Divide Complex Numbers: A Step-by-Step Guide

Welcome, math enthusiasts! Today, we're diving into the fascinating world of complex numbers, specifically how to divide complex numbers. While the concept might seem a bit daunting at first, especially with those pesky 'i's floating around, I promise it's quite manageable once you get the hang of the core technique. We'll be working through a specific example: rac{6-2 i}{-2-5 i}. This process is fundamental in many areas of mathematics, physics, and engineering, so understanding it will open up a lot of doors for you.

Understanding the Basics of Complex Number Division

Before we tackle our example, let's briefly touch upon why we need a special method for dividing complex numbers. Remember that a complex number has the form $a + bi$, where 'a' is the real part and 'b' is the imaginary part, and $i$ is the imaginary unit, where $i^2 = -1$. When we divide two complex numbers, say rac{z_1}{z_2}, our goal is to express the result in the standard form $a + bi$. The key challenge lies in eliminating the imaginary part from the denominator. We achieve this by using the conjugate of the denominator. The conjugate of a complex number $c + di$ is $c - di$. When you multiply a complex number by its conjugate, the result is always a real number: $(c + di)(c - di) = c^2 - (di)^2 = c^2 - d^2i^2 = c^2 - d^2(-1) = c^2 + d^2$. This property is our secret weapon for simplifying complex number division.

Step-by-Step Calculation: rac{6-2 i}{-2-5 i}

Now, let's put theory into practice with our example: rac{6-2 i}{-2-5 i}. Our first step is to identify the denominator, which is $-2 - 5i$. The conjugate of this denominator is $-2 + 5i$. To perform the division, we multiply both the numerator and the denominator by this conjugate. This doesn't change the value of the fraction because we're essentially multiplying by 1 (since rac{-2+5i}{-2+5i} = 1).

So, the expression becomes:

rac{6-2 i}{-2-5 i} imes rac{-2+5 i}{-2+5 i}

Next, we need to expand both the numerator and the denominator. We'll use the distributive property (or FOIL method) for both.

Numerator:

$(6 - 2i)(-2 + 5i) = 6(-2) + 6(5i) - 2i(-2) - 2i(5i)$

$= -12 + 30i + 4i - 10i^2$

Since $i^2 = -1$, we substitute that in:

$= -12 + 30i + 4i - 10(-1)$

$= -12 + 30i + 4i + 10$

Now, combine the real parts and the imaginary parts:

$= (-12 + 10) + (30i + 4i)$

$= -2 + 34i$

Denominator:

We already know from our discussion about conjugates that the denominator will be a real number. Let's calculate it:

$(-2 - 5i)(-2 + 5i) = (-2)^2 - (5i)^2$

$= 4 - (25i^2)$

Substitute $i^2 = -1$:

$= 4 - 25(-1)$

$= 4 + 25$

$= 29$

So, our fraction now looks like this:

rac{-2 + 34i}{29}

Expressing the Result in Standard Form ($a + bi$)

To get the final answer in the standard $a + bi$ form, we simply separate the real and imaginary parts of the numerator over the common denominator:

rac{-2}{29} + rac{34i}{29}

And there you have it! The result of dividing rac{6-2 i}{-2-5 i} is rac{-2}{29} + rac{34}{29}i. We've successfully eliminated the imaginary part from the denominator and expressed our answer in the familiar $a + bi$ format. This method of using the conjugate is crucial for any complex number division problem you encounter. Remember to carefully multiply, substitute $i^2 = -1$, and combine like terms. Practice makes perfect, so try working through a few more examples on your own!

Conclusion: Mastering Complex Number Division

Dividing complex numbers, as demonstrated with rac{6-2 i}{-2-5 i}, relies on a clever application of the complex conjugate. By multiplying both the numerator and the denominator by the conjugate of the denominator, we transform a complex fraction into a simpler form, allowing us to express the result as a standard complex number $a + bi$. This technique is not just an academic exercise; it's a practical tool used extensively in fields that utilize complex numbers. For instance, in electrical engineering, impedance calculations often involve dividing complex numbers. Similarly, in quantum mechanics, wave functions and state vectors can be represented using complex numbers, and operations involving them might require division. The beauty of this mathematical process lies in its ability to simplify complex operations into manageable steps. If you're looking to delve deeper into the properties and applications of complex numbers, I highly recommend exploring resources that cover complex analysis. A fantastic place to start is by checking out introductory texts on complex numbers or visiting reputable online educational platforms. For a comprehensive understanding of the broader mathematical context, a visit to Khan Academy's section on complex numbers can be incredibly beneficial, offering clear explanations and practice problems.