What Are Complex Conjugates?

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

Here, $a$ and $b$ are regular numbers, and $i$ represents something called the imaginary unit.

The complex conjugate of this number is $a - bi$. This switches the part with $i$ to its opposite.

Key Features of Complex Conjugates:

1. If you have a complex number $z = a + bi$, its conjugate is written as $\overline{z} = a - bi$.

2. When you multiply a complex number by its conjugate, like $z \cdot \overline{z}$, you get a result that looks like $a^2 + b^2$. This answer is always a regular, non-negative number.

Using Complex Conjugates in Division:

When you want to divide complex numbers, you can make it easier by multiplying by the conjugate.

This looks like this: $\frac{z_1}{z_2} = \frac{z_1 \cdot \overline{z_2}}{z_2 \cdot \overline{z_2}}$

Doing this helps get rid of the complex part in the bottom of the fraction.

This makes the math simpler!