Dividing Complex Numbers A Comprehensive Guide To (6-5i) By (7+6i)
Complex number division, at first glance, might seem daunting. However, by understanding the underlying principles and applying a systematic approach, it becomes a manageable and even elegant process. In this comprehensive guide, we will dissect the problem of dividing complex numbers, specifically focusing on the expression (6-5i)/(7+6i). We will delve into the theory behind complex number division, explore the concept of conjugates, and meticulously walk through the steps required to arrive at the solution. This detailed explanation aims to not only provide the answer but also to equip you with a solid understanding of how to tackle similar problems in the future.
The journey begins with recognizing the standard form of a complex number: a + bi, where 'a' represents the real part and 'b' represents the imaginary part, and 'i' is the imaginary unit, defined as the square root of -1. When dividing complex numbers, our goal is to eliminate the imaginary part from the denominator, effectively expressing the result in the standard a + bi form. This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number (a + bi) is (a - bi), where the sign of the imaginary part is flipped. This seemingly simple trick leverages the difference of squares identity: (a + b)(a - b) = a² - b². In the context of complex numbers, this identity becomes particularly powerful because i² = -1, which allows us to eliminate the imaginary term.
In our specific case, the denominator is (7 + 6i). Therefore, its conjugate is (7 - 6i). The core strategy involves multiplying both the numerator (6 - 5i) and the denominator (7 + 6i) by this conjugate (7 - 6i). This crucial step ensures that we are not changing the value of the expression, as we are essentially multiplying by 1. The subsequent steps involve careful expansion, simplification, and separation of the real and imaginary components. By meticulously following this process, we can transform the complex division problem into a series of simpler algebraic manipulations, ultimately leading to the desired result in the standard complex number form.
The Conjugate Key: Eliminating Imaginary Denominators
The cornerstone of complex number division lies in the concept of the conjugate. The conjugate of a complex number is formed by simply changing the sign of its imaginary part. For instance, the conjugate of a + bi is a - bi, and vice versa. This seemingly simple manipulation is the key to eliminating the imaginary part from the denominator when performing division.
The power of the conjugate stems from a fundamental algebraic identity: the difference of squares. When a complex number is multiplied by its conjugate, the result is always a real number. To illustrate this, let's consider the general case: (a + bi)(a - bi). Expanding this product, we get a² - abi + abi - b²i². The middle terms, -abi and +abi, cancel each other out. Furthermore, since i² = -1, the last term becomes -b²(-1) = +b². Therefore, the entire expression simplifies to a² + b², which is a real number.
This principle is directly applicable to our problem, (6-5i)/(7+6i). The denominator is (7 + 6i), and its conjugate is (7 - 6i). When we multiply (7 + 6i) by its conjugate (7 - 6i), we are effectively applying the difference of squares identity. The result will be 7² + 6², which equals 49 + 36 = 85, a real number. This is precisely what we aim for – a real number in the denominator.
The next step involves multiplying the numerator (6 - 5i) by the same conjugate (7 - 6i). This ensures that we are multiplying the entire fraction by 1, thus preserving its value. However, the multiplication in the numerator will result in another complex number, which we will need to simplify. The subsequent sections will delve into the detailed expansion and simplification of both the numerator and the denominator, showcasing how the conjugate effectively transforms a complex division problem into a more manageable algebraic exercise. Understanding the role of the conjugate is not just about memorizing a trick; it's about grasping the fundamental principles that govern complex number arithmetic, providing a solid foundation for tackling more advanced problems in the future.
Step-by-Step Solution: Dividing (6-5i) by (7+6i)
Now, let's embark on the step-by-step solution to divide the complex number (6 - 5i) by (7 + 6i). As established earlier, the key is to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of (7 + 6i) is (7 - 6i). This strategic multiplication will eliminate the imaginary component from the denominator, allowing us to express the final answer in the standard a + bi form. This process meticulously transforms the problem into a series of manageable algebraic steps.
Step 1: Multiply by the Conjugate
Multiply both the numerator and the denominator by the conjugate of the denominator:
This step is crucial as it sets the stage for simplifying the expression. We are essentially multiplying by 1, ensuring that the value of the complex number remains unchanged while strategically manipulating its form.
Step 2: Expand the Numerator
Expand the numerator (6 - 5i)(7 - 6i) using the distributive property (often referred to as the FOIL method):
(6 * 7) + (6 * -6i) + (-5i * 7) + (-5i * -6i)
This expansion meticulously breaks down the multiplication, ensuring that each term is accounted for. The resulting expression will contain both real and imaginary terms, which will be further simplified in the subsequent steps.
Step 3: Simplify the Numerator
Simplify the expanded numerator:
42 - 36i - 35i + 30i²
Recall that i² = -1. Substitute this value into the expression:
42 - 36i - 35i + 30(-1)
Combine the real terms (42 and -30) and the imaginary terms (-36i and -35i):
(42 - 30) + (-36 - 35)i
This simplification process groups like terms, making it easier to identify and combine the real and imaginary components.
Step 4: Further Simplify the Numerator
This gives us:
12 - 71i
The numerator is now simplified to the form (12 - 71i), a complex number in the standard a + bi format. This simplified numerator will be placed over the simplified denominator in the next steps.
Step 5: Expand the Denominator
Expand the denominator (7 + 6i)(7 - 6i). This is where the power of the conjugate comes into play. As discussed earlier, multiplying a complex number by its conjugate results in a real number. Apply the distributive property:
(7 * 7) + (7 * -6i) + (6i * 7) + (6i * -6i)
This expansion mirrors the process used for the numerator, ensuring that all terms are considered.
Step 6: Simplify the Denominator
Simplify the expanded denominator:
49 - 42i + 42i - 36i²
Notice that the imaginary terms (-42i and +42i) cancel each other out, as expected. Substitute i² = -1:
49 - 36(-1)
This cancellation is the hallmark of using the conjugate, effectively eliminating the imaginary component from the denominator.
Step 7: Further Simplify the Denominator
This gives us:
49 + 36 = 85
The denominator is now a real number, 85. This is the key outcome of multiplying by the conjugate, simplifying the division problem significantly.
Step 8: Combine Numerator and Denominator
Now, we have the simplified numerator (12 - 71i) and the simplified denominator (85). Combine them to form the resulting complex number:
This fraction represents the result of the complex number division, but it can be further refined to the standard a + bi form.
Step 9: Separate Real and Imaginary Parts
Separate the real and imaginary parts:
This step expresses the complex number in its standard form, where the real part (12/85) and the imaginary part (-71/85) are clearly identified. This final form is the desired outcome of the complex number division.
Final Answer
Therefore, (6 - 5i) / (7 + 6i) = 12/85 - (71/85)i
This step-by-step solution meticulously outlines the process of dividing complex numbers, emphasizing the importance of the conjugate and the careful application of algebraic principles. Each step builds upon the previous one, leading to the final solution in a clear and logical manner. Understanding this process empowers you to confidently tackle similar complex number division problems.
Real and Imaginary Components: The Final Result
After performing the division and simplifying the expression, we arrive at the final answer in the standard complex number form: a + bi. In this format, 'a' represents the real component and 'b' represents the imaginary component. Understanding how to identify and interpret these components is crucial for working with complex numbers and applying them in various mathematical and scientific contexts. In our specific problem, (6 - 5i) / (7 + 6i), the final result is:
From this expression, we can clearly identify the real and imaginary components. The real component, 'a', is 12/85, which is a rational number. It represents the part of the complex number that lies along the real number line. The imaginary component, 'b', is -71/85, also a rational number. It is the coefficient of the imaginary unit 'i' and represents the part of the complex number that lies along the imaginary axis. The negative sign indicates that this component lies in the negative direction of the imaginary axis.
These real and imaginary components together define the complex number's position in the complex plane, a two-dimensional plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part. The complex number 12/85 - (71/85)i can be visualized as a point in this plane with coordinates (12/85, -71/85). This geometric interpretation is fundamental in understanding the properties and applications of complex numbers.
The ability to separate and interpret the real and imaginary components is essential for various operations with complex numbers, such as addition, subtraction, multiplication, and exponentiation. For example, when adding complex numbers, we simply add the real components together and the imaginary components together. Similarly, understanding these components is crucial in applications such as electrical engineering, quantum mechanics, and fluid dynamics, where complex numbers are used to model various phenomena. In electrical engineering, for instance, complex numbers are used to represent alternating currents and voltages, where the real part represents the resistive component and the imaginary part represents the reactive component. Grasping the significance of real and imaginary components is therefore not just a mathematical exercise but a crucial skill for understanding and applying complex numbers in diverse fields.
In conclusion, the division of complex numbers, exemplified by the problem (6 - 5i) / (7 + 6i), is a process that, while seemingly intricate at first, becomes clear and manageable with a systematic approach and a solid understanding of the underlying principles. The key to this process lies in the strategic use of the conjugate, a concept that allows us to eliminate the imaginary component from the denominator and express the result in the standard complex number form a + bi. This form, with its distinct real and imaginary components, is fundamental to understanding and applying complex numbers in various fields.
Throughout this guide, we have meticulously broken down the division process into a series of clear and concise steps. We began by understanding the significance of the conjugate and its role in transforming the division problem. We then proceeded to the detailed step-by-step solution, which involved multiplying both the numerator and the denominator by the conjugate, expanding the products, simplifying the expressions, and finally separating the real and imaginary components. This methodical approach not only provides the solution to the specific problem but also equips you with a framework for tackling similar complex number division problems in the future.
Furthermore, we emphasized the importance of interpreting the real and imaginary components of the final result. These components define the complex number's position in the complex plane and are crucial for various operations with complex numbers and their applications in diverse fields such as electrical engineering, quantum mechanics, and fluid dynamics. Mastering the division of complex numbers is therefore not just about memorizing a procedure; it's about developing a deeper understanding of complex number arithmetic and its broader implications.
By diligently working through the steps outlined in this guide and practicing with additional examples, you can confidently master complex number division and unlock the power of complex numbers in various mathematical and scientific contexts. The ability to manipulate complex numbers effectively is a valuable asset in many fields, and this comprehensive guide has provided you with the tools and knowledge necessary to achieve that mastery. Therefore, embrace the challenge, practice the techniques, and embark on your journey to becoming proficient in the world of complex numbers.
