Multiplying Negative Numbers — Definition, Formula & Examples
Multiplying negative numbers follows two key rules: a negative number times a positive number gives a negative result, and a negative number times a negative number gives a positive result.
For any real numbers and where and : the product , the product , and the product . These rules preserve the distributive property and the structure of the real number system.
Key Formula
(-a) \times (-b) = a \times b$$
$$(-a) \times b = -(a \times b)
Where:
- = A positive real number (the absolute value of the first factor)
- = A positive real number (the absolute value of the second factor)
How It Works
The sign of a product depends on counting how many negative factors appear. If you multiply an even number of negative numbers together, the result is positive. If you multiply an odd number of negative numbers together, the result is negative. To compute the product, find the product of the absolute values first, then apply the sign rule. For example, has three negative factors (odd count), so the result is .
Worked Example
Problem: Calculate .
Step 1: Identify the signs of both factors. Both are negative.
Step 2: Apply the sign rule: negative times negative equals positive.
Step 3: Multiply the absolute values.
Step 4: Combine the sign with the product of absolute values.
Answer:
Another Example
Problem: Calculate .
Step 1: Identify the signs. The first factor is negative, the second is positive.
Step 2: Apply the sign rule: negative times positive equals negative.
Step 3: Multiply the absolute values and attach the negative sign.
Answer:
Visualization
Why It Matters
Multiplying negative numbers is a foundational skill used constantly in pre-algebra and algebra courses, especially when solving equations, simplifying expressions, and working with coordinates. In physics, negative values represent direction, so correctly multiplying them determines whether a force or velocity points left or right. Getting the sign rules wrong leads to cascading errors in nearly every calculation that follows.
Common Mistakes
Mistake: Thinking that a negative times a negative gives a negative result.
Correction: Two negatives cancel out when multiplied. A negative times a negative always gives a positive result. Remember: an even count of negative factors yields a positive product.
Mistake: Confusing the rule for multiplication with the rule for addition (e.g., thinking ).
Correction: Adding two negatives gives a more negative number: . Multiplying two negatives gives a positive: . Keep the operations and their sign rules separate.