What's A Negative Times A Negative

The seemingly simple question of "what's a negative times a negative?" often sparks curiosity and sometimes confusion. The answer, a positive, is a cornerstone of arithmetic and algebra, influencing everything from solving equations to understanding complex numbers. This exploration will delve into the "why" behind this rule, providing various explanations and real-world examples to solidify your understanding.

Unpacking the Concept: Multiplying Negatives

Multiplication, at its core, is repeated addition. For example, 3 x 4 means adding 4 to itself three times (4 + 4 + 4 = 12). When we introduce negative numbers, this basic understanding needs a slight adjustment. A negative number can be thought of as the opposite of a positive number. Therefore, multiplying by a negative number involves a concept of "repeated subtraction" or, more precisely, repeated addition of the opposite.

Visualizing Multiplication with Number Lines

A number line provides a powerful visual aid to grasp the concept of multiplying negative numbers.

• Positive x Positive: 3 x 2 means taking three steps of size 2 in the positive direction from zero, landing at +6.

• Positive x Negative: 3 x (-2) means taking three steps of size 2 in the negative direction from zero, landing at -6.

• Negative x Positive: -3 x 2 can be interpreted as the opposite of 3 x 2. We know 3 x 2 is 6, so -3 x 2 is -6. Think of it as "taking away" three sets of two.

• Negative x Negative: -3 x (-2) This is where it gets interesting. It means the opposite of taking away three sets of -2. Taking away three sets of -2 would move us from 0 to +6. Since we're doing the opposite of that action, we end up at +6.

The "Opposite Of" Interpretation

Another way to think about a negative sign is as "the opposite of." So, -(-2) means "the opposite of -2," which is +2. When multiplying -3 x (-2), you can read it as "-3 times the opposite of 2." The opposite of 2 is -2, and -3 times -2, understood as the opposite of taking away 3 sets of -2 results in positive 6.

Why Does a Negative Times a Negative Equal a Positive? Different Perspectives

Several explanations can clarify why a negative times a negative equals a positive.

1. The Pattern Approach: Maintaining Consistency

Consider the following pattern:

3 x (-2) = -6 2 x (-2) = -4 1 x (-2) = -2 0 x (-2) = 0 -1 x (-2) = ? -2 x (-2) = ? -3 x (-2) = ?

Notice that as the first number decreases by 1, the result increases by 2. To maintain this pattern, the results of the last three equations must be:

-1 x (-2) = 2 -2 x (-2) = 4 -3 x (-2) = 6

This pattern demonstrates that a negative times a negative must be a positive to maintain consistency within the multiplication rules.

2. The Distributive Property of Multiplication

The distributive property states that a(b + c) = ab + ac. We can use this property to prove that -1 x -1 = 1.

We know that -1 + 1 = 0. Let's multiply both sides of this equation by -1:

-1(-1 + 1) = -1(0)

Using the distributive property:

(-1 x -1) + (-1 x 1) = 0

We know that -1 x 1 = -1. Therefore:

(-1 x -1) + (-1) = 0

To make this equation true, (-1 x -1) must equal 1:

1 + (-1) = 0

Therefore, -1 x -1 = 1. This can be extended to any negative numbers.

3. The Analogy of Debts and Removal of Debts

Think of positive numbers as money you have and negative numbers as debts you owe. Multiplication can then be thought of as "groups of."

• 3 x 2: Three groups of $2 (positive) means you have $6.
• 3 x (-2): Three groups of $2 debt means you owe $6 (negative).
• -3 x 2: This is a bit trickier, but can be interpreted as "taking away" three groups of $2. If someone takes away three groups of $2 from you, you are $6 poorer (negative).
• -3 x (-2): This is the most insightful. It means "taking away" three groups of a $2 debt. If someone removes three $2 debts from you, you are $6 richer (positive). The removal of debt is a gain.

This analogy provides a practical, real-world connection to the concept of multiplying negatives.

4. Mathematical Proof using Axioms

While less intuitive, a rigorous mathematical proof can also demonstrate why a negative times a negative equals a positive. This proof relies on fundamental axioms of arithmetic. Here's a simplified version:

• Axiom 1: For any number 'a', a + 0 = a (Identity Property of Addition)
• Axiom 2: For any number 'a', there exists a number '-a' such that a + (-a) = 0 (Additive Inverse Property)
• Axiom 3: a * 0 = 0 (Multiplication by Zero)
• Axiom 4: a * (b + c) = ab + ac (Distributive Property)

The Goal: Prove that (-a) * (-b) = a * b

1. Start with 0 = a + (-a) (Additive Inverse Property)
2. Multiply both sides by -b: -b * 0 = -b * (a + (-a))
3. Simplify: 0 = -b * (a + (-a)) (Multiplication by Zero)
4. Apply the Distributive Property: 0 = (-b * a) + (-b * -a)
5. Rewrite -b * a as -(ba): 0 = -(ba) + (-b * -a)
6. Add (ba) to both sides: ba = (-b * -a)
7. Therefore: (-b * -a) = b * a (This proves that a negative times a negative equals a positive)

This formal proof provides the most rigorous justification for the rule.

Real-World Applications of Multiplying Negatives

The concept of multiplying negatives isn't just abstract mathematics; it has practical applications in various fields:

• Physics: In physics, negative numbers often represent direction. For example, velocity in one direction can be positive, while velocity in the opposite direction is negative. Calculating changes in velocity can involve multiplying negative numbers.

  • Example: If an object slows down (negative acceleration) while moving in a negative direction, multiplying these two negative values results in a positive value, indicating a positive change in kinetic energy (an increase in kinetic energy even though the object is slowing down in the negative direction).

• Finance: Negative numbers represent debt or losses. Understanding how negative numbers interact is crucial for managing finances.

  • Example: Consider an investment that loses money consistently for a period, and then an intervention turns things around. Losing $100 per month for three months is -3 * 100 = -$300. Actively reducing the loss each month (a negative change in a negative value) by, say, $20 (represented as -1 * -20) effectively increases profits.

• Computer Programming: Negative numbers are used in various programming contexts, such as representing offsets, temperatures below zero, or changes in quantities.

  • Example: In game development, consider character movement. Moving -5 units/second for -2 seconds (meaning, subtracting/undoing the movement that happened over the last 2 seconds) results in a positive change in position. It means that if the character had been moving in reverse, "undoing" that reverses the direction.

• Temperature: Temperature scales often use negative numbers to represent temperatures below zero. Calculating temperature changes can involve multiplying negative numbers.

  • Example: If the temperature is decreasing at a rate of -2 degrees per hour, the change in temperature after -3 hours (thinking backward in time) would be (-2) * (-3) = 6 degrees. This means that 3 hours ago, the temperature was 6 degrees higher.

• Coordinate Geometry: Working with coordinates on a graph often involves negative numbers. Calculations involving distances, slopes, and areas can require multiplying negative numbers.

  • Example: If finding the area of a geometric shape and the formula produces a negative result, multiplying by -1 (effectively -1 * negative area) gives the positive magnitude of the area.

Common Misconceptions and How to Avoid Them

• Confusing Multiplication with Addition: A common mistake is to apply the rules of addition to multiplication. Remember, a negative plus a negative is always negative (e.g., -2 + -3 = -5), but a negative times a negative is positive (e.g., -2 x -3 = 6).

• Forgetting the Order of Operations: Always follow the order of operations (PEMDAS/BODMAS). Multiplication should be performed before addition or subtraction.

• Overgeneralizing the Rule: The rule applies specifically to multiplication (and division). It doesn't apply directly to other operations.

To avoid these misconceptions:

• Practice consistently with different examples.
• Use visual aids like number lines to reinforce the concept.
• Relate the concept to real-world situations to make it more concrete.
• Clearly distinguish between the rules for addition and multiplication of negative numbers.

FAQ: Frequently Asked Questions

• Why is it important to understand this rule? Understanding why a negative times a negative equals a positive is fundamental to algebra and higher-level mathematics. It is also used in physics, computer science, and other STEM fields. It provides the basis for solving equations, manipulating expressions, and understanding abstract concepts.

• Does the same rule apply to division? Yes, a negative divided by a negative is also positive. The rules for multiplication and division regarding signs are the same. A negative divided by a positive, or a positive divided by a negative, results in a negative quotient.

• What about more than two negative numbers multiplied together? The sign of the result depends on the number of negative factors. If there are an even number of negative factors, the result is positive. If there are an odd number of negative factors, the result is negative.

  • Example: -1 x -1 x -1 = -1 (odd number of negative factors)
  • Example: -1 x -1 x -1 x -1 = 1 (even number of negative factors)

• How can I explain this to a child? Using the debt analogy (as described above) is often the most effective way to explain this to children. Relate it to removing debts and becoming richer. The number line visual also helps.

• Are there any exceptions to this rule? No, the rule that a negative times a negative is a positive is a fundamental rule of arithmetic and algebra, and there are no exceptions to it within the standard number system.

Conclusion: Mastering the Negatives

The seemingly simple rule that a negative times a negative equals a positive is a cornerstone of mathematics. By understanding the underlying principles, visualizing the concept, and connecting it to real-world applications, you can solidify your grasp of this essential rule. Don't just memorize the rule; understand why it works. This will empower you to tackle more complex mathematical concepts with confidence. The journey to mastering mathematics is paved with understanding fundamental concepts like this one. Continue exploring, questioning, and practicing, and you will find yourself navigating the world of numbers with greater ease and understanding.