Dividing Complex Numbers: Finding The First Step In Quotient Of Z/w

In the realm of complex numbers, division might seem a bit perplexing at first. Unlike real numbers, we can't directly divide by a complex number in the same way. Instead, we employ a clever technique involving the conjugate of the denominator. This article delves into the initial step of finding the quotient of two complex numbers, using the specific example of z = 5 - 9i and w = 6 + i. Understanding this foundational step is crucial for mastering complex number arithmetic and its applications in various fields such as electrical engineering, quantum mechanics, and signal processing. So, let's embark on this journey to demystify complex number division.

Understanding Complex Numbers

Before diving into the division process, let's briefly recap what complex numbers are. A complex number is expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1. The 'a' part is called the real part, and the 'b' part is called the imaginary part. Complex numbers extend the real number system, allowing us to work with the square roots of negative numbers. They can be visualized on a complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This geometric representation provides a powerful way to understand complex number operations.

Complex numbers are not just abstract mathematical concepts; they have profound practical applications. In electrical engineering, they are used to analyze alternating current (AC) circuits. In quantum mechanics, they are essential for describing wave functions. In signal processing, they help in analyzing and manipulating signals. The ability to perform arithmetic operations on complex numbers, including division, is therefore fundamental in these diverse fields. The elegance of complex numbers lies in their ability to simplify complex problems, often providing solutions that would be difficult or impossible to obtain using real numbers alone.

The Challenge of Division

When we encounter the division of two complex numbers, such as z/w, where z = 5 - 9i and w = 6 + i, we face a unique challenge. We cannot simply divide the real and imaginary parts separately. The presence of 'i' in the denominator makes the expression non-standard. To address this, we need to transform the denominator into a real number. This is where the concept of the complex conjugate comes into play. The complex conjugate of a complex number a + bi is a - bi. Multiplying a complex number by its conjugate eliminates the imaginary part in the denominator, paving the way for a clean division. This technique is analogous to rationalizing the denominator when dealing with surds in real number arithmetic.

Division of complex numbers is not merely a mechanical process; it's a technique rooted in the fundamental properties of complex numbers. The complex conjugate acts as a tool to manipulate the expression without changing its inherent value. By understanding the underlying principles, we can appreciate the elegance and efficiency of this method. The ability to divide complex numbers opens up a new realm of mathematical possibilities, enabling us to solve problems that were previously intractable. The significance of this operation extends beyond pure mathematics, impacting various scientific and engineering disciplines.

The Conjugate: Our Key to Division

The key to dividing complex numbers lies in the concept of the complex conjugate. For a complex number w = a + bi, its conjugate, denoted as w̄, is a - bi. The magic happens when we multiply a complex number by its conjugate. Let's consider w multiplied by w̄: (a + bi)(a - bi) = a² - (bi)² = a² - (b² * i²) = a² - (b² * -1) = a² + b². Notice that the result is a real number, as the imaginary unit 'i' has been eliminated. This property is the cornerstone of complex number division. By multiplying both the numerator and denominator of a complex fraction by the conjugate of the denominator, we transform the denominator into a real number, making the division straightforward.

The complex conjugate is not just a mathematical trick; it's a fundamental concept that reveals the inherent structure of complex numbers. It's a reflection across the real axis in the complex plane, providing a geometric interpretation of this algebraic operation. The conjugate allows us to manipulate complex numbers in a way that preserves their mathematical integrity. This is crucial in various applications, where the properties of complex numbers need to be maintained throughout calculations. The use of the conjugate in division is a testament to the power of mathematical tools in simplifying complex problems. It's a prime example of how a seemingly abstract concept can have concrete applications.

The First Step: Multiplying by the Conjugate

Now, let's apply this knowledge to our specific problem: finding the quotient of z/w, where z = 5 - 9i and w = 6 + i. The first crucial step is to multiply both the numerator and the denominator by the conjugate of the denominator, w. The conjugate of w = 6 + i is w̄ = 6 - i. So, we multiply the fraction (5 - 9i) / (6 + i) by (6 - i) / (6 - i). This gives us: [(5 - 9i) / (6 + i)] * [(6 - i) / (6 - i)]. This seemingly simple step is the key to unlocking the division of complex numbers. It sets the stage for simplifying the expression and obtaining the quotient in the standard complex number form (a + bi).

This initial step is not just a mechanical manipulation; it's a strategic move that transforms the problem into a manageable form. By multiplying by the conjugate, we are essentially clearing the imaginary component from the denominator. This allows us to separate the real and imaginary parts of the quotient, making it easier to express the result in the standard complex number format. The importance of this step cannot be overstated; it's the foundation upon which the entire division process rests. Without it, we would be stuck with a complex expression that is difficult to interpret and use. The elegance of this method lies in its simplicity and effectiveness. It's a testament to the power of mathematical techniques in solving seemingly complex problems.

The Correct Expression

Therefore, the expression that shows the first step in finding the quotient of z/w is:

(5 - 9i) / (6 + i) * (6 - i) / (6 - i)

This expression correctly represents the multiplication of both the numerator and the denominator by the conjugate of the denominator. It's the foundational step that allows us to proceed with the division process. The next steps would involve expanding the products in the numerator and the denominator, simplifying the expression, and finally expressing the quotient in the standard form a + bi. However, this initial step is the most crucial, as it sets the direction for the rest of the solution.

The beauty of this expression lies in its clarity and precision. It encapsulates the essence of complex number division in a concise form. It's a visual representation of the strategy we employ to tackle this type of problem. The expression serves as a roadmap, guiding us through the subsequent steps. It's a reminder that mathematical solutions are often built upon a series of well-defined steps, each contributing to the final result. This expression is not just a mathematical notation; it's a symbol of the power of mathematical thinking in unraveling complex problems.

Next Steps: Simplifying and Finding the Quotient

Once we have the expression (5 - 9i) / (6 + i) * (6 - i) / (6 - i), the next step is to expand the products in both the numerator and the denominator. In the numerator, we have (5 - 9i)(6 - i), which we can expand using the distributive property (or the FOIL method). This gives us 5 * 6 + 5 * (-i) + (-9i) * 6 + (-9i) * (-i) = 30 - 5i - 54i + 9i². Remember that i² = -1, so we can substitute that in: 30 - 5i - 54i - 9 = 21 - 59i. In the denominator, we have (6 + i)(6 - i), which is a difference of squares: 6² - i² = 36 - (-1) = 37. Now we have the quotient (21 - 59i) / 37. To express this in the standard form a + bi, we simply divide both the real and imaginary parts by the denominator: (21/37) - (59/37)i. This is the final quotient of z/w.

This process of simplifying and finding the quotient highlights the importance of algebraic manipulation in complex number arithmetic. The ability to expand products, simplify expressions, and substitute known values is crucial for arriving at the correct solution. The steps involved are not just mechanical; they require a deep understanding of the properties of complex numbers and algebraic operations. The final result, (21/37) - (59/37)i, is a complex number in the standard form, making it easy to interpret and use in further calculations. The journey from the initial expression to the final quotient showcases the power of mathematical techniques in transforming complex problems into simple solutions. It's a testament to the elegance and efficiency of mathematics.

Conclusion

In conclusion, the first step in finding the quotient of two complex numbers involves multiplying both the numerator and the denominator by the conjugate of the denominator. This crucial step transforms the problem into a manageable form, allowing us to simplify the expression and obtain the quotient in the standard complex number form. Understanding the concept of the complex conjugate and its role in division is fundamental to mastering complex number arithmetic. This skill is not only essential in mathematics but also in various fields such as electrical engineering, quantum mechanics, and signal processing, where complex numbers play a vital role. The journey from understanding the problem to finding the solution highlights the power of mathematical thinking and the elegance of complex number arithmetic.