# Polar Form of Complex Number/Examples

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## Examples of Polar Form of Complex Number

### Example: $i$

The imaginary unit $i$ can be expressed in polar form as $\polar {1, \dfrac \pi 2}$.

### Example: $-i$

The imaginary number $-i$ can be expressed in polar form as $\polar {1, \dfrac {3 \pi} 2}$.

### Example: $-1$

The real number $-1$ can be expressed as a complex number in polar form as $\polar {1, \pi}$.

### Example: $-4$

The real number $-4$ can be expressed as a complex number in polar form as $\polar {4, \pi}$.

### Example: $\sqrt 2 i$

The imaginary number $\sqrt 2 i$ can be expressed in polar form as $\polar {\sqrt 2, \dfrac \pi 2}$.

### Example: $2 - 2 i$

The complex number $2 - 2 i$ can be expressed as a complex number in polar form as $\polar {2 \sqrt 2, \dfrac {7 \pi} 4}$.

### Example: $-1 + \sqrt 3 i$

The complex number $-1 + \sqrt 3 i$ can be expressed as a complex number in polar form as $\polar {2, \dfrac {2 \pi} 3}$.

### Example: $2 + 2 \sqrt 3 i$

The complex number $2 + 2 \sqrt 3 i$ can be expressed as a complex number in polar form as $\polar {4, \dfrac \pi 3}$.

### Example: $2 \sqrt 2 + 2 \sqrt 2 i$

The complex number $2 \sqrt 2 + 2 \sqrt 2 i$ can be expressed as a complex number in polar form as $\polar {4, \dfrac \pi 4}$.

### Example: $\dfrac {\sqrt 3} 2 - \dfrac {3 i} 2$

The complex number $\dfrac {\sqrt 3} 2 - \dfrac {3 i} 2$ can be expressed as a complex number in polar form as $\polar {\sqrt 3, \dfrac {5 \pi} 3}$.

### Example: $-5 + 5 i$

The complex number $-5 + 5 i$ can be expressed as a complex number in polar form as $\polar {5 \sqrt 2, \dfrac {3 \pi} 4}$.

### Example: $-\sqrt 6 - \sqrt 2 i$

The complex number $-\sqrt 6 - \sqrt 2 i$ can be expressed as a complex number in polar form as $\polar {2 \sqrt 2, \dfrac {7 \pi} 6}$.

### Example: $-3 i$

The complex number $-3 i$ can be expressed as a complex number in polar form as $\polar {3, \dfrac {3 \pi} 2}$.

### Example: $-2 \sqrt 3 - 2 i$

The complex number $-2 \sqrt 3 - 2 i$ can be expressed as a complex number in polar form as $\polar {4, \dfrac {7 \pi} 6}$.

### Example: $2 + i$

The complex number $2 + i$ can be expressed as a complex number in polar form as $\polar {\sqrt 5, \arctan {\dfrac 1 2} }$.

### Example: $-3 - 4 i$

The complex number $-3 - 4 i$ can be expressed as a complex number in polar form as $\polar {5, \pi + \arctan {\dfrac 4 3} }$.

### Example: $1 - 2 i$

The complex number $1 - 2 i$ can be expressed as a complex number in polar form as $\polar {\sqrt 5, -\arctan 2}$.

### Example: $6 \paren {\cos 240 \degrees + i \sin 240 \degrees}$

The complex number $6 \paren {\cos 240 \degrees + i \sin 240 \degrees}$ can be expressed as:

$6 \paren {\cos 240 \degrees + i \sin 240 \degrees} = 6 \cis 240 \degrees = 6 \cis \dfrac {4 \pi} 3 = 6 e^{4 \pi / 3}$

and depicted in the complex plane as:

### Example: $4 \cis \dfrac {3 \pi} 5$

The complex number $4 \paren {\cos \dfrac {3 \pi} 5 + i \sin \dfrac {3 \pi} 5}$ can be expressed as:

$4 \paren {\cos \dfrac {3 \pi} 5 + i \sin \dfrac {3 \pi} 5} = 4 \paren {\cos 108 \degrees + i \sin 108 \degrees} = 6 \cis 108 \degrees = 6 \cis \dfrac {3 \pi} 5 = 6 e^{3 \pi / 5}$

and depicted in the complex plane as:

### Example: $2 \cis \dfrac {-\pi} 4$

The complex number $2 \paren {\cos \dfrac {-\pi} 4 + i \sin \dfrac {-\pi} 4}$ can be expressed as:

$4 \paren {\cos \dfrac {-\pi} 4 + i \sin \dfrac {-\pi} 4} = 2 \paren {\cos 315 \degrees + i \sin 315 \degrees} = 2 \cis 315 \degrees = 2 \cis \dfrac {-\pi} 4 = 2 e^{-\pi / 4}$

and depicted in the complex plane as:

### Example: $6 \cis 135 \degrees$

The complex number $\polar {6, 135 \degrees}$ can be expressed in Cartesian form as:

$6 \cis 135 \degrees = -3 \sqrt 2 + 3 \sqrt 2 i$

and depicted in the complex plane as:

### Example: $12 \cis 90 \degrees$

The complex number $\polar {12, 90 \degrees}$ can be expressed in Cartesian form as:

$12 \cis 90 \degrees = 12 i$

and depicted in the complex plane as:

### Example: $4 \cis 315 \degrees$

The complex number $\polar {4, 315 \degrees}$ can be expressed in Cartesian form as:

$4 \cis 315 \degrees = 2 \sqrt 2 - 2 \sqrt 2 i$

and depicted in the complex plane as:

### Example: $2 \cis \dfrac {5 \pi} 4$

The complex number $\polar {2, \dfrac {5 \pi} 4}$ can be expressed in Cartesian form as:

$2 \cis \dfrac {5 \pi} 4 = -\sqrt 2 - \sqrt 2 i$

and depicted in the complex plane as:

### Example: $5 \cis \dfrac {7 \pi} 6$

The complex number $\polar {5, \dfrac {7 \pi} 6}$ can be expressed in Cartesian form as:

$5 \cis \dfrac {7 \pi} 6 = -5 \dfrac {\sqrt 3} 2 - \dfrac 5 2 i$

and depicted in the complex plane as:

### Example: $3 \cis \dfrac {-2 \pi} 3$

The complex number $\polar {3, -\dfrac {2 \pi} 3}$ can be expressed in Cartesian form as:

$3 \, \map \cis {-\dfrac {2 \pi} 3} = -\dfrac 3 2 - \dfrac {3 \sqrt 3} 2 i$

and depicted in the complex plane as: