Properties of Modulus

Properties of Modulus

## Formula

$$|z_1z_2|=|z_1||z_2|$$

$$\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$$

$$|z^n|=|z|^n$$

$$|\bar z|=|z|$$

$$\left|\frac{1}{z}\right|=\frac{1}{|z|}$$

## Triangle Inequality

$$|z_1+z_2|\le|z_1|+|z_2|$$

## Reverse Triangle Inequality

$$\left||z_1|-|z_2|\right|\le|z_1-z_2|$$

## Conditions / Special Cases

Equality in the triangle inequality occurs when $z_1$ and $z_2$ have the same direction or one of them is zero.

## Usage

- Used in inequalities and problems involving the magnitude of complex expressions.