Complex Number Quotient: (4-3i) / Conjugate Explained

by Luna Greco 54 views

Hey guys! Today, we're diving into the fascinating world of complex numbers. Specifically, we're going to tackle a problem that involves finding the quotient of a complex number divided by its conjugate. Don't worry if that sounds intimidating – we'll break it down step-by-step so it's super easy to understand. Our main goal here is to not just give you the answer, but to really understand the underlying concepts and how to apply them. Complex numbers might seem abstract, but they are a fundamental part of mathematics and have tons of real-world applications, especially in fields like electrical engineering and quantum mechanics. So, let's jump right in and get our hands dirty with some complex number arithmetic!

Understanding Complex Numbers and Conjugates

Before we dive into the problem, let's make sure we're all on the same page about what complex numbers and conjugates actually are. 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. Now, what's this i thing? Well, i is defined as the square root of -1. This might seem a bit weird at first because, in the regular world of real numbers, you can't take the square root of a negative number. That's where the "imaginary" part comes in! The a part of a + bi is called the real part, and the b part is called the imaginary part. Think of it like this: a complex number is a combination of a real number and an imaginary number.

Okay, so we've got complex numbers down. Now, what about conjugates? The conjugate of a complex number is simply found by changing the sign of the imaginary part. So, if you have a complex number a + bi, its conjugate is a - bi. See? Super simple! The real part stays the same, but the imaginary part flips its sign. For example, the conjugate of 2 + 3i is 2 - 3i, and the conjugate of -1 - i is -1 + i. Understanding conjugates is crucial because they have some really neat properties, especially when it comes to dividing complex numbers. One of the most important properties is that when you multiply a complex number by its conjugate, you always get a real number. This is because the imaginary parts cancel out, which makes our lives much easier when we're dealing with division. We'll see this in action in just a bit!

The Problem: (4 - 3i) Divided by its Conjugate

Alright, let's get to the heart of the problem. We're asked to find the quotient of the complex number 4 - 3i divided by its conjugate. Remember, the quotient is just the result you get when you divide one number by another. So, in this case, we need to divide 4 - 3i by its conjugate. First things first, what is the conjugate of 4 - 3i? Well, following our definition from earlier, we just flip the sign of the imaginary part. So, the conjugate of 4 - 3i is 4 + 3i. Now we know what we're dividing by!

The problem now becomes: (4 - 3i) / (4 + 3i). Now, you might be thinking, "How do I actually divide complex numbers?" It's not as straightforward as dividing regular numbers, but it's not too tricky either. The key is to use the conjugate! To divide complex numbers, we multiply both the numerator (the top part of the fraction) and the denominator (the bottom part) by the conjugate of the denominator. This might seem a little strange, but it's a clever trick that gets rid of the imaginary part in the denominator, making the division much easier. Why does this work? Because, as we mentioned earlier, when you multiply a complex number by its conjugate, you get a real number. So, by multiplying the denominator by its conjugate, we're essentially turning it into a real number. This allows us to simplify the fraction and express the quotient in the standard a + bi form.

Step-by-Step Solution

Okay, let's put this into action and solve the problem step-by-step. We have (4 - 3i) / (4 + 3i). The first step is to multiply both the numerator and the denominator by the conjugate of the denominator, which is (4 - 3i). So, we get: [(4 - 3i) / (4 + 3i)] * [(4 - 3i) / (4 - 3i)]. Notice that we're multiplying by a fraction that's equal to 1, so we're not actually changing the value of the expression, just its form. Now, we need to multiply out the numerators and the denominators. Let's start with the numerator: (4 - 3i) * (4 - 3i). We can use the FOIL method (First, Outer, Inner, Last) to do this:

  • First: 4 * 4 = 16
  • Outer: 4 * (-3i) = -12i
  • Inner: (-3i) * 4 = -12i
  • Last: (-3i) * (-3i) = 9i²

So, the numerator becomes 16 - 12i - 12i + 9i². Remember that i² is equal to -1, so we can substitute that in: 16 - 12i - 12i + 9(-1) = 16 - 24i - 9 = 7 - 24i. Now, let's move on to the denominator: (4 + 3i) * (4 - 3i). Again, we use the FOIL method:

  • First: 4 * 4 = 16
  • Outer: 4 * (-3i) = -12i
  • Inner: (3i) * 4 = 12i
  • Last: (3i) * (-3i) = -9i²

So, the denominator becomes 16 - 12i + 12i - 9i². Notice that the -12i and +12i terms cancel each other out, which is exactly what we wanted! Again, we substitute i² = -1: 16 - 9(-1) = 16 + 9 = 25. Now we have our quotient: (7 - 24i) / 25. To express this in the standard a + bi form, we simply divide both the real and imaginary parts by 25: 7/25 - (24/25)i. And that's our answer!

The Answer and Why It's Important

So, the quotient of the complex number 4 - 3i divided by its conjugate is 7/25 - (24/25)i, which corresponds to option D. Awesome! We've successfully navigated through the world of complex number division. But why is this important? Well, understanding how to divide complex numbers is crucial for many advanced mathematical concepts and applications. As we mentioned earlier, complex numbers are used extensively in electrical engineering, particularly in circuit analysis and signal processing. They're also fundamental in quantum mechanics, where they're used to describe the wave functions of particles. Beyond these specific applications, working with complex numbers helps develop your mathematical intuition and problem-solving skills, which are valuable in any field. The process of multiplying by the conjugate, in particular, is a powerful technique that can be applied in various algebraic manipulations.

Key Takeaways and Practice

Let's recap the key takeaways from this problem. First, remember what complex numbers and conjugates are. A complex number is of the form a + bi, and its conjugate is a - bi. The key to dividing complex numbers is to multiply both the numerator and the denominator by the conjugate of the denominator. This eliminates the imaginary part from the denominator, allowing you to express the quotient in the standard a + bi form. Practice makes perfect, so try working through some similar problems on your own. You can change the complex number or try dividing a complex number by another complex number (not just its conjugate). The more you practice, the more comfortable you'll become with these concepts. Remember, mathematics is like learning a language – the more you use it, the better you'll get! Keep exploring the fascinating world of complex numbers, and you'll be amazed at the power and beauty of mathematics.

What is the quotient of the complex number 2 + 3i divided by its conjugate?

Options

A. -5/13 + 12/13i B. -5/13 - 12/13i C. 5/13 - 12/13i D. 5/13 + 12/13i