Dividing Complex Numbers A Step By Step Guide To (5-5i) ÷ (3-i)

Navigating the realm of complex numbers can sometimes feel like traversing uncharted waters. When faced with dividing complex numbers, the process might seem daunting at first. However, by understanding the fundamental principles and applying a systematic approach, even the most intricate divisions become manageable. This article delves into the process of dividing complex numbers, specifically focusing on the quotient of (5 - 5i) ÷ (3 - i). We will break down the steps involved, explore the underlying concepts, and ultimately arrive at the correct solution, equipping you with the knowledge and confidence to tackle similar problems.

What are Complex Numbers?

Before diving into the division, it's essential to have a solid grasp of what complex numbers are. A complex number is essentially a number that can be expressed in the form a + bi, where a and b are real numbers, and i represents the imaginary unit. The imaginary unit i is defined as the square root of -1 (√-1). The a component is known as the real part, while the b component is known as the imaginary part. Complex numbers extend the real number system by incorporating this imaginary unit, allowing us to represent solutions to equations that have no real roots. For example, the equation x² + 1 = 0 has no real solutions because the square of any real number is non-negative. However, introducing the imaginary unit allows us to express the solutions as x = ±i.

Complex numbers are not merely abstract mathematical constructs; they have significant applications in various fields, including electrical engineering, quantum mechanics, and fluid dynamics. In electrical engineering, complex numbers are used to represent alternating current circuits, where the imaginary part corresponds to the phase difference between voltage and current. In quantum mechanics, complex numbers are essential for describing the wave function of a particle. In fluid dynamics, they are used to analyze fluid flow patterns. Understanding complex numbers, therefore, opens doors to comprehending and solving problems in diverse scientific and engineering disciplines.

The Complex Conjugate: A Key Concept

One crucial concept in working with complex numbers, particularly in division, is the complex conjugate. The complex conjugate of a complex number a + bi is obtained by simply changing the sign of the imaginary part, resulting in a - bi. The complex conjugate plays a pivotal role in the division process because it allows us to eliminate the imaginary part from the denominator, transforming the division problem into a more manageable form. When a complex number is multiplied by its complex conjugate, the result is always a real number. This property is the cornerstone of complex number division.

To illustrate this, let's consider a complex number, say 2 + 3i. Its complex conjugate would be 2 - 3i. If we multiply these two complex numbers together, we get:

(2 + 3i)(2 - 3i) = 22 + 2(-3i) + 3i2 + 3i(-3i) = 4 - 6i + 6i - 9i²

Since i² is equal to -1, the expression simplifies to:

4 - 9(-1) = 4 + 9 = 13

As you can see, the result is a real number. This principle of multiplying by the complex conjugate to eliminate the imaginary part in the denominator is the foundation of complex number division. Without understanding this concept, navigating complex number division would be a significantly more challenging endeavor.

Step-by-Step Solution: Dividing (5 - 5i) by (3 - i)

Now, let's tackle the problem at hand: dividing the complex number (5 - 5i) by (3 - i). The key to performing this division is to multiply both the numerator and the denominator by the complex conjugate of the denominator. This process effectively eliminates the imaginary component from the denominator, simplifying the expression.

1. Identify the Complex Conjugate: The denominator is (3 - i). Its complex conjugate is (3 + i). Remember, we simply change the sign of the imaginary part.
2. Multiply Numerator and Denominator: Multiply both the numerator (5 - 5i) and the denominator (3 - i) by the complex conjugate (3 + i):

[(5 - 5i) / (3 - i)] * [(3 + i) / (3 + i)]

3. Expand the Numerator: Apply the distributive property (also known as FOIL - First, Outer, Inner, Last) to expand the numerator:

(5 - 5i)(3 + i) = 53 + 5i - 5i3 - 5ii = 15 + 5i - 15i - 5i²

Remember that i² = -1, so we can substitute:

15 + 5i - 15i - 5(-1) = 15 + 5i - 15i + 5 = 20 - 10i

4. Expand the Denominator: Similarly, expand the denominator:

(3 - i)(3 + i) = 33 + 3i - i3 - ii = 9 + 3i - 3i - i²

Again, substitute i² = -1:

9 - (-1) = 9 + 1 = 10

5. Simplify the Expression: Now we have the complex number in the form (20 - 10i) / 10. We can simplify this by dividing both the real and imaginary parts by the denominator:

(20 - 10i) / 10 = 20/10 - (10i)/10 = 2 - i

Therefore, the quotient of (5 - 5i) ÷ (3 - i) is 2 - i. This corresponds to option A in the given choices.

Common Mistakes to Avoid

When dividing complex numbers, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and ensure accurate calculations. One frequent error is forgetting to multiply both the numerator and the denominator by the complex conjugate. This is a crucial step, as it's the only way to eliminate the imaginary part from the denominator. Multiplying only the numerator will not achieve the desired simplification and will lead to an incorrect result.

Another common mistake is incorrectly applying the distributive property (FOIL method) when expanding the products in the numerator and denominator. Carelessly multiplying terms or forgetting to multiply all terms can result in errors in the subsequent steps. It's essential to meticulously apply the distributive property, ensuring that each term is multiplied correctly. A small mistake in this step can propagate through the rest of the calculation, leading to a wrong final answer.

Additionally, errors can occur when simplifying the expression after expanding. Forgetting that i² = -1 and not substituting it correctly can lead to an incorrect simplification. Remember to always replace i² with -1, as this is a fundamental property of complex numbers. Failing to do so will prevent you from combining like terms and arriving at the correct simplified form.

Finally, pay close attention to the signs of the terms, especially when dealing with negative signs and imaginary units. A simple sign error can significantly alter the outcome. Double-checking your calculations and being mindful of the signs throughout the process can help you avoid these errors.

Conclusion: Mastering Complex Number Division

Dividing complex numbers, while seemingly intricate at first, becomes a straightforward process with a clear understanding of the underlying principles. The key takeaway is the use of the complex conjugate to eliminate the imaginary part from the denominator. By multiplying both the numerator and denominator by the complex conjugate, we transform the division into a manageable form that can be easily simplified.

Throughout this article, we have meticulously walked through the steps of dividing (5 - 5i) ÷ (3 - i), demonstrating the application of the complex conjugate and the simplification process. We also highlighted common mistakes to avoid, equipping you with the knowledge to tackle similar problems with confidence.

Complex numbers are more than just abstract mathematical entities; they have practical applications in various fields. Mastering operations with complex numbers, including division, opens doors to understanding and solving problems in diverse scientific and engineering disciplines. So, continue practicing, exploring, and delving deeper into the fascinating world of complex numbers!

Therefore, the correct answer to the question "What is the quotient of the complex numbers (5-5i) ÷ (3-i)?" is A. 2 - i.