why is a negative times a negative positive

2 min read 15-04-2025

why is a negative times a negative positive

The rule that a negative number multiplied by a negative number equals a positive number often trips up students. It might seem counterintuitive, but there are several ways to understand why this is true. This explanation will explore the reasoning behind this fundamental mathematical concept.

Understanding the Number Line and Multiplication

Let's start with the basics. Multiplication is essentially repeated addition. For example, 3 x 4 means adding 3 four times (3 + 3 + 3 + 3 = 12). The number line provides a visual representation of numbers, with positive numbers to the right of zero and negative numbers to the left.

Visualizing Negative Multiplication

Consider -3 x 4. This means adding -3 four times: (-3) + (-3) + (-3) + (-3) = -12. This shows that a negative number multiplied by a positive number results in a negative number. The multiplication moves you further to the left on the number line.

Now, let's consider -3 x -4. We can't directly interpret this as repeated addition in the same way. However, we can use the concept of patterns and properties of multiplication to understand the result.

Patterns in Multiplication

Observe the following pattern:

3 x 4 = 12
3 x 3 = 9
3 x 2 = 6
3 x 1 = 3
3 x 0 = 0

Notice that as the second factor decreases by 1, the product decreases by 3. Let's continue this pattern:

3 x -1 = -3
3 x -2 = -6
3 x -3 = -9
3 x -4 = -12

Again, we see a consistent pattern. Now let's look at a similar pattern with -3 as the first factor:

-3 x 4 = -12
-3 x 3 = -9
-3 x 2 = -6
-3 x 1 = -3
-3 x 0 = 0

Following the pattern:

-3 x -1 = 3
-3 x -2 = 6
-3 x -3 = 9
-3 x -4 = 12

See the pattern? As the second factor decreases by 1, the product increases by 3. This consistent pattern demonstrates that a negative times a negative must be positive to maintain the established mathematical relationships.

The Distributive Property

The distributive property further solidifies this concept. The distributive property states that a(b + c) = ab + ac. Let's use this to illustrate:

Let's say we want to solve (-1) * (-5). We can rewrite -5 as (0-5). Using the distributive property:

(-1) * (0 - 5) = (-1) * 0 - (-1) * 5 = 0 - (-5) = 5

This shows that (-1) multiplied by a negative number results in the opposite (positive) of that number.

The Inverse Operation

Multiplication and division are inverse operations. If we divide a negative number by a negative number, we get a positive number. Maintaining consistency between these inverse operations supports the rule that a negative times a negative equals a positive.

Conclusion

While not immediately intuitive, the rule that a negative times a negative equals a positive is consistent with the patterns of multiplication, the distributive property, and the relationship between multiplication and division. Understanding these principles helps solidify this seemingly paradoxical mathematical truth. The consistent patterns and properties of numbers make this rule essential for maintaining the integrity and logical consistency of mathematical operations.