To divide complex numbers, we use a method called multiplying by the conjugate.

Let's look at two complex numbers:

• ( z_1 = a + bi )
• ( z_2 = c + di )

Here, ( a ), ( b ), ( c ), and ( d ) are regular numbers (called real numbers).

When we want to divide these complex numbers, we write it like this:

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

Here’s how to divide using conjugates:

1. Find the Conjugate: First, we need the conjugate of ( z_2 ), which is ( c - di ).

2. Multiply by the Conjugate: Next, we multiply both the top (numerator) and bottom (denominator) by this conjugate:

$$\frac{a + bi}{c + di} \cdot \frac{c - di}{c - di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$$

3. Simplify the Bottom: Now we simplify the bottom:

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

This happens because it forms a pattern called the difference of squares.

4. Expand the Top: Then we expand the top:

$$(a + bi)(c - di) = ac + adi - bci - b(d^2)i = (ac + bd) + (ad - bc)i$$

5. Final Result: So in the end, we get:

$$\frac{z_1}{z_2} = \frac{(ac + bd) + (ad - bc)i}{c^2 + d^2}$$

This way, we can effectively divide complex numbers, and we end up with a new complex number in the form ( x + yi ).