When multiplying a long string of numbers (like (-2) × (-3) × (-4) × (-5)), simply count the negative signs:
• EVEN count of negatives (2, 4, 6...): Result is POSITIVE (+).
• ODD count of negatives (1, 3, 5...): Result is NEGATIVE (-).

🔬 INTERACTIVE SIGN SIMULATOR

Live Signed Multiplication & Division Solver

Select any sign combination below to see how signs interact in real time:

Problem: (+6) × (+7) = +42 (Both Positive)Positive × Positive

Sign Evaluation

+ & + ➔ +

Matching Signs = POSITIVE

Absolute Number Math

Value without sign

Final Result

+42

Signed Answer

Step 1: Check the Signs

🔍 Signs: (+) and (+)

Both numbers have the SAME positive sign. Same signs always yield a POSITIVE result.

⚠️ EXAM MISTAKE RADAR

3 Mistakes Students Make on Signed Numbers Exams

1. Confusing Addition with Multiplication

Writing -3 - 5 = +8 ❌. Adding negatives stays negative (-3 + -5 = -8). Only multiplying negatives turns positive (-3 × -5 = +15)!

2. The Parenthesis Exponent Trap (-4² vs (-4)²)

In (-4)², the negative is squared: (-4) × (-4) = +16. But in -4², exponents apply first: -(4 × 4) = -16!

3. Division by Zero (0 ÷ -5 vs -5 ÷ 0)

0 ÷ -5 = 0 (zero divided by anything is 0). But -5 ÷ 0 = UNDEFINED (division by zero is mathematically impossible!).

🧠 ACTIVE RECALL QUIZ

Test Your Understanding in 60 Seconds

Select your answers and click "Check Answers" to test your mastery:

Question 1: What is (-8) × (-6)?

Question 2: What is (-72) ÷ (+9)?

Question 3: Evaluate (-1) × (-2) × (-3) × (-4):

📄 Free Printable Practice

Print Signed Integer Practice Worksheets

Generate 20 customizable practice problems for multiplying and dividing positive and negative numbers with instant PDF answer keys.

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PANMATHS.COM • INTEGER CHEAT SHEET

Multiplying & Dividing Signed Integers (Grades 6–8)

Name: _______________________________

Date: _______________ Period: ________

1. SAME SIGNS = POSITIVE (+)

(+) × (+) = (+)  |  (-) × (-) = (+)

(+6) × (+7) = +42 | (-6) × (-7) = +42

2. DIFFERENT SIGNS = NEGATIVE (-)

(+) × (-) = (-)  |  (-) × (+) = (-)

(-6) × (+7) = -42 | (+54) ÷ (-9) = -6

⚡ MULTI-FACTOR NEGATIVE SIGN COUNTING RULE

Count total minus signs: EVEN count (2, 4, 6...) ➔ POSITIVE (+) | ODD count (1, 3, 5...) ➔ NEGATIVE (-)

Operation Type	Problem	Sign Rule	Final Result
Double Negative Multiplied	(-8) × (-9)	Same signs = (+)	+72
Opposite Signs Multiplied	(-7) × (+8)	Different signs = (-)	-56
Double Negative Divided	(-64) ÷ (-8)	Same signs = (+)	+8
Multi-Factor Chain	(-2) × (-3) × (-4)	3 negatives = Odd = (-)	-24

🚫 ADDITION VS MULTIPLICATION

-5 - 3 = -8! (Adding negatives stays negative). Only multiplying gives (-5) × (-3) = +15!

🔄 PARENTHESES EXPONENTS

(-5)² = +25 (negative is inside). But -5² = -25 (exponent applies first)!

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