Division of Complex Numbers

Division of Complex Numbers

## Formula

If:

$$z_1=a+ib,\qquad z_2=c+id,\qquad z_2\ne0$$

then:

$$\frac{z_1}{z_2}=\frac{a+ib}{c+id}$$

Multiplying numerator and denominator by the conjugate:

$$\frac{a+ib}{c+id}=\frac{(a+ib)(c-id)}{c^2+d^2}$$

Therefore:

$$\frac{a+ib}{c+id}=\frac{ac+bd}{c^2+d^2}+i\frac{bc-ad}{c^2+d^2}$$

## Conditions / Special Cases

The denominator cannot be zero:

$$c^2+d^2\ne0$$

## Usage

- Used to express the quotient of two complex numbers in standard form.