Simplifying Complex Number Expressions Divide 6-5i By 7+6i
Complex numbers, with their real and imaginary parts, might seem daunting at first. However, they follow specific rules and operations, making them manageable with the right approach. In this comprehensive guide, we'll delve into the process of simplifying the expression (6 - 5i) / (7 + 6i), breaking down each step for clarity. Mastering complex number simplification is crucial for various fields, including electrical engineering, quantum mechanics, and applied mathematics. Before we jump into the solution, let's solidify our understanding of complex numbers.
At its core, a complex number is composed of two parts: a real part and an imaginary part. It is generally written in the form a + bi, where 'a' represents the real part, 'b' represents the imaginary part, and 'i' is the imaginary unit. The imaginary unit, denoted by 'i', is defined as the square root of -1 (i = √-1). This foundational concept is essential because it extends the number system beyond real numbers, allowing us to work with the square roots of negative numbers. The real part of a complex number exists on the real number line, while the imaginary part exists on the imaginary number line, perpendicular to the real axis. The combination of these two parts forms a complex plane, where every complex number can be represented as a point. Understanding this representation helps in visualizing operations with complex numbers, such as addition, subtraction, multiplication, and division. Furthermore, complex numbers introduce unique properties, such as the cyclical nature of 'i' powers (i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1), which is crucial for simplifying complex expressions. The conjugate of a complex number, which plays a vital role in division, is obtained by changing the sign of the imaginary part. For instance, the conjugate of a + bi is a - bi. This conjugate helps eliminate the imaginary part from the denominator, making the division operation feasible. The magnitude of a complex number, calculated as √(a^2 + b^2), represents the distance from the origin to the point representing the complex number in the complex plane. This magnitude is important in various applications, such as signal processing and electrical circuit analysis. Operations with complex numbers follow specific rules. Addition and subtraction involve combining the real and imaginary parts separately. Multiplication involves using the distributive property and the fact that i^2 = -1. Division, which is our focus here, requires multiplying both the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator. This process transforms the denominator into a real number, making the complex number simplification possible. By mastering these basics, you gain the ability to manipulate complex numbers effectively, opening doors to solving complex mathematical and real-world problems.
Understanding the Problem: Dividing Complex Numbers
The core of our task is to simplify the expression (6 - 5i) / (7 + 6i). This involves dividing one complex number (6 - 5i) by another (7 + 6i). Direct division isn't possible in the way we handle real numbers. Instead, we employ a clever trick: multiplying both the numerator and the denominator by the conjugate of the denominator. Why this works will become clear as we proceed. The conjugate of a complex number (a + bi) is (a - bi). The key property here is that when a complex number is multiplied by its conjugate, the result is a real number. This eliminates the imaginary part from the denominator, which is what we aim for in division. Before we dive into the specific steps, let’s consider the general approach for dividing complex numbers. If we have two complex numbers, say z1 = a + bi and z2 = c + di, then the division z1 / z2 is performed by multiplying both the numerator and the denominator by the conjugate of z2, which is c - di. This process can be represented as: (a + bi) / (c + di) = [(a + bi) * (c - di)] / [(c + di) * (c - di)]. The denominator then becomes (c^2 + d^2), which is a real number. The numerator is expanded using the distributive property, and the real and imaginary parts are collected. The final simplified form is then expressed as a single complex number in the form A + Bi, where A and B are real numbers. This method is crucial for any division operation involving complex numbers, as it ensures that the result is expressed in standard complex number form. Now, let’s apply this method to our specific problem. We need to identify the conjugate of the denominator, which is (7 + 6i). The conjugate of (7 + 6i) is (7 - 6i). We will then multiply both the numerator and denominator of our expression by (7 - 6i). This step is critical, as it sets the stage for eliminating the imaginary part from the denominator. The next step involves carefully expanding the numerator and the denominator using the distributive property, also known as the FOIL method (First, Outer, Inner, Last). This expansion is where the arithmetic of complex number multiplication comes into play, and it’s crucial to ensure accuracy to arrive at the correct final answer. The expanded expression will then be simplified by combining like terms and using the property i^2 = -1. This property is fundamental to simplifying expressions involving the imaginary unit 'i'.
Step-by-Step Solution: Multiplying by the Conjugate
The initial step in simplifying (6 - 5i) / (7 + 6i) involves multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of (7 + 6i) is (7 - 6i). So, we multiply: [(6 - 5i) / (7 + 6i)] * [(7 - 6i) / (7 - 6i)]. This crucial step transforms the division problem into a multiplication problem involving complex numbers, which is easier to handle. By multiplying both the numerator and the denominator by the same expression, we are essentially multiplying by 1, which does not change the value of the original expression. This is a standard technique used in algebra and complex number manipulation to rationalize denominators or, in this case, to eliminate the imaginary part from the denominator. The next stage is to expand both the numerator and the denominator separately. For the numerator (6 - 5i) * (7 - 6i), we use the distributive property (FOIL method): First terms: 6 * 7 = 42; Outer terms: 6 * (-6i) = -36i; Inner terms: (-5i) * 7 = -35i; Last terms: (-5i) * (-6i) = 30i^2. Combining these, we get 42 - 36i - 35i + 30i^2. For the denominator (7 + 6i) * (7 - 6i), we again use the distributive property: First terms: 7 * 7 = 49; Outer terms: 7 * (-6i) = -42i; Inner terms: 6i * 7 = 42i; Last terms: (6i) * (-6i) = -36i^2. This gives us 49 - 42i + 42i - 36i^2. Notice how the middle terms (-42i and +42i) cancel each other out, which is a direct consequence of multiplying a complex number by its conjugate. This cancellation is exactly why we use the conjugate in division, as it eliminates the imaginary term from the denominator, leaving us with a real number. The next crucial step is to simplify the expressions obtained after expansion. This involves combining like terms and, most importantly, using the property i^2 = -1. This property is the cornerstone of simplifying complex number expressions, as it allows us to convert imaginary terms involving i^2 into real numbers, which can then be combined with other real terms. This conversion is essential for expressing the final answer in the standard form of a complex number, a + bi. Let's proceed with this simplification in the subsequent steps.
Simplifying the Expression: Combining Real and Imaginary Terms
Having expanded both the numerator and denominator, our next critical step is to simplify the expression by combining real and imaginary terms. Let's first address the numerator, which we expanded to 42 - 36i - 35i + 30i^2. We recall that i^2 = -1, so we substitute -1 for i^2: 42 - 36i - 35i + 30(-1). This simplifies to 42 - 36i - 35i - 30. Now, we combine the real terms (42 and -30) and the imaginary terms (-36i and -35i): Real part: 42 - 30 = 12; Imaginary part: -36i - 35i = -71i. Therefore, the simplified numerator is 12 - 71i. Next, we simplify the denominator, which we expanded to 49 - 42i + 42i - 36i^2. Again, we substitute -1 for i^2: 49 - 42i + 42i - 36(-1). This simplifies to 49 - 42i + 42i + 36. Notice that the imaginary terms (-42i and +42i) cancel each other out, as expected when multiplying a complex number by its conjugate. This is a key benefit of using the conjugate in complex number division. Now, we combine the real terms (49 and 36): Real part: 49 + 36 = 85. Thus, the simplified denominator is 85, which is a real number, fulfilling our goal of eliminating the imaginary part from the denominator. Now that we have simplified the numerator to 12 - 71i and the denominator to 85, we can rewrite the entire expression as (12 - 71i) / 85. This is a significant step forward, but to express the complex number in standard form (a + bi), we need to separate the real and imaginary parts. This separation is crucial for clearly representing the complex number and for further operations or analyses that might be required. To separate the real and imaginary parts, we divide each term in the numerator by the denominator: (12/85) - (71i/85). This gives us the complex number in the form a + bi, where a = 12/85 and b = -71/85. Now we have expressed the result in the standard form of a complex number, with a clear real part and a clear imaginary part.
Final Result: Expressing the Complex Number in Standard Form
Having simplified the expression, we arrive at the final result: (12/85) - (71/85)i. This is the standard form of a complex number, a + bi, where the real part is 12/85 and the imaginary part is -71/85. This form is crucial because it clearly presents the complex number's components, making it easier to understand and use in further calculations or applications. Expressing complex numbers in standard form allows for consistent and accurate communication of mathematical results. The real part (12/85) represents the component of the complex number that lies along the real axis in the complex plane, while the imaginary part (-71/85) represents the component along the imaginary axis. These two components together define the position of the complex number in the complex plane. To summarize the steps we took to arrive at this solution: We started with the expression (6 - 5i) / (7 + 6i). We multiplied both the numerator and the denominator by the conjugate of the denominator, which was (7 - 6i). This step is crucial for eliminating the imaginary part from the denominator. We expanded both the numerator and the denominator using the distributive property (FOIL method). This involved multiplying each term in the first complex number by each term in the second complex number. We simplified the expanded expression by combining like terms and using the property i^2 = -1. This property is fundamental in complex number arithmetic, as it allows us to convert imaginary terms involving i^2 into real numbers. We separated the real and imaginary parts to express the final answer in the standard form of a complex number, a + bi. This involved dividing each term in the simplified numerator by the simplified denominator. The result, (12/85) - (71/85)i, is now in its simplest form. This complex number represents a specific point in the complex plane, with its real part (12/85) and imaginary part (-71/85) defining its coordinates. This final simplified form is essential for various applications, including electrical engineering, physics, and mathematics, where complex numbers are used extensively.
Conclusion: Mastering Complex Number Simplification
In conclusion, we've successfully simplified the complex number expression (6 - 5i) / (7 + 6i) to its standard form, (12/85) - (71/85)i. This process highlights the importance of understanding complex number properties and operations, particularly the use of conjugates in division. Mastering complex number simplification is a foundational skill in various scientific and engineering disciplines. The key takeaway from this exercise is the technique of multiplying both the numerator and the denominator by the conjugate of the denominator. This method effectively eliminates the imaginary part from the denominator, allowing us to express the complex number in the standard form a + bi. This standard form is crucial for further analysis, calculations, and applications involving complex numbers. The steps involved in this simplification process are applicable to any complex number division problem. Understanding and applying these steps consistently will ensure accuracy and efficiency in simplifying complex number expressions. Complex numbers are not just abstract mathematical concepts; they have real-world applications in various fields. In electrical engineering, complex numbers are used to represent alternating currents and voltages. In quantum mechanics, they are essential for describing wave functions. In signal processing, they are used for analyzing and manipulating signals. Therefore, mastering complex number operations, including simplification, is vital for anyone pursuing these fields. Furthermore, the process of simplifying complex expressions reinforces fundamental algebraic skills, such as the distributive property, combining like terms, and using key mathematical identities. These skills are transferable to other areas of mathematics and problem-solving. Practice is key to mastering complex number simplification. Working through various examples and exercises will solidify your understanding of the concepts and techniques involved. As you gain more experience, you will be able to recognize patterns and apply the appropriate strategies more efficiently. The journey through complex numbers, while potentially challenging at first, unveils a fascinating and powerful mathematical world. With a solid understanding of the fundamentals and consistent practice, you can confidently tackle complex number problems and appreciate their significance in various scientific and engineering contexts. The ability to manipulate and simplify complex numbers opens doors to more advanced mathematical concepts and real-world applications.
