Solving U + V Given U = 1 + I And V = 1 - I A Step By Step Guide

In the realm of mathematics, particularly when dealing with complex numbers, we often encounter scenarios where we need to perform basic arithmetic operations. This article delves into a problem involving the addition of two complex numbers. Our main objective is to determine the sum of u and v, given that u is defined as 1 + i and v is defined as 1 - i, where i represents the imaginary unit (√-1). This exploration will not only provide the solution but also offer a clear, step-by-step explanation, enhancing your understanding of complex number arithmetic. Understanding the fundamental operations with complex numbers is crucial for various applications in fields such as electrical engineering, quantum mechanics, and signal processing. This article aims to demystify the process and empower you to tackle similar problems with confidence. Let's embark on this mathematical journey together, unraveling the intricacies of complex number addition.

Breaking Down Complex Numbers: Understanding the Components

Before we dive into the solution, it's crucial to understand the fundamentals of complex numbers. A complex number is generally expressed in the form a + bi, where a represents the real part and bi represents the imaginary part. The imaginary unit, denoted by i, is defined as the square root of -1. In our problem, u = 1 + i and v = 1 - i. Here, for u, the real part is 1, and the imaginary part is 1i (or simply i). For v, the real part is also 1, but the imaginary part is -1i (or -i). Visualizing complex numbers on the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part, can provide a geometric understanding of these numbers. This foundation is essential for performing operations like addition, subtraction, multiplication, and division with complex numbers. The ability to decompose complex numbers into their real and imaginary components is key to simplifying and solving mathematical expressions involving them. This concept is not only crucial for this specific problem but also for a broader understanding of complex number theory and its applications in various scientific and engineering disciplines. Mastering these fundamentals will undoubtedly enhance your problem-solving skills in mathematics and related fields.

The Mechanics of Complex Number Addition: A Step-by-Step Guide

Now, let's move on to the core of the problem: adding the two complex numbers, u and v. The addition of complex numbers follows a straightforward rule: we add the real parts together and the imaginary parts together separately. This is akin to combining like terms in algebraic expressions. In our case, we have u = 1 + i and v = 1 - i. To find u + v, we add the real parts (1 + 1) and the imaginary parts (i + (-i)). This can be written as (1 + 1) + (i - i). Performing the addition, we get 2 + 0i. Since 0i is equal to 0, the final result is simply 2. This demonstrates the simplicity of complex number addition when approached systematically. By separating the real and imaginary components, we can perform the operation with ease. This method ensures accuracy and clarity in our calculations. Understanding this process is essential for tackling more complex problems involving complex numbers. It lays the groundwork for understanding other operations such as subtraction, multiplication, and division, which build upon this fundamental principle. The ability to confidently add complex numbers is a cornerstone of complex number arithmetic.

Solving the Problem: u + v = ?

With the understanding of complex number addition firmly in place, let's now explicitly solve for u + v. We are given that u = 1 + i and v = 1 - i. Following the principle of complex number addition, we add the real parts and the imaginary parts separately: u + v = (1 + i) + (1 - i) = (1 + 1) + (i - i). Performing the arithmetic, we get 2 + 0i. As 0i is simply 0, the result simplifies to 2. Therefore, u + v = 2. This result is a real number, which is a special case in complex number operations. When the imaginary parts cancel each other out during addition, the result is a real number. This highlights the relationship between real and complex numbers, where real numbers are a subset of complex numbers. The solution demonstrates the power of applying fundamental principles to solve mathematical problems. By systematically breaking down the problem into manageable steps, we arrived at the correct answer with clarity and confidence. This approach is applicable to a wide range of mathematical problems, emphasizing the importance of a strong foundation in basic concepts.

Analyzing the Answer Choices: Identifying the Correct Solution

Now that we have determined that u + v = 2, let's examine the provided answer choices to identify the correct option. The choices are:

A. 2 B. 0 C. -2i D. -2 E. 2i

By comparing our solution with the given options, it's clear that option A, 2, is the correct answer. The other options represent different possibilities, such as a zero result, a purely imaginary number (-2i or 2i), or a negative real number (-2). These options highlight common misconceptions or errors that might arise if the addition is performed incorrectly. For instance, option B (0) might result from incorrectly canceling both the real and imaginary parts. Options C and E (-2i and 2i) suggest a misunderstanding of how real and imaginary parts combine during addition. Option D (-2) could be a result of incorrectly handling the signs. By systematically working through the problem and understanding the principles of complex number addition, we were able to confidently identify the correct answer and avoid these potential pitfalls. This process of analyzing answer choices is a valuable skill in mathematics, as it reinforces understanding and helps in eliminating incorrect options.

Conclusion: Solidifying Understanding of Complex Number Addition

In conclusion, we have successfully solved the problem of finding u + v when u = 1 + i and v = 1 - i. By understanding the fundamentals of complex numbers and applying the principles of complex number addition, we determined that u + v = 2. This exercise underscores the importance of a solid foundation in basic mathematical concepts. The ability to break down complex problems into simpler steps is crucial for success in mathematics and related fields. Complex number addition, as demonstrated in this article, is a fundamental operation that serves as a building block for more advanced concepts. By mastering these basics, you can confidently tackle more challenging problems involving complex numbers. This article has not only provided a solution to the specific problem but also aimed to enhance your understanding of the underlying principles. The step-by-step approach, coupled with explanations of key concepts, empowers you to apply this knowledge to a broader range of mathematical problems. Remember, practice and consistent application of these principles are key to building proficiency in mathematics. We encourage you to explore further examples and problems to solidify your understanding of complex number arithmetic and its applications.