Dividing Complex Numbers A Step By Step Solution For (4-i) By (3+4i)

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Hey guys! Let's dive into the fascinating world of complex numbers and tackle the problem of dividing them. Complex numbers, with their real and imaginary parts, might seem a bit intimidating at first, but trust me, once you get the hang of it, it's super straightforward. In this article, we'll break down the process of dividing complex numbers, using the specific example of (4-i) / (3+4i). We'll walk through each step, making sure you understand the logic behind it all. So, grab your imaginary calculators (just kidding, you won't need one!), and let's get started!

Understanding Complex Numbers

Before we jump into the division, let's quickly recap what complex numbers are all about. A complex number is basically a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. This imaginary unit, i, is defined as the square root of -1. This means that i² = -1. The a part is called the real part of the complex number, and the bi part is called the imaginary part. For example, in the complex number 3 + 2i, 3 is the real part, and 2i is the imaginary part. Complex numbers are incredibly useful in various fields, including electrical engineering, quantum mechanics, and applied mathematics.

Why do we need complex numbers?

Complex numbers might seem a bit abstract, but they're essential for solving problems that real numbers alone can't handle. Think about finding the square root of a negative number. With real numbers, this is impossible, but with complex numbers, we can easily do it using the imaginary unit i. This opens up a whole new world of mathematical possibilities. Furthermore, complex numbers are not just theoretical constructs; they have real-world applications. In electrical engineering, they're used to analyze alternating current (AC) circuits. In quantum mechanics, they're fundamental to describing the behavior of particles. And in fields like signal processing and control systems, complex numbers provide a powerful tool for modeling and understanding complex phenomena. So, while they might seem a bit unusual at first, complex numbers are a crucial part of the mathematical toolkit.

The Division Challenge: (4-i) / (3+4i)

Alright, now let's get to the heart of the matter: dividing complex numbers. We're faced with the challenge of simplifying the expression (4-i) / (3+4i). Directly dividing these numbers isn't something we can do in the traditional sense. Instead, we use a clever trick: multiplying both the numerator and the denominator by the conjugate of the denominator. This might sound like a mouthful, but it's actually quite simple. The conjugate of a complex number a + bi is a - bi. Basically, we just change the sign of the imaginary part. So, the conjugate of 3 + 4i is 3 - 4i. Multiplying by the conjugate helps us eliminate the imaginary part from the denominator, making the division much easier.

Why use the conjugate?

The trick of multiplying by the conjugate is all about getting rid of the imaginary part in the denominator. When we multiply a complex number by its conjugate, something magical happens: the imaginary terms cancel out. Let's see why this works. If we have a complex number a + bi, its conjugate is a - bi. When we multiply them together, we get (a + bi)(a - bi). Using the distributive property (or the FOIL method), we get a² - abi + abi - (bi)². Notice that the -abi and +abi terms cancel each other out. And since i² = -1, the last term becomes -b²(-1) = b². So, the whole expression simplifies to a² + b², which is a real number! This is why multiplying by the conjugate is so effective for dividing complex numbers. It transforms the denominator into a real number, making the division process much more manageable. So, remember this key concept: the conjugate is your best friend when dividing complex numbers.

Step-by-Step Solution

Let's break down the division of (4-i) / (3+4i) into manageable steps:

  1. Identify the Conjugate: As we discussed, the conjugate of the denominator (3 + 4i) is (3 - 4i).

  2. Multiply Numerator and Denominator: Multiply both the numerator and the denominator by the conjugate:

    (4 - i) / (3 + 4i) * (3 - 4i) / (3 - 4i)

  3. Expand the Numerator: Use the distributive property (FOIL method) to expand the numerator:

    (4 - i) * (3 - 4i) = 4 * 3 + 4 * (-4i) + (-i) * 3 + (-i) * (-4i)

    = 12 - 16i - 3i + 4i²

  4. Simplify the Numerator: Remember that i² = -1. Substitute this into the expression:

    12 - 16i - 3i + 4(-1) = 12 - 16i - 3i - 4

    Combine the real and imaginary terms:

    (12 - 4) + (-16i - 3i) = 8 - 19i

  5. Expand the Denominator: Similarly, expand the denominator using the distributive property:

    (3 + 4i) * (3 - 4i) = 3 * 3 + 3 * (-4i) + 4i * 3 + 4i * (-4i)

    = 9 - 12i + 12i - 16i²

  6. Simplify the Denominator: Again, substitute i² = -1:

    9 - 12i + 12i - 16(-1) = 9 + 16

    = 25

  7. Combine the Results: Now we have the simplified numerator and denominator:

    (8 - 19i) / 25

  8. Express in Standard Form: Finally, express the result in the standard complex number form (a + bi):

    8/25 - (19/25)i

Breaking it down further

Let's take a closer look at each step to ensure we're crystal clear on the process. The most crucial part of dividing complex numbers is getting rid of the imaginary component in the denominator. This is why we multiply both the numerator and the denominator by the conjugate. Think of it like rationalizing the denominator with radicals – we're doing something similar here, but with imaginary numbers. When we expand the numerator, we use the FOIL method (First, Outer, Inner, Last) to multiply each term correctly. This ensures we don't miss any terms and get the correct result. Simplifying the numerator involves combining like terms (real and imaginary) and remembering that i² is equal to -1. This substitution is essential for reducing the expression. The denominator, when multiplied by its conjugate, always results in a real number, which simplifies the fraction significantly. Finally, we express the result in the standard complex number form, separating the real and imaginary parts, so it's clear and easy to understand.

The Answer

So, after all that calculation, we've found that (4-i) / (3+4i) simplifies to 8/25 - (19/25)i. Looking at the options provided, the correct answer is:

B. 8/25 - (19/25)i

Double-checking your work

It's always a good idea to double-check your work, especially when dealing with complex numbers, where it's easy to make a small mistake. One way to verify your answer is to multiply your result by the original denominator. If you've done everything correctly, you should get back the original numerator. In this case, we can multiply (8/25 - (19/25)i) by (3 + 4i) and see if we get (4 - i). This extra step can give you confidence in your answer and help you catch any errors you might have made. Another strategy is to use a complex number calculator or an online tool to verify your calculations. These tools can quickly perform the division and give you the result, allowing you to compare it with your own solution. By employing these double-checking methods, you can ensure accuracy and build a solid understanding of the process.

Key Takeaways

Dividing complex numbers might seem tricky at first, but it becomes much easier with practice. The key is to remember the conjugate and how it helps eliminate the imaginary part from the denominator. Here's a quick recap of the main steps:

  1. Find the conjugate of the denominator.
  2. Multiply both the numerator and the denominator by the conjugate.
  3. Expand and simplify both the numerator and the denominator.
  4. Express the result in the standard form (a + bi).

Practice makes perfect

The more you practice dividing complex numbers, the more comfortable you'll become with the process. Try working through different examples with varying degrees of complexity. You can find practice problems in textbooks, online resources, or even create your own. The important thing is to reinforce the steps and understand the underlying concepts. As you gain experience, you'll start to recognize patterns and develop a better intuition for how complex numbers behave. You'll also become more efficient at performing the calculations and avoiding common mistakes. So, don't be afraid to tackle challenging problems and push yourself to improve. With consistent practice, you'll master the art of dividing complex numbers and feel confident in your ability to handle these types of problems.

Conclusion

And there you have it! We've successfully divided the complex number (4-i) by (3+4i). Remember, the key is to multiply by the conjugate of the denominator. With a bit of practice, you'll be dividing complex numbers like a pro. Keep practicing, and you'll master this important concept in no time! You got this!