Conjugate of a Complex Number

Conjugate of a Complex Number

## Definition / Concept

The conjugate of a complex number is obtained by changing the sign of its imaginary part.

## Formula

If:

$$z=a+ib$$

then:

$$\bar z=a-ib$$

## Conditions / Special Cases

$$\overline{\bar z}=z$$

$$\overline{z_1+z_2}=\bar z_1+\bar z_2$$

$$\overline{z_1z_2}=\bar z_1\bar z_2$$

$$\overline{\left(\frac{z_1}{z_2}\right)}=\frac{\bar z_1}{\bar z_2}$$

## Usage

- Used to simplify division and calculate modulus.

- Used extensively in complex-number identities.