Dividing complex numbers might look hard at first, but once you understand complex conjugates, it gets easier. Let me break it down for you step by step.

What Are Complex Numbers and Conjugates?

A complex number looks like this: $a + bi$.

• $a$ is the real part.
• $b$ is the imaginary part.
• $i$ is the imaginary unit, which means $i^2 = -1$.

The conjugate of a complex number like $z = a + bi$ is $z^* = a - bi$. This conjugate is really important!

How to Divide Complex Numbers

Let’s say we want to divide two complex numbers:

$$\frac{z_1}{z_2} = \frac{a + bi}{c + di}$$

Here’s how to do it, step by step:

1. Start with Your Numbers

   • You have two complex numbers: $z_1 = a + bi$ and $z_2 = c + di$.

2. Find the Conjugate of the Bottom Number

   • The conjugate of $z_2$ is $z_2^* = c - di$.

3. Multiply by the Conjugate

   • To remove the imaginary unit in the denominator, you need to multiply both the top and bottom by the conjugate of the bottom:
   $$\frac{z_1}{z_2} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$$

4. Expand the Top (Numerator)

   • Use the distributive property to expand:

   $(a + bi)(c - di) = ac - adi + bci - bdi^2$

   Since $i^2 = -1$, this becomes:

   $ac + bd + (bc - ad)i$

5. Expand the Bottom (Denominator)

   • For the bottom part, use the difference of squares:

   $(c + di)(c - di) = c^2 - (di)^2 = c^2 + d^2$

6. Put It All Together

   • Now, combine what you got:
   $$\frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$$

7. Separate the Real and Imaginary Parts

   • You can write this in a clearer way:
   $$\frac{z_1}{z_2} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i$$

And that’s it! You have successfully divided two complex numbers using their conjugates. This method is really helpful because it gets rid of imaginary numbers in the bottom, making your answers easier to understand.

Practice Makes Perfect! The best way to get good at this is to practice with different complex numbers. So try some examples out! You’ll find that with practice, dividing complex numbers will become a lot more manageable!