Is The Complex Number Subtraction Correct? A Detailed Explanation

Hey guys! Let's dive into a math problem today and see if we can figure out if the solution presented is correct. We're looking at complex number subtraction, and it's crucial to understand the rules to avoid making mistakes. Complex numbers, at first glance, might seem a bit intimidating with their real and imaginary parts, but trust me, once you grasp the basics, they're actually quite fun to work with! This article breaks down the process, explains the logic behind each step, and helps you verify the given solution. We'll use a conversational tone, so it feels like we're just chatting about math together. Let's get started!

The Problem: Complex Number Subtraction

The problem we're tackling today involves subtracting one complex number from another. Specifically, we're examining the following calculation:

$(11 + 2i) - (3 - 10i)$

The provided solution breaks down the steps as follows:

$\begin{aligned} (11 & +2 i)-(3-10 i) \\ & =11+2 i-3-10 i \\ & =(11-3)+(2 i-10 i) \\ & =8-8 i \end{aligned}$

Our mission is to meticulously analyze each step and determine if the final answer, $8 - 8i$, is indeed correct. To do this, we need to remember the fundamental principles of complex number arithmetic. So, let's put on our math hats and get to work!

Understanding Complex Numbers

Before we dissect the solution, let's quickly recap what complex numbers are all about. A complex number is essentially a combination of a real number and an imaginary number. It's typically expressed in the form a + bi, where:

• a represents the real part
• b represents the imaginary part
• i is the imaginary unit, defined as the square root of -1 ($i = \sqrt{-1}$)

Think of it like this: the real part is something you can visualize on a regular number line, while the imaginary part introduces a new dimension, dealing with the square root of negative numbers. Now, when we perform operations like addition or subtraction with complex numbers, we treat the real and imaginary parts separately.

The Subtraction Process: A Detailed Look

When subtracting complex numbers, the key is to treat the real and imaginary parts as separate terms. This is similar to how you would combine like terms in algebraic expressions. To subtract one complex number from another (say, (a + bi) - (c + di)), you perform the following operations:

1. Distribute the negative sign: This is crucial! The minus sign in front of the second complex number needs to be distributed to both its real and imaginary parts.

2. Combine the real parts: Subtract the real part of the second complex number from the real part of the first complex number (a - c).

3. Combine the imaginary parts: Subtract the imaginary part of the second complex number from the imaginary part of the first complex number (bi - di).

4. Write the result: The final answer will be in the form (a - c) + (b - d)i.

Analyzing the Given Solution Step-by-Step

Now that we've refreshed our understanding of complex number subtraction, let's dissect the provided solution step-by-step to see if it follows these principles correctly:

Step 1: Distributing the Negative Sign

The original problem is:

$(11 + 2i) - (3 - 10i)$

The first step in the solution shows the distribution of the negative sign:

$= 11 + 2i - 3 + 10i$

• Is this correct? Yes! The negative sign in front of the parenthesis has been correctly distributed. It's extremely important to remember that subtracting a negative is the same as adding a positive, hence -(-10i) becomes +10i. This is a common area for errors, so double-checking this step is a great habit to form. We see that the signs have changed appropriately, and this sets us up for the next step in combining like terms.

Step 2: Grouping Real and Imaginary Parts

The next step groups the real and imaginary terms together:

$= (11 - 3) + (2i + 10i)$

• Is this correct? Absolutely! This step correctly groups the real parts (11 and -3) and the imaginary parts (2i and 10i). This makes the subtraction and addition much easier to visualize and perform. Grouping like terms is a fundamental algebraic technique, and it applies perfectly to complex numbers as well. By reorganizing the terms in this way, we prepare ourselves to do the final arithmetic.

Step 3: Performing the Arithmetic

The final step performs the subtraction and addition:

$= 8 + 12i$

• Is this correct? No! This is where the error lies. While the real part is calculated correctly (11 - 3 = 8), the imaginary part is incorrect. 2i + 10i should equal 12i, not -8i as shown in the original problem. This is a simple arithmetic mistake, but it changes the entire answer. It's a great example of why carefully reviewing each step is crucial, even if the overall approach is correct.

The Correct Solution

Based on our analysis, the provided solution has a small but significant error in the final step. Let's walk through the corrected solution:

1. $(11 + 2i) - (3 - 10i)$
2. $= 11 + 2i - 3 + 10i$ (Distribute the negative sign)
3. $= (11 - 3) + (2i + 10i)$ (Group real and imaginary parts)
4. $= 8 + 12i$ (Perform the arithmetic)

Therefore, the correct answer is $8 + 12i$.

Why the Original Solution Was Incorrect

The original solution incorrectly calculated the imaginary part. The error was in the final step where it stated 2i + 10i = -8i. The correct calculation is 2i + 10i = 12i. This highlights the importance of carefully checking each step in a mathematical problem, even the seemingly simple ones. A small mistake can lead to a completely different answer. Always double-check your arithmetic!

Common Mistakes in Complex Number Subtraction

Complex number arithmetic can be tricky, and there are a few common pitfalls to watch out for. Being aware of these mistakes can help you avoid them in your own calculations:

• Forgetting to Distribute the Negative Sign: This is probably the most frequent error. Remember, the negative sign in front of the parentheses applies to both the real and imaginary parts of the complex number being subtracted. Make sure you change the signs of all terms inside the parentheses.

• Incorrectly Combining Real and Imaginary Parts: You can only combine real parts with real parts and imaginary parts with imaginary parts. Don't mix them up! Treat 'i' like a variable when combining terms.

• Arithmetic Errors: Simple addition or subtraction errors can easily creep in, especially when dealing with negative numbers. Double-check your calculations, particularly in the final steps.

• Misunderstanding the Imaginary Unit: Remember that 'i' is the square root of -1. Don't treat it like a regular variable in all situations (e.g., when multiplying complex numbers, i² = -1).

Conclusion: The Importance of Careful Calculation

In conclusion, while the provided solution demonstrated a good understanding of the complex number subtraction process – correctly distributing the negative sign and grouping like terms – a small arithmetic error in the final step led to an incorrect answer. The correct answer is $8 + 12i$, not $8 - 8i$. This exercise underscores the importance of meticulous calculation and careful review in mathematics. Even a single mistake can alter the outcome significantly.

So, guys, next time you're working with complex numbers (or any math problem for that matter!), remember to take your time, double-check your work, and stay sharp! Happy calculating!