Solving (9-6i) * M = 9-6i Find The Value Of M

Introduction

In the realm of mathematics, particularly when dealing with complex numbers, equations often present intriguing challenges. This article delves into solving the equation (9-6i) * m = 9-6i, where our goal is to determine the value of 'm'. This exploration will not only provide the solution but also elucidate the underlying principles of complex number arithmetic and algebraic manipulation. Understanding these concepts is crucial for anyone venturing into advanced mathematical studies, as complex numbers form the bedrock of various fields, including electrical engineering, quantum mechanics, and signal processing.

Understanding the Problem

Before diving into the solution, let's first dissect the problem at hand. The equation (9-6i) * m = 9-6i involves complex numbers, which are numbers that can be 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. In our equation, 9-6i is a complex number, and 'm' represents the unknown value we seek to find. The challenge lies in isolating 'm' while adhering to the rules of complex number arithmetic. This process will involve techniques such as division and understanding the properties of complex conjugates.

Solving the Equation: A Step-by-Step Approach

To solve the equation (9-6i) * m = 9-6i, we need to isolate 'm'. This can be achieved by dividing both sides of the equation by (9-6i). This gives us:

m = (9-6i) / (9-6i)

At first glance, this might seem straightforward – any number divided by itself is 1. However, it's crucial to rigorously demonstrate this, especially when dealing with complex numbers. To do this, we can perform the division explicitly. Dividing a complex number by another complex number involves multiplying both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of a complex number a + bi is a - bi. In our case, the complex conjugate of 9-6i is 9+6i. Let's proceed with the multiplication:

m = (9-6i) / (9-6i) * (9+6i) / (9+6i)

This step is crucial because multiplying by the conjugate eliminates the imaginary part from the denominator, making the division process simpler. Now, we multiply out the numerators and denominators:

Numerator: (9-6i) * (9+6i) = 99 + 96i - 6i9 - 6i6i = 81 + 54i - 54i - 36i^2

Denominator: (9-6i) * (9+6i) = 99 + 96i - 6i9 - 6i6i = 81 + 54i - 54i - 36i^2

Notice that the terms +54i and -54i cancel each other out in both the numerator and the denominator. Also, recall that i^2 = -1. Substituting this into our expressions, we get:

Numerator: 81 - 36(-1) = 81 + 36 = 117

Denominator: 81 - 36(-1) = 81 + 36 = 117

Therefore, m = 117 / 117 = 1

The Solution: m = 1

After carefully performing the complex number division, we arrive at the solution: m = 1. This result signifies that when the complex number (9-6i) is multiplied by 1, it remains unchanged, which aligns with the fundamental properties of multiplication. This solution underscores the importance of understanding complex number arithmetic and the techniques used to manipulate complex expressions.

Alternative Approach: Direct Division

While the method above meticulously demonstrates the complex number division process, there's a more direct approach to solving this particular equation. Since the equation is (9-6i) * m = 9-6i, and we are looking for 'm', we can directly divide both sides by (9-6i), as we did initially:

m = (9-6i) / (9-6i)

Without delving into the complex conjugate multiplication, we can recognize that any non-zero number (or complex number, in this case) divided by itself is 1. Therefore:

m = 1

This approach reaffirms our previous solution and highlights the simplicity of the problem when viewed from a broader perspective. However, it's crucial to understand the more detailed method involving complex conjugates, as it is applicable to a wider range of complex number division problems.

Common Pitfalls and How to Avoid Them

When working with complex numbers, there are several common pitfalls that students often encounter. One of the most frequent mistakes is incorrectly applying the distributive property when multiplying complex numbers. It's crucial to remember that the imaginary unit 'i' behaves like a variable, but with the added property that i^2 = -1. Another common error is neglecting to multiply by the complex conjugate when dividing complex numbers. This step is essential to eliminate the imaginary part from the denominator and simplify the expression. To avoid these pitfalls, it's imperative to practice complex number arithmetic extensively and to double-check each step in the calculation process.

The Significance of Complex Numbers

Complex numbers, though initially appearing abstract, play a pivotal role in various scientific and engineering disciplines. In electrical engineering, they are used to analyze alternating current (AC) circuits, where voltage and current oscillate sinusoidally. In quantum mechanics, complex numbers are fundamental to describing the wave functions of particles. Signal processing, a field that deals with the analysis and manipulation of signals, also relies heavily on complex numbers. Understanding complex numbers is, therefore, not just an academic exercise but a crucial skill for anyone pursuing a career in these fields. This application highlights the relevance of mastering complex number operations, such as the one we addressed in the equation (9-6i) * m = 9-6i.

Practice Problems

To solidify your understanding of complex number arithmetic and equation-solving, consider tackling these practice problems:

1. Solve for 'z' in the equation (2 + 3i) * z = 4 - i.
2. Find 'x' if (1 - i) * x = 2 + 2i.
3. Determine 'y' in the equation (5 + 5i) * y = 10.

These problems will provide valuable practice in applying the techniques discussed in this article. Remember to use the methods of complex conjugate multiplication and algebraic manipulation to arrive at the solutions.

Conclusion

In conclusion, solving the equation (9-6i) * m = 9-6i demonstrates the application of fundamental principles in complex number arithmetic. Through careful algebraic manipulation and an understanding of complex conjugates, we determined that m = 1. This exercise not only provides a solution to a specific problem but also reinforces the broader importance of complex numbers in various scientific and engineering disciplines. By mastering these concepts and practicing regularly, you can build a strong foundation for tackling more complex mathematical challenges in the future.