Important Conjugate Identities

Important Conjugate Identities

## Formula

$$z+\bar z=2\operatorname{Re}(z)$$

$$z-\bar z=2i\operatorname{Im}(z)$$

$$z\bar z=a^2+b^2$$

$$\frac{1}{z}=\frac{\bar z}{z\bar z}$$

For:

$$z=a+ib$$

we get:

$$\frac{1}{z}=\frac{a-ib}{a^2+b^2}$$

## Conditions / Special Cases

For $z\ne0$:

$$z\bar z>0$$

## Usage

- Used to extract real and imaginary parts.

- Used to find reciprocals and simplify complex expressions.