Adding Complex Numbers A Step-by-Step Guide To Summing (-4 + I) And (10 - 5i)

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In the realm of complex numbers, addition is a fundamental operation that combines two complex numbers into a single complex number. This article delves into the process of adding two specific complex numbers: (-4 + i) and (10 - 5i). Complex numbers, which are expressions of the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (√-1), play a crucial role in various fields such as mathematics, physics, and engineering. Understanding how to perform arithmetic operations on complex numbers is essential for solving problems in these domains. The sum of complex numbers involves adding their real and imaginary parts separately. This straightforward process allows us to manipulate complex numbers with ease, making them a powerful tool for mathematical analysis and problem-solving. Let's embark on a step-by-step journey to compute the sum of the given complex numbers and explore the underlying principles of complex number addition.

Understanding Complex Numbers

Before diving into the addition process, it's crucial to grasp the essence of complex numbers. A complex number is composed of two parts: a real part and an imaginary part. The real part is a standard real number, while the imaginary part is a real number multiplied by the imaginary unit 'i'. The imaginary unit 'i' is defined as the square root of -1, a concept that extends the number system beyond the real numbers. Complex numbers are typically expressed in the form a + bi, where 'a' represents the real part and 'b' represents the imaginary part. For instance, in the complex number 3 + 4i, 3 is the real part, and 4 is the imaginary part. Understanding this structure is key to performing operations on complex numbers. When adding complex numbers, we treat the real and imaginary parts as separate components, combining like terms. This approach ensures that the result is also a complex number in the standard form a + bi. The concept of complex numbers provides a framework for solving equations and modeling phenomena that cannot be described using real numbers alone. Their applications span across various scientific and engineering disciplines, highlighting their importance in mathematical problem-solving.

Step-by-Step Addition

To add the complex numbers (-4 + i) and (10 - 5i), we follow a simple procedure: combine the real parts and combine the imaginary parts. The real parts are -4 and 10, and their sum is -4 + 10 = 6. The imaginary parts are i (which is 1i) and -5i, and their sum is 1i - 5i = -4i. Therefore, the sum of the two complex numbers is 6 - 4i. This process mirrors the addition of algebraic expressions, where like terms are grouped together. The result, 6 - 4i, is another complex number, adhering to the closure property of complex number addition. This property ensures that the sum of any two complex numbers is also a complex number. The step-by-step approach simplifies the addition process, making it easy to perform complex number arithmetic. By breaking down the complex numbers into their real and imaginary components, we can apply the rules of real number addition to each part separately. This method provides a clear and concise way to compute the sum of complex numbers, which is fundamental in various mathematical and scientific contexts.

Detailed Calculation

Let's break down the calculation into more detail for clarity. We start with the expression (-4 + i) + (10 - 5i). To add these complex numbers, we group the real parts together and the imaginary parts together. This gives us (-4 + 10) + (i - 5i). Now, we perform the addition for each group separately. For the real parts, -4 + 10 equals 6. For the imaginary parts, i - 5i is the same as 1i - 5i, which equals -4i. Combining these results, we get 6 - 4i. This detailed calculation illustrates the straightforward nature of complex number addition. By treating the real and imaginary parts as separate terms, we can apply the familiar rules of arithmetic to each part. The result, 6 - 4i, represents a new complex number that is the sum of the original two complex numbers. This methodical approach ensures accuracy and clarity in complex number operations. Understanding this process is crucial for tackling more complex problems involving complex numbers in various fields of study.

Identifying the Correct Option

Now that we've calculated the sum of (-4 + i) and (10 - 5i) to be 6 - 4i, we can identify the correct option from the given choices. The options are:

A. -3 + 5i B. -3 - 5i C. 6 - 6i D. 6 - 4i

Comparing our result, 6 - 4i, with the options, we see that option D matches our calculated sum. Therefore, the correct answer is D. 6 - 4i. This step highlights the importance of careful calculation and comparison when working with complex numbers. The ability to accurately perform complex number arithmetic is essential for solving problems in mathematics, physics, and engineering. By systematically adding the real and imaginary parts, we arrived at the correct answer and confidently identified the matching option. This process reinforces the fundamental principles of complex number addition and its application in problem-solving.

In conclusion, the sum of the complex numbers (-4 + i) and (10 - 5i) is 6 - 4i. We arrived at this result by adding the real parts (-4 and 10) to get 6 and adding the imaginary parts (i and -5i) to get -4i. This process demonstrates the fundamental principle of complex number addition, which involves treating the real and imaginary parts separately. The correct option, D. 6 - 4i, confirms our calculation. Understanding how to perform arithmetic operations on complex numbers is crucial in various fields, including mathematics, physics, and engineering. Complex numbers provide a powerful tool for solving problems that cannot be addressed using real numbers alone. By mastering the basics of complex number addition, individuals can confidently tackle more complex problems involving these fascinating mathematical entities. The step-by-step approach outlined in this article provides a clear and concise method for adding complex numbers, ensuring accuracy and clarity in calculations. This skill is essential for anyone working with complex numbers in any capacity.