Complex Number Addition Find Z Where W1 + Z = W2

In the realm of complex numbers, we often encounter scenarios where we need to find the missing piece to complete an equation. This article delves into one such problem, focusing on the addition of complex numbers. We will explore how to determine the complex number that, when added to a given complex number w1, results in another complex number w2. This involves understanding the fundamental operations with complex numbers, particularly addition and subtraction. This problem is a foundational concept in complex number theory and is crucial for understanding more advanced topics.

Given two complex numbers, w1 = -2 + 3i and w2 = 4 - 4i, our objective is to find a complex number, let's call it z, such that when z is added to w1, the result is w2. In mathematical terms, we are looking for z that satisfies the equation: w1 + z = w2. This problem tests our understanding of complex number addition and subtraction, as well as our ability to manipulate complex number equations.

Step 1: Setting up the Equation

The first step in solving this problem is to translate the word problem into a mathematical equation. We are given that w1 + z = w2. We know the values of w1 and w2, so we can substitute them into the equation: (-2 + 3i) + z = (4 - 4i). This equation now represents the relationship between the complex numbers and the unknown z that we need to find.

Step 2: Isolating the Unknown Complex Number (z)

To find the value of z, we need to isolate it on one side of the equation. This is done by subtracting w1 from both sides of the equation. Subtracting a complex number is similar to subtracting algebraic expressions, where we subtract the real and imaginary parts separately. So, we have: z = w2 - w1. This step is crucial as it sets up the final calculation for finding z.

Step 3: Performing the Subtraction

Now that we have isolated z, we can substitute the values of w1 and w2 into the equation: z = (4 - 4i) - (-2 + 3i). To perform the subtraction, we subtract the real parts and the imaginary parts separately. This means we subtract the real part of w1 from the real part of w2, and the imaginary part of w1 from the imaginary part of w2. It's important to pay close attention to the signs to avoid errors in the calculation.

Step 4: Calculating the Real Part

The real part of z is obtained by subtracting the real part of w1 from the real part of w2. The real part of w2 is 4, and the real part of w1 is -2. So, the real part of z is 4 - (-2), which simplifies to 4 + 2 = 6. This calculation gives us the real component of the complex number z.

Step 5: Calculating the Imaginary Part

The imaginary part of z is found by subtracting the imaginary part of w1 from the imaginary part of w2. The imaginary part of w2 is -4, and the imaginary part of w1 is 3. Therefore, the imaginary part of z is -4 - 3 = -7. This calculation gives us the imaginary component of the complex number z, which is crucial for determining the correct answer.

Step 6: Expressing z in Complex Number Form

Now that we have the real and imaginary parts of z, we can express z in the standard complex number form, which is a + bi, where a is the real part and b is the imaginary part. In our case, the real part is 6 and the imaginary part is -7. So, z = 6 - 7i. This is the complex number that, when added to w1, results in w2.

Step 7: Verifying the Solution (Optional but Recommended)

To ensure our solution is correct, we can verify it by adding z to w1 and checking if the result is w2. Adding (6 - 7i) to (-2 + 3i) gives us (6 - 2) + (-7 + 3)i, which simplifies to 4 - 4i. This is indeed equal to w2, so our solution is correct. Verification is a good practice to avoid errors and build confidence in the answer.

We have determined that the complex number z that can be added to w1 to produce w2 is 6 - 7i. Now, let's look at the available options and identify the correct one:

A. 6 - i

B. -6 - i

C. 6 - 7i

D. -6 + 7i

Comparing our solution (6 - 7i) with the options, we can see that option C matches our result.

The correct answer is C. 6 - 7i. This complex number, when added to w1 = -2 + 3i, will produce w2 = 4 - 4i. Understanding complex number addition and subtraction is essential for solving problems like this, and this solution demonstrates the step-by-step process to arrive at the correct answer.

When working with complex numbers, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and improve your accuracy in solving complex number problems. Here are some of the most common mistakes:

Sign Errors

One of the most frequent errors occurs when dealing with negative signs. When subtracting complex numbers, it's crucial to distribute the negative sign correctly. For example, when subtracting (-2 + 3i) from (4 - 4i), you need to subtract both the real and imaginary parts. A common mistake is to only subtract the real part or to forget to distribute the negative sign to the imaginary part. Always double-check the signs when performing addition or subtraction of complex numbers.

Incorrectly Combining Real and Imaginary Parts

Complex numbers have a real part and an imaginary part, and these parts should be treated separately. A common mistake is to try to combine the real and imaginary parts as if they were like terms. For instance, you cannot add 6 and -7i to get a single term. The complex number 6 - 7i must remain as it is. Remember, real parts are added/subtracted with real parts, and imaginary parts are added/subtracted with imaginary parts.

Forgetting the Imaginary Unit (i)

The imaginary unit i is a crucial component of complex numbers. Forgetting to include i with the imaginary part is a common mistake. When you calculate the imaginary part, make sure to include i in your final answer. For example, if you calculate the imaginary part to be -7, the correct representation is -7i. Always include the imaginary unit when expressing the imaginary part of a complex number.

Errors in Basic Arithmetic

Sometimes, the mistake is not in the complex number concept itself but in the basic arithmetic calculations. Simple addition or subtraction errors can lead to an incorrect answer. For instance, a mistake in subtracting -2 from 4 (e.g., writing 4 - (-2) = 2 instead of 6) can throw off the entire solution. Double-check your arithmetic calculations, especially when dealing with negative numbers.

Not Verifying the Solution

As mentioned earlier, verifying your solution is a good practice. Not doing so can lead to missing errors. After finding the complex number z, add it to w1 and check if you get w2. This simple step can catch any mistakes you might have made in the process. Always verify your solution to ensure accuracy.

Misunderstanding Complex Number Operations

Complex number operations have specific rules. For example, when adding or subtracting complex numbers, you add or subtract the real and imaginary parts separately. When multiplying complex numbers, you use the distributive property and remember that i^2 = -1. Misunderstanding these rules can lead to errors. Ensure you have a solid understanding of complex number operations.

In conclusion, finding a complex number to add to a given complex number to produce another involves understanding complex number arithmetic, particularly addition and subtraction. By setting up the equation, isolating the unknown, and performing the operations carefully, we can arrive at the correct solution. The key steps include understanding how to manipulate complex number equations, performing the arithmetic correctly, and expressing the result in the standard complex number form. The correct answer to our problem is 6 - 7i, which corresponds to option C. Remember to always verify your solution to ensure accuracy and avoid common mistakes.

Understanding the concepts and techniques discussed in this article is crucial for success in complex number problems. By mastering these fundamentals, you will be well-equipped to tackle more advanced topics in complex number theory and mathematics in general. Practice is key, so work through similar problems to reinforce your understanding and skills. With dedication and attention to detail, you can excel in complex number manipulations and problem-solving.