Solving For Real Numbers A And B In A Complex Equation

JU07/16/2025 08, 2025 by THE IDEN 55 views

Determining Real Numbers a and b Given a Complex Equation

This article will delve into the process of finding the real numbers a and b when given a complex equation. The equation we're tackling is: (√5 + i) / (2 - i) = a + bi. This type of problem falls under the domain of complex number manipulation, which is a fundamental concept in mathematics, especially in algebra and calculus. To solve this, we'll need to use techniques like multiplying by the conjugate and separating real and imaginary parts. The key takeaway here is understanding how to manipulate complex numbers to isolate the real and imaginary components, allowing us to determine the values of a and b. This skill is crucial not only for academic purposes but also for various applications in engineering, physics, and computer science, where complex numbers are used to model and solve real-world problems. Complex numbers, with their real and imaginary components, provide a powerful tool for representing and analyzing phenomena that cannot be described by real numbers alone. From electrical circuits to quantum mechanics, complex numbers play a vital role in understanding and predicting the behavior of complex systems. Thus, mastering the manipulation of complex numbers, as demonstrated in this article, is an invaluable asset for anyone pursuing a career in STEM fields. Moreover, the ability to solve such equations enhances one's problem-solving skills, which are transferable to other areas of life. By breaking down the complex equation into smaller, manageable steps, we can systematically arrive at the solution, illustrating the importance of a structured approach to problem-solving. The challenge presented by this equation is not merely mathematical; it also tests one's ability to think critically and apply previously learned concepts in a novel situation. This is a skill that is highly valued in both academic and professional settings, where individuals are often faced with complex problems that require creative and analytical solutions. Therefore, understanding and mastering the techniques presented in this article will not only help you solve this particular equation but also equip you with the skills necessary to tackle a wide range of mathematical and real-world challenges. The process of finding a and b involves a series of steps that require careful attention to detail and a solid understanding of complex number arithmetic. From multiplying by the conjugate to simplifying the resulting expression, each step is crucial in arriving at the correct solution. By working through this example, you will gain a deeper appreciation for the intricacies of complex numbers and their applications. This knowledge will serve you well in your future studies and career endeavors, providing you with a strong foundation for tackling more advanced mathematical concepts and real-world problems.

Let's embark on the step-by-step journey to solve the complex equation (√5 + i) / (2 - i) = a + bi and find the real values of a and b. Our initial hurdle is the complex denominator (2 - i). To eliminate this and simplify the expression, we employ a clever technique: multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of (2 - i) is (2 + i). This operation is crucial because it transforms the denominator into a real number, making it easier to isolate the real and imaginary parts of the complex number. Mathematically, multiplying a complex number by its conjugate results in the sum of the squares of its real and imaginary parts, effectively eliminating the imaginary component from the denominator. This is a fundamental principle in complex number arithmetic and is widely used in various mathematical and engineering applications. By multiplying both the numerator and the denominator by the same value (the conjugate), we ensure that the value of the overall expression remains unchanged. This is a key concept in algebraic manipulation, where we aim to simplify expressions without altering their fundamental value. The next step involves carefully expanding the numerator and the denominator using the distributive property. This requires meticulous attention to detail, as each term must be multiplied correctly to avoid errors. The expansion process reveals the underlying structure of the complex number and allows us to identify and combine like terms. This is a crucial skill in algebra and is essential for simplifying complex expressions. Once we have expanded the numerator and the denominator, we can proceed to simplify the expression by combining like terms. This involves adding the real parts together and the imaginary parts together. This step is crucial for isolating the real and imaginary components of the complex number, which is necessary for determining the values of a and b. After simplifying the numerator and the denominator, we will have a complex number in the form (x + yi) / z, where x, y, and z are real numbers. To express this in the standard form a + bi, we simply divide both the real and imaginary parts of the numerator by the denominator. This step is straightforward but essential for obtaining the final answer in the desired format. By separating the real and imaginary parts, we can directly compare the resulting expression with the given form a + bi and identify the values of a and b. This is the final step in the solution process and provides us with the answer to the problem. The entire process, from multiplying by the conjugate to separating the real and imaginary parts, demonstrates the power and elegance of complex number manipulation. By mastering these techniques, you will be well-equipped to tackle a wide range of problems involving complex numbers.

Step 1: Multiply by the Conjugate

To initiate the process of finding a and b, our primary task is to eliminate the complex component from the denominator. This is achieved by multiplying both the numerator and the denominator of the given expression by the conjugate of the denominator. In our case, the denominator is (2 - i), and its conjugate is (2 + i). The conjugate of a complex number is formed by simply changing the sign of the imaginary part. This seemingly simple step is the cornerstone of simplifying complex fractions and allows us to transform the denominator into a real number. The underlying principle behind this technique is the difference of squares identity: (x - y) (x + y) = x² - y². When applied to complex numbers, this identity results in the elimination of the imaginary term in the denominator. By multiplying (2 - i) by its conjugate (2 + i), we obtain 2² - (i)², which simplifies to 4 - (-1) = 5. This transformation is crucial for isolating the real and imaginary parts of the complex number. Multiplying both the numerator and the denominator by the same value ensures that we are not changing the value of the overall expression. This is a fundamental principle in algebraic manipulation and is essential for maintaining the integrity of the equation. The numerator becomes (√5 + i) (2 + i), which we will expand in the next step. The denominator, as we have already seen, becomes 5. This transformation sets the stage for further simplification and allows us to express the complex number in the standard form a + bi. The act of multiplying by the conjugate is not merely a mathematical trick; it is a powerful technique that simplifies complex expressions and reveals their underlying structure. By mastering this technique, you will gain a deeper understanding of complex numbers and their properties. This understanding will be invaluable in various mathematical and engineering applications, where complex numbers are used to model and solve real-world problems. The careful and deliberate application of this technique is a testament to the importance of precision in mathematical calculations. Even a small error in this step can lead to a significantly different result. Therefore, it is crucial to pay close attention to detail and ensure that each step is performed correctly. By practicing this technique, you will develop the skills necessary to handle complex expressions with confidence and accuracy.

Step 2: Expand and Simplify

Following the multiplication by the conjugate, the next crucial step involves expanding the numerator and simplifying the entire expression. Our numerator is now (√5 + i) (2 + i). To expand this, we apply the distributive property (also known as the FOIL method), ensuring that each term in the first complex number is multiplied by each term in the second complex number. This process is fundamental to algebraic manipulation and is essential for simplifying expressions involving multiple terms. The expansion yields: (√5)(2) + (√5)(i) + (i)(2) + (i)². Remembering that i² = -1 is paramount in this step. This substitution is the key to simplifying complex numbers, as it allows us to combine the real and imaginary terms. Without this substitution, we would not be able to express the complex number in the standard form a + bi. After substituting i² = -1, the numerator becomes 2√5 + √5i + 2i - 1. Now, we group the real and imaginary terms together. The real terms are 2√5 and -1, and the imaginary terms are √5i and 2i. Combining these terms, we get (2√5 - 1) + (√5 + 2)i. This step is crucial for isolating the real and imaginary parts of the complex number. The denominator, as we calculated in the previous step, is 5. Therefore, the entire expression becomes [(2√5 - 1) + (√5 + 2)i] / 5. To express this in the standard form a + bi, we divide both the real and imaginary parts of the numerator by the denominator. This gives us: a = (2√5 - 1) / 5 and b = (√5 + 2) / 5. These are the real values of a and b that satisfy the given equation. The process of expanding and simplifying the numerator highlights the importance of careful algebraic manipulation. Each step must be performed with precision to avoid errors. The ability to identify and combine like terms is a fundamental skill in algebra and is essential for simplifying complex expressions. By mastering these techniques, you will be well-equipped to tackle a wide range of mathematical problems. The final result, a = (2√5 - 1) / 5 and b = (√5 + 2) / 5, demonstrates the power of complex number arithmetic. By manipulating the given expression using algebraic techniques, we were able to isolate the real and imaginary parts and determine the values of a and b. This is a testament to the elegance and utility of complex numbers in mathematics.

Step 3: Separate Real and Imaginary Parts

The culmination of our simplification process arrives at the crucial step of separating the real and imaginary parts. After expanding and simplifying, we've reached the expression [(2√5 - 1) + (√5 + 2)i] / 5. To isolate a and b, we must divide both the real and imaginary components by the denominator, 5. This is a straightforward application of the distributive property of division over addition. By separating the real and imaginary parts, we can directly equate them to a and b in the given form a + bi. This step is the key to unlocking the solution to the problem. Dividing the real part (2√5 - 1) by 5, we obtain a = (2√5 - 1) / 5. This is the real component of the complex number. Similarly, dividing the imaginary part (√5 + 2)i by 5, we obtain b = (√5 + 2) / 5. This is the imaginary component of the complex number. Therefore, we have successfully determined the values of a and b that satisfy the given equation. a = (2√5 - 1) / 5 and b = (√5 + 2) / 5 represent the solution to the problem. This separation of real and imaginary parts is a fundamental concept in complex number arithmetic. It allows us to analyze and manipulate complex numbers in a way that is analogous to working with real numbers. By understanding this concept, you can solve a wide range of problems involving complex numbers. The ability to separate real and imaginary parts is also crucial in various applications of complex numbers, such as in electrical engineering and quantum mechanics. In these fields, complex numbers are used to model and analyze real-world phenomena, and the separation of real and imaginary parts allows us to extract meaningful information from these models. The final values of a and b provide a complete solution to the problem. They represent the real and imaginary components of the complex number that is equal to the given expression. This solution demonstrates the power of complex number manipulation and the importance of careful algebraic techniques. By following a systematic approach, we were able to simplify the given expression and isolate the real and imaginary parts, leading us to the correct answer. This process highlights the importance of precision and attention to detail in mathematical calculations. Each step must be performed with care to avoid errors and ensure that the final solution is accurate. The successful separation of real and imaginary parts is a testament to your understanding of complex number arithmetic and your ability to apply algebraic techniques effectively. This skill will serve you well in your future mathematical endeavors.

In conclusion, after meticulously navigating through the steps of multiplying by the conjugate, expanding, simplifying, and finally separating the real and imaginary parts, we have successfully determined the values of a and b. The real number a is (2√5 - 1) / 5, and the real number b is (√5 + 2) / 5. These values represent the solution to the equation (√5 + i) / (2 - i) = a + bi. This journey through complex number manipulation has underscored the importance of several key mathematical concepts. The use of the conjugate to eliminate the imaginary component from the denominator is a powerful technique that simplifies complex fractions. The distributive property, applied during the expansion of the numerator, is a fundamental tool in algebraic manipulation. The recognition that i² = -1 is crucial for simplifying expressions involving imaginary numbers. And the final step of separating the real and imaginary parts allows us to isolate the values of a and b. These concepts are not only essential for solving this particular problem but also for understanding and applying complex numbers in various other contexts. Complex numbers, with their unique properties and applications, play a vital role in many areas of science and engineering. From electrical circuits to quantum mechanics, complex numbers provide a powerful tool for modeling and analyzing complex systems. Therefore, mastering the manipulation of complex numbers, as demonstrated in this article, is an invaluable skill for anyone pursuing a career in these fields. The values of a and b that we have found are not just abstract mathematical results; they represent the components of a complex number that satisfies a specific equation. This equation may arise in a variety of real-world applications, such as in the analysis of alternating current circuits or in the study of wave phenomena. Therefore, the ability to solve such equations is not merely an academic exercise; it is a practical skill that can be applied to solve real-world problems. The process of finding a and b has also highlighted the importance of a systematic approach to problem-solving. By breaking down the problem into smaller, manageable steps, we were able to navigate through the complexities of the equation and arrive at the solution. This structured approach is a valuable skill that can be applied to solve problems in any field. The final values of a and b stand as a testament to the power of mathematical reasoning and the elegance of complex number arithmetic. They represent the culmination of a series of logical steps, each building upon the previous one, leading us to a precise and satisfying solution. This journey through complex numbers has not only provided us with the answer to a specific problem but has also deepened our understanding of the fundamental principles of mathematics.

In summary, determining the real numbers a and b from the complex equation (√5 + i) / (2 - i) = a + bi involves a series of strategic algebraic manipulations. The cornerstone of the solution lies in the technique of multiplying both the numerator and the denominator by the conjugate of the denominator. This crucial step effectively eliminates the imaginary component from the denominator, paving the way for simplification. The conjugate, in this instance, is (2 + i), and multiplying by it leverages the difference of squares pattern to transform the denominator into a real number. Following the multiplication by the conjugate, the next vital step is the meticulous expansion of the numerator. This involves applying the distributive property (often remembered by the acronym FOIL) to ensure that each term is correctly multiplied. This process generates a sum of terms that include both real and imaginary components. The key to simplification at this stage is the recognition and substitution of i² with -1, which is a fundamental property of imaginary numbers. This substitution allows us to combine like terms, further isolating the real and imaginary parts of the complex number. With the numerator expanded and simplified, the next phase involves separating the real and imaginary components. This is achieved by dividing both the real and imaginary parts of the simplified numerator by the real denominator. This division effectively expresses the complex number in the standard form a + bi, where a and b are the real and imaginary coefficients, respectively. The final step is the explicit identification of a and b. By comparing the simplified complex number with the form a + bi, we can directly equate the corresponding real and imaginary parts. This provides us with the values of a and b that satisfy the original equation. The values we found are a = (2√5 - 1) / 5 and b = (√5 + 2) / 5. This entire process demonstrates the elegance and power of complex number arithmetic. It highlights the importance of understanding fundamental algebraic principles, such as the distributive property and the properties of conjugates. Moreover, it underscores the significance of careful and methodical manipulation to arrive at a solution. The ability to solve such equations is not only a valuable skill in mathematics but also in various fields of science and engineering where complex numbers are used to model and solve real-world problems.

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