Solving Complex Numbers 7+(9+27i)-2(25-11i) A Step-by-Step Guide
In the realm of mathematics, complex numbers extend the familiar number line by incorporating an imaginary unit, denoted as 'i', where i² = -1. These numbers, expressed in the form a + bi, where 'a' and 'b' are real numbers, open up a fascinating world of algebraic operations. Understanding complex number operations is fundamental in various fields, including electrical engineering, quantum mechanics, and signal processing. This article delves into the intricacies of adding, subtracting, and multiplying complex numbers, providing a comprehensive guide to mastering these essential skills. We will dissect the given expression, 7 + (9 + 27i) - 2(25 - 11i), meticulously walking through each step to arrive at the correct equivalent complex number.
The journey begins with a clear understanding of the structure of complex numbers. The real part, 'a', represents the number's position on the real number line, while the imaginary part, 'b', indicates its position along the imaginary axis. This two-dimensional representation allows for a richer mathematical landscape, where solutions to equations that are impossible in the real number system become readily apparent. The beauty of complex numbers lies in their ability to model phenomena that involve both magnitude and phase, making them indispensable tools in various scientific and engineering disciplines. As we embark on this exploration, remember that complex number operations follow specific rules, ensuring that the results remain within the complex number system. Our focus will be on simplifying the given expression by applying these rules methodically, leading us to the correct answer among the provided options. The goal is not just to find the solution but to gain a deeper appreciation for the elegance and power of complex number arithmetic.
To effectively tackle the expression 7 + (9 + 27i) - 2(25 - 11i), we'll employ a systematic approach, breaking it down into manageable steps. The first step involves simplifying the expression by distributing the scalar multiplication. This means multiplying the term outside the parenthesis, in this case, -2, with each term inside the parenthesis (25 - 11i). This distribution is crucial as it allows us to remove the parenthesis and combine like terms, a fundamental aspect of simplifying algebraic expressions. Distributing the multiplication correctly ensures that we maintain the integrity of the equation and avoid any errors that might arise from incorrect operations. The result of this distribution will give us a new expression that is easier to work with.
Next, we focus on combining the real and imaginary terms separately. This involves identifying the real parts (the numbers without the 'i') and the imaginary parts (the numbers multiplied by 'i') and then adding or subtracting them accordingly. This process is akin to combining like terms in any algebraic expression, but with the added complexity of dealing with the imaginary unit. By carefully grouping and combining these terms, we gradually simplify the expression, bringing us closer to the final solution. The key here is to treat the real and imaginary parts as distinct entities, ensuring that they are not mixed up during the simplification process. This methodical approach not only leads to the correct answer but also enhances our understanding of how complex numbers behave under algebraic operations. As we progress through these steps, the expression will transform, revealing its underlying structure and ultimately leading us to the equivalent complex number.
Now, let's execute the calculations step-by-step to arrive at the solution. Starting with the original expression, 7 + (9 + 27i) - 2(25 - 11i), we first distribute the -2 across the terms inside the parenthesis: -2 * 25 = -50 and -2 * -11i = +22i. This transforms the expression into 7 + 9 + 27i - 50 + 22i. The next crucial step is to combine the real terms and the imaginary terms separately. The real terms are 7, 9, and -50, which when combined, give us 7 + 9 - 50 = -34.
Similarly, we combine the imaginary terms, which are 27i and 22i. Adding these gives us 27i + 22i = 49i. Now, we have the simplified complex number: -34 + 49i. This result represents the equivalent complex number of the original expression. By meticulously following the order of operations and carefully combining like terms, we have successfully navigated the complexities of complex number arithmetic. This process not only provides the answer but also reinforces the importance of precision and attention to detail in mathematical calculations. The final result, -34 + 49i, aligns with one of the provided options, confirming our solution and demonstrating the power of systematic problem-solving in mathematics.
Having meticulously performed the calculations, we have arrived at the simplified complex number -34 + 49i. Now, it's time to identify the correct option among the choices provided. Comparing our result with the given options: A. -34 + 5i, B. 66 - 49i, C. -34 + 49i, and D. 66 + 5i, it becomes clear that option C, -34 + 49i, perfectly matches our calculated result. This confirms the accuracy of our step-by-step approach and highlights the importance of careful calculation and attention to detail in solving mathematical problems.
The process of elimination further solidifies our choice. Options A, B, and D differ significantly from our result, either in the real part or the imaginary part, or both. This underscores the significance of each step in the calculation process and the potential for errors if any step is overlooked or performed incorrectly. The correct identification of option C not only provides the answer to the question but also serves as a validation of our understanding of complex number operations. This exercise demonstrates the power of mathematical reasoning and the satisfaction of arriving at the correct solution through a systematic and logical approach.
The process of simplifying expressions with complex numbers, as we've demonstrated, hinges on a few key concepts. Understanding the structure of a complex number as a combination of a real and an imaginary part is paramount. The real part behaves like any ordinary number, while the imaginary part is always a multiple of the imaginary unit 'i'. Operations such as addition, subtraction, and multiplication follow specific rules that ensure the result remains a complex number. Addition and subtraction involve combining the real and imaginary parts separately, while multiplication requires the distributive property and the understanding that i² = -1. These fundamental rules are the building blocks for more complex operations and applications of complex numbers.
Complex numbers are not merely abstract mathematical constructs; they have wide-ranging applications in various fields. In electrical engineering, they are used to analyze alternating current (AC) circuits, where voltage and current have both magnitude and phase. In quantum mechanics, complex numbers are essential for describing wave functions, which govern the behavior of particles at the atomic level. Signal processing, another field that heavily relies on complex numbers, uses them to analyze and manipulate signals, such as audio and video. The ability to represent and manipulate complex numbers is therefore a crucial skill for professionals in these fields. Moreover, complex numbers play a vital role in various mathematical areas, including fractal geometry and number theory, showcasing their versatility and importance in the broader mathematical landscape. The mastery of complex number operations opens doors to understanding and solving problems in a diverse range of scientific and engineering disciplines.
In conclusion, we have successfully navigated the complexities of simplifying the expression 7 + (9 + 27i) - 2(25 - 11i) by employing a step-by-step approach. This journey has not only led us to the correct answer, -34 + 49i, but has also reinforced the fundamental principles of complex number arithmetic. Mastering complex number arithmetic is crucial for anyone venturing into fields like electrical engineering, quantum mechanics, or advanced mathematics. The ability to confidently manipulate complex numbers unlocks a deeper understanding of these disciplines and allows for the solution of a wider range of problems.
The systematic approach we've taken – distributing multiplication, combining real and imaginary terms separately, and carefully performing each calculation – is a valuable technique that can be applied to other mathematical problems as well. This process underscores the importance of precision and attention to detail in mathematical calculations. Moreover, understanding the underlying concepts of complex numbers, such as the nature of the imaginary unit and the rules for operations, is key to building a solid foundation in this area. By practicing and applying these principles, one can develop a strong command of complex number arithmetic, paving the way for further exploration of mathematical concepts and their applications in the real world. The world of complex numbers, though seemingly abstract, is a powerful tool for solving real-world problems, and mastering its arithmetic is a worthwhile endeavor.
