Multiplying Negative Numbers: A Quick Guide

Hey guys! Let's dive into a super important concept in math: multiplying negative numbers. It might seem a bit tricky at first, but I promise, once you get the hang of it, it's actually pretty straightforward. We're going to break down the problem $(-6) * (-2)$ step by step, so you’ll be a pro in no time!

Understanding the Basics of Negative Numbers

Before we jump into the multiplication itself, let's quickly recap what negative numbers are all about. Think of the number line. Zero is right in the middle, positive numbers stretch out to the right, and negative numbers go off to the left. A negative number is simply a number less than zero. It represents the opposite of a positive number. For example, if +5 represents having five dollars, then -5 represents owing five dollars.

Negative numbers are everywhere in real life! Temperature is a classic example. If the temperature is -10°C, that means it's ten degrees below freezing. Another example is in finances. If your bank account has a balance of -$50, that means you're $50 overdrawn. Understanding these real-world applications can really help make negative numbers click.

Now, when we start multiplying these negative numbers, that's where things can get interesting. The key rule to remember is this: a negative times a negative equals a positive. Keep that locked in your memory, and you're already halfway there!

Step-by-Step Multiplication of (-6) * (-2)

Okay, let's tackle our problem: $(-6) * (-2)$. Here’s how we're going to break it down:

1. Ignore the Signs (For Now): First, let's just focus on the absolute values of the numbers. That means we're looking at 6 and 2. Forget about the minus signs for a second.
2. Multiply the Absolute Values: Multiply these numbers together: $6 * 2 = 12$. Simple enough, right?
3. Apply the Rule: Now, here's the crucial part. Remember our rule? A negative times a negative equals a positive. Since we're multiplying -6 and -2, we know the answer will be positive.
4. Write the Answer: So, the final answer is +12, or simply 12. That's it!

So, $(-6) * (-2) = 12$. We took two negative numbers, multiplied their absolute values, and then applied the rule that a negative times a negative gives us a positive result. Easy peasy!

The Rule: Negative Times Negative Equals Positive

Let's drill down on this rule a bit more because it's super important. Why does a negative times a negative equal a positive? There are a few ways to think about it.

One way is to think of multiplication as repeated addition. For example, $3 * 2$ means adding 2 to itself three times: $2 + 2 + 2 = 6$. Now, what does $3 * (-2)$ mean? It means adding -2 to itself three times: $(-2) + (-2) + (-2) = -6$. So, multiplying a positive by a negative results in a negative.

But what about a negative times a negative? Think of it as the opposite of multiplying by a negative. So, $(-3) * (-2)$ means the opposite of adding -2 to itself three times. The opposite of $(-2) + (-2) + (-2) = -6$ is +6. That's why a negative times a negative is a positive.

Another way to think about it is through patterns. Consider this:

• $3 * (-2) = -6$
• $2 * (-2) = -4$
• $1 * (-2) = -2$
• $0 * (-2) = 0$

Notice that as the first number decreases by 1, the result increases by 2. If we continue the pattern:

• $-1 * (-2) = 2$
• $-2 * (-2) = 4$
• $-3 * (-2) = 6$

The pattern clearly shows that a negative times a negative results in a positive. Understanding these patterns can make the rule much more intuitive.

Examples of Multiplying Negative Numbers

To really nail this down, let's look at a few more examples:

• Example 1: $(-4) * (-5)$
  • Multiply the absolute values: $4 * 5 = 20$
  • Apply the rule: Negative times negative equals positive.
  • Answer: 20
• Example 2: $(-10) * (-3)$
  • Multiply the absolute values: $10 * 3 = 30$
  • Apply the rule: Negative times negative equals positive.
  • Answer: 30
• Example 3: $(-1) * (-1)$
  • Multiply the absolute values: $1 * 1 = 1$
  • Apply the rule: Negative times negative equals positive.
  • Answer: 1

As you can see, the process is the same every time. Multiply the absolute values, and then remember that a negative times a negative is always positive. Keep practicing, and it'll become second nature!

Common Mistakes to Avoid

Even though the rule is simple, it's easy to make mistakes, especially when you're just starting out. Here are a few common pitfalls to watch out for:

• Forgetting the Rule: The most common mistake is simply forgetting that a negative times a negative equals a positive. Always double-check your signs!
• Confusing with Addition/Subtraction: Sometimes, people get this mixed up with adding or subtracting negative numbers. Remember, when you add a negative number, it's like subtracting a positive number (e.g., $5 + (-3) = 5 - 3 = 2$). Multiplication is different!
• Incorrectly Applying the Rule with More Than Two Numbers: If you're multiplying more than two numbers, remember to apply the rule step by step. For example, $(-2) * (-3) * (-4)$. First, $(-2) * (-3) = 6$. Then, $6 * (-4) = -24$. Notice that an even number of negative signs results in a positive answer, while an odd number of negative signs results in a negative answer.

Practice Problems

Ready to test your skills? Try these practice problems:

1. $(-7) * (-2) = ?$
2. $(-5) * (-8) = ?$
3. $(-9) * (-4) = ?$
4. $(-12) * (-3) = ?$
5. $(-6) * (-6) = ?$

Answers:

1. 14
2. 40
3. 36
4. 36
5. 36

How did you do? If you got them all right, awesome! If not, don't worry. Just go back and review the steps, and try again. Practice makes perfect!

Conclusion

Multiplying negative numbers doesn't have to be scary. Just remember the key rule: a negative times a negative equals a positive. Break down the problem into smaller steps, focus on the absolute values first, and then apply the rule. With a little practice, you'll be multiplying negative numbers like a math whiz! Keep up the great work, and happy calculating!