Negative Times A Negative Equals A

Negative Times a Negative Equals a Positive: Understanding the Math Behind It

Why does a negative number multiplied by a negative number equal a positive number? This seemingly counterintuitive rule of mathematics often stumps students. This article will explore the reasoning behind this fundamental concept, providing a clear and intuitive explanation, along with practical examples. Understanding this principle is crucial for mastering algebra and beyond.

Understanding the Concept through Patterns:

One of the simplest ways to grasp this concept is by observing patterns in multiplication. Let's consider a series of multiplications involving decreasing negative numbers:

• 3 x (-3) = -9
• 2 x (-3) = -6
• 1 x (-3) = -3
• 0 x (-3) = 0
• -1 x (-3) = ?

Notice that as the first number decreases by one, the product increases by three. Following this pattern, -1 x (-3) must equal 3. This pattern visually demonstrates that multiplying a negative by a negative results in a positive.

The Number Line Approach:

Another useful approach is visualizing multiplication on a number line. Multiplication can be seen as repeated addition. For example, 3 x 2 means adding 2 three times (2 + 2 + 2 = 6).

Now, let's consider -3 x 2. This means adding -2 three times (-2 + -2 + -2 = -6).

What about -3 x -2? This can be interpreted as subtracting -2 three times. Subtracting a negative is the same as adding a positive. Therefore, -3 x -2 = 6.

Algebraic Explanation:

A more formal explanation utilizes the distributive property of multiplication. Let's consider the following expression:

(-a) * (-b)

We can rewrite -a as 0 - a. Substituting this into our expression:

(0 - a) * (-b)

Applying the distributive property:

0 * (-b) - a * (-b)

Since 0 multiplied by any number is 0, the expression simplifies to:

-a * (-b)

Now, we know that a negative multiplied by a positive is a negative. So:

• a * (-b) = ab

Therefore, (-a) * (-b) = ab, demonstrating that the product of two negative numbers is always positive.

Real-World Applications:

Understanding negative times a negative applies to various real-world scenarios. For instance, consider the concept of debt. If you have a debt (-$100) and you eliminate (-1) that debt, the net effect is an increase of $100 (+$100).

Conclusion:

The rule that a negative multiplied by a negative equals a positive might seem unconventional at first, but it's a consistent and logical consequence of the established rules of arithmetic. Understanding this concept through patterns, the number line, or algebraic proof provides a solid foundation for more advanced mathematical concepts. By grasping these explanations, you can conquer this mathematical hurdle and confidently navigate more complex calculations.