Subtracting Complex Numbers Exploring Z1 - Z2

In this article, we delve into the realm of complex number subtraction, focusing on a specific problem involving two complex numbers, z1 and z2. Complex numbers, a fundamental concept in mathematics, extend the real number system by incorporating an imaginary unit, denoted as i, where i² = -1. Understanding operations with complex numbers, such as subtraction, is crucial in various fields, including electrical engineering, quantum mechanics, and signal processing. This exploration aims to provide a comprehensive understanding of how to subtract complex numbers and to solve the given problem step by step.

Setting the Stage: Defining z1 and z2

Before diving into the subtraction process, let's clearly define the complex numbers we'll be working with. We are given z1 = 4 - 4√3i and z2 = -16i. Each complex number consists of two parts: a real part and an imaginary part. In z1, the real part is 4, and the imaginary part is -4√3. Similarly, in z2, the real part is 0 (since it's not explicitly written), and the imaginary part is -16. To effectively subtract these complex numbers, it's essential to treat the real and imaginary parts separately. This approach aligns with the fundamental rules of complex number arithmetic, ensuring accurate results. Understanding the composition of complex numbers is the first step towards mastering their manipulation.

The Subtraction Process: A Step-by-Step Guide

The core concept behind subtracting complex numbers lies in treating the real and imaginary parts as separate components. To find z1 - z2, we subtract the real part of z2 from the real part of z1, and similarly, we subtract the imaginary part of z2 from the imaginary part of z1. This process can be represented mathematically as follows:

(z1 - z2) = (Real(z1) - Real(z2)) + (Imaginary(z1) - Imaginary(z2))i

Applying this to our given complex numbers, we have:

• Real part: 4 - 0 = 4
• Imaginary part: -4√3 - (-16) = -4√3 + 16

Combining these, we get z1 - z2 = 4 + (-4√3 + 16)i. This step is crucial as it breaks down the subtraction into manageable parts, minimizing the chances of error. By keeping the real and imaginary components distinct, we ensure the final result accurately reflects the difference between the two complex numbers. This method is universally applicable to any complex number subtraction, making it a fundamental skill in complex number arithmetic.

Putting It All Together: Calculating z1 - z2

Now, let's substitute the values of z1 and z2 into our subtraction formula and simplify the expression. We have z1 = 4 - 4√3i and z2 = -16i. Therefore, z1 - z2 = (4 - 4√3i) - (-16i). The next step involves distributing the negative sign in front of the parentheses, which changes the sign of the terms inside. This gives us z1 - z2 = 4 - 4√3i + 16i. To proceed further, we group the real and imaginary parts together. The real part remains 4, and we combine the imaginary parts: -4√3i + 16i. This combination can be written as (16 - 4√3)i. Thus, the final result is z1 - z2 = 4 + (16 - 4√3)i. This meticulous step-by-step approach ensures accuracy and clarity in the solution.

Final Result: Expressing the Solution in Standard Form

After performing the subtraction and simplifying the expression, we arrive at the final result: z1 - z2 = 4 + (16 - 4√3)i. To express this in a more standard form, we can factor out a 4 from the imaginary part: z1 - z2 = 4 + 4(4 - √3)i. This form is often preferred as it clearly separates the real and imaginary components and can make further calculations or comparisons easier. Now, let's compare our result with the given options to identify the correct answer. We have z1 - z2 = 4 + 4(4 - √3)i, which can also be written as 4 + 4(√3 - 4)(-1)i or 4 + 4(√3 - 4)(-i) or 4 + 4(√3 - 4)i. Comparing this with the provided options, we can identify the correct choice. This final step solidifies our understanding of the solution and its representation.

Identifying the Correct Option: Matching the Result

Having calculated z1 - z2 = 4 + (16 - 4√3)i, our next step is to match this result with the provided options. The options are:

• A. 4 - 12√3i
• B. 4 + 12√3i
• C. 4 - 4(√3 - 4)i
• D. 4 + 4(√3 - 4)i

Comparing our calculated result with these options, we can see that option D, which is 4 + 4(4 - √3)i or 4 + 4(√3 - 4)i, matches our result. To ensure clarity, let's rewrite our result as 4 + (16 - 4√3)i = 4 + 4(4 - √3)i. This clearly demonstrates the equivalence between our solution and option D. Therefore, option D is the correct answer. This step-by-step comparison ensures that we accurately identify the correct solution from the given choices.

Common Mistakes to Avoid in Complex Number Subtraction

When dealing with complex number subtraction, several common mistakes can occur if one isn't careful. One frequent error is incorrectly distributing the negative sign when subtracting complex numbers. For instance, in our problem, subtracting z2 means subtracting both its real and imaginary parts. Failing to distribute the negative sign properly can lead to an incorrect imaginary part in the result. Another common mistake is combining the real and imaginary parts incorrectly. Remember, real and imaginary terms are distinct and should not be added together directly. They should remain separate components of the complex number. Additionally, errors can arise from incorrect simplification of the imaginary part, especially when dealing with radicals like √3. Always double-check your arithmetic, particularly when simplifying expressions involving square roots. By being aware of these potential pitfalls, you can significantly improve your accuracy when subtracting complex numbers.

Real-World Applications of Complex Number Subtraction

Complex number subtraction, while seemingly abstract, has significant real-world applications across various fields. In electrical engineering, complex numbers are used to represent alternating current (AC) circuits, where the real part corresponds to resistance and the imaginary part corresponds to reactance. Subtracting complex impedances, which are represented as complex numbers, allows engineers to analyze circuit behavior and optimize designs. In physics, particularly in quantum mechanics, complex numbers are essential for describing wave functions. Subtracting complex wave functions can help determine interference patterns and probabilities of quantum events. Signal processing, another field reliant on complex numbers, uses subtraction to filter out noise and extract relevant information from signals. These examples illustrate that complex number subtraction is not just a mathematical exercise but a crucial tool in solving practical problems in science and engineering. Understanding these applications can provide a deeper appreciation for the importance of mastering complex number arithmetic.

Conclusion: Mastering Complex Number Subtraction

In conclusion, subtracting complex numbers involves treating the real and imaginary parts separately, ensuring careful distribution of negative signs, and accurately simplifying the resulting expression. Through a step-by-step approach, we successfully calculated z1 - z2 = 4 + 4(4 - √3)i and identified the correct option. Avoiding common mistakes, such as incorrect distribution and improper simplification, is crucial for accuracy. Furthermore, understanding the real-world applications of complex number subtraction in fields like electrical engineering, physics, and signal processing underscores its significance. By mastering this fundamental operation, you'll be well-equipped to tackle more advanced concepts in mathematics and related disciplines. This exploration provides a solid foundation for further studies in complex analysis and its applications.