Why Is Negative Times Negative Positive

Why is a Negative Times a Negative Positive? Unraveling the Mystery of Multiplication

Many of us learned the rule "negative times negative equals positive" in school, but few truly understand why this is the case. It's not just an arbitrary rule; it's a logical consequence of how we define multiplication and negative numbers. This article will delve into the reasoning behind this fundamental mathematical principle, exploring various approaches to illuminate its truth.

Understanding the Basics: Multiplication as Repeated Addition

Before tackling negative numbers, let's solidify our understanding of multiplication with positive integers. Multiplication is essentially repeated addition. For example, 3 x 4 means adding 3 to itself four times: 3 + 3 + 3 + 3 = 12.

Introducing the Negative: The Concept of Opposites

Negative numbers represent the opposite of their positive counterparts. Think of a number line; -3 is the opposite of 3, located equidistant from zero but in the opposite direction. This "oppositeness" is key to understanding the rules of multiplication involving negative numbers.

Exploring Negative Times Positive

Let's start with a simpler case: a negative number multiplied by a positive number. Consider -3 x 4. Using the repeated addition concept, this would mean adding -3 to itself four times: (-3) + (-3) + (-3) + (-3) = -12. This demonstrates that a negative number multiplied by a positive number results in a negative number.

The Crucial Step: Negative Times Negative

Now, the big question: why is -3 x -4 = 12? Several approaches help explain this:

1. Maintaining Consistency: The Distributive Property

The distributive property states that a(b + c) = ab + ac. Let's apply this to the expression -3 x (4 + -4). We know that 4 + (-4) = 0. Therefore:

-3 x (4 + -4) = -3 x 0 = 0

Now, using the distributive property:

-3 x (4 + -4) = (-3 x 4) + (-3 x -4) = -12 + (-3 x -4)

Since the entire expression equals 0, we must have:

-12 + (-3 x -4) = 0

This means -3 x -4 must equal 12 to make the equation true. This approach highlights that the rule of negative times negative being positive is necessary to maintain consistency within the established rules of mathematics.

2. Patterns and Number Lines: Visualizing the Concept

Observe the pattern when multiplying by -1:

• 4 x -1 = -4
• 3 x -1 = -3
• 2 x -1 = -2
• 1 x -1 = -1
• 0 x -1 = 0

Following this pattern, we should see that:

• -1 x -1 = 1

This visually demonstrates the progression towards positive values when multiplying negative numbers.

3. The Concept of 'Undoing': Inverse Operations

Multiplication and division are inverse operations. If we divide a number by itself, we get 1 (excluding zero). This concept extends to negative numbers.

Consider the expression (-3) / (-3). We know this equals 1. Since multiplication and division are inverse operations, if we multiply -3 by something to get 1, that something must be positive, reinforcing the idea that -3 multiplied by a positive number (in this case, -3 * -1/3 = 1) results in a positive number. This line of reasoning provides a unique perspective on why the rule holds.

Conclusion: A Consistent Mathematical System

The rule that a negative times a negative is positive isn't arbitrary; it's a fundamental consequence of maintaining consistency within the mathematical system we've built. It ensures that the distributive property, the concept of opposites, and the relationship between multiplication and division all work harmoniously, creating a robust and logical framework for mathematical operations. Understanding these underlying principles provides a deeper appreciation for this seemingly simple, yet crucial, mathematical rule.