# Complex Modulus/Examples

## Examples of Complex Modulus

### Complex Modulus of $i$

$\cmod i = \cmod {-i} = 1$

### Complex Modulus of $-5$

$\left\vert{-5}\right\vert = 5$

### Complex Modulus of $1 + i$

$\cmod {1 + i} = \sqrt 2$

### Complex Modulus of $4 + 3 i$

$\cmod {4 + 3 i} = 5$

### Complex Modulus of $-4 + 2 i$

$\cmod {-4 + 2 i} = 2 \sqrt 5$

### Complex Modulus of $3iz - z^2$

Let:

$w = 3 i z - z^2$

where $z = x + i y$.

Then:

$\cmod w^2 = x^4 + y^4 + 2 x^2 y^2 - 6 x^2 y - 6 y^3 + 9 x^2 + 9 y^2$

### Complex Modulus of $\tan \theta + i$

$\left\vert{\tan \theta + i}\right\vert = \left\vert{\sec \theta}\right\vert$

where:

$\theta \in \R$ is a real number
$\tan \theta$ denotes the tangent function
$\sec \theta$ denotes the secant function.

### Complex Modulus of $\dfrac {1 + 2 i t - t^2} {1 + t^2}$

$\cmod {\dfrac {1 + 2 i t - t^2} {1 + t^2} } = 1$

where:

$t \in \R$ is a real number.

Let $z_1 = 4 - 3 i$ and $z_2 = -1 + 2 i$.

### Complex Modulus of $z_1 + z_2$

$\cmod {z_1 + z_2} = \sqrt {10}$

### Complex Modulus of $z_1 - z_2$

$\cmod {z_1 - z_2} = 5 \sqrt 2$

### Complex Modulus of $2 \overline z_1 - 3 \overline z_2 - 2$

$\cmod {2 \overline {z_1} - 3 \overline {z_2} - 2} = 15$