Multiplying Integers- Rules and Examples

What Multiplying Integers Actually Means

Multiplying integers is just repeated addition on steroids. If you have 3 × 4, you're adding 3 four times: 3 + 3 + 3 + 3 = 12. Simple enough.

But what happens when negative numbers enter the picture? That's where most people get tripped up. The good news: there are only two rules you need to memorize, and once you get them down, you can handle any integer multiplication problem.

The Two Rules That Govern Everything

Integer multiplication boils down to two questions:

• What sign should the answer have? (positive or negative)
• What's the absolute value of the result? (just multiply the numbers ignoring signs)

Here's how you determine the sign:

Rule 1: Same Signs Give a Positive Result

When you multiply two positive numbers, you get a positive result. When you multiply two negative numbers, you also get a positive result.

Positive × Positive = Positive

Negative × Negative = Positive

This trips people up constantly. Negative times negative doesn't stay negative—it flips to positive. Remember that.

Rule 2: Different Signs Give a Negative Result

When the signs don't match, your answer is negative.

Positive × Negative = Negative

Negative × Positive = Negative

Quick Reference Table

Examples Worked Out

Positive × Positive

7 × 5 = 35

Both numbers are positive. The answer is positive. Nothing complicated here.

Negative × Negative

(-7) × (-5) = 35

Same sign (both negative), so the result is positive. The 7 and 5 multiply to 35, and the negatives cancel out.

Positive × Negative

7 × (-5) = -35

Mixed signs. The 7 and 5 multiply to 35, but the answer carries the negative sign from the (-5).

Negative × Positive

(-7) × 5 = -35

Same situation as above. Different signs mean negative result.

Multiplying More Than Two Integers

When you have chains of numbers, work from left to right or pair them up. The sign depends on how many negative numbers are in the mix:

• Even number of negatives → result is positive
• Odd number of negatives → result is negative

Example: (-2) × (-3) × (-4) × 5

You have three negatives. Three is odd, so the final answer is negative.

Multiply the absolute values: 2 × 3 × 4 × 5 = 120

Apply the negative sign: -120

How to Multiply Integers: Step-by-Step

Here's the process you should follow every time:

1. Ignore the signs. Multiply the absolute values of all numbers first.
2. Count the negative signs. Determine if you have an even or odd count.
3. Apply the sign. Even negatives = positive. Odd negatives = negative.

Practice problem: (-6) × 3 × (-2) × (-1)

Step 1: 6 × 3 × 2 × 1 = 36

Step 2: Three negatives. Three is odd.

Step 3: Odd negatives = negative

Answer: -36

Common Mistakes to Avoid

• Forgetting that two negatives make a positive. This is the most frequent error. When you see (-3) × (-4), don't automatically write -12. Write 12.
• Confusing multiplication with addition. When you multiply by a negative, you're not subtracting—you're reversing direction. Don't add the negative; multiply through.
• Dropping parentheses too early. Keep the signs attached to their numbers until you've finished counting negatives.

Multiplying by Zero

Zero is the wildcard. Any integer multiplied by zero equals zero.

7 × 0 = 0

(-7) × 0 = 0

0 × 0 = 0

This rule overrides everything else. Doesn't matter how many negatives, how big the numbers—multiply by zero and you get zero.

Why This Matters

Integer multiplication isn't just abstract math. You use it constantly: calculating profit and loss, determining debt, measuring temperature changes, tracking inventory that goes up and down. The sign tells you direction. The magnitude tells you size.

Master these rules now and you'll handle fractions, decimals, and algebraic expressions with integers without breaking a sweat. The foundation matters.