# Operations on Complex Numbers in Polar Form - Calculator

An online calculator to add, subtract, multiply and divide complex numbers in polar form is presented.
In what follows, the imaginary unit $i$ is defined as: $i^2 = -1$ or $i = \sqrt{-1}$.

## Complex Numbers in Polar Form

Complex numbers may be represented in standard from as
$Z = a + i b$ where $a$ and $b$ are real numbers and in polar form as
$Z = \rho \: \; \angle \; \: \theta$ , where $\rho$ is the magnitude of $Z$ and $\theta$ its argument in degrees or radians.
with the following relationships
Given $Z = a + i b$, we have $\rho = \sqrt {a^2+b^2}$ and $\theta = \arctan \left(\dfrac{b}{a}\right)$ taking into account the quadrant where the point $(a,b)$ is located.
Given $Z = \rho \: \; \angle \; \: \theta$ , we have $a = \rho \cos \theta$ and $a = \rho \sin \theta$

## Formulas to Add, Subtract, Multiply and Divide Complex Numbers in Polar Form

### Adding Complex Numbers in Polar Form

$z_1$ and $z_2$ are two complex numbers given by
$z_1 = \rho_1 \; \angle \; \theta_1$ and $z_2 = \rho_2 \; \angle \; \theta_2$
Write $Z_1$ and $Z_2$ in standard complex forms
$Z_1 = \rho_1 \cos \theta_1 + i \; \rho_1 \sin \theta_1$
$Z_2 = \rho_2 \cos \theta_2 + i \; \rho_2 \sin \theta_2$
$Z_1 + Z_2 = \rho_1 \cos \theta_1 + \rho_2 \cos \theta_2 + i \; ( \rho_1 \sin \theta_1 + \rho_2 \sin \theta_2)$
in polar form
$Z_1 + Z_2 = \rho \; \; \angle \; \theta$
where
$\rho = \sqrt {(\rho_1 \cos \theta_1 + \rho_2 \cos \theta_2)^2 + (\rho_1 \sin \theta_1 + \rho_2 \sin \theta_2)^2}$
and
$\theta = \arctan (\dfrac{\rho_1 \sin \theta_1 + \rho_2 \sin \theta_2}{\rho_1 \cos \theta_1 + \rho_2 \cos \theta_2})$

### Subtracting Complex Numbers in Polar Form

In standard complex form
$Z_1 - Z_2 = \rho_1 \cos \theta_1 - \rho_2 \cos \theta_2 + i \; ( \rho_1 \sin \theta_1 - \rho_2 \sin \theta_2)$
in polar form
$Z_1 - Z_2 = \rho \; \; \angle \; \theta$
where
$\rho = \sqrt {(\rho_1 \cos \theta_1 - \rho_2 \cos \theta_2)^2 + (\rho_1 \sin \theta_1 - \rho_2 \sin \theta_2)^2}$
and
$\theta = \arctan (\dfrac{\rho_1 \sin \theta_1 - \rho_2 \sin \theta_2}{\rho_1 \cos \theta_1 - \rho_2 \cos \theta_2})$

It is much easier to multiply and divide complex numbers in polar form.

### Multiplying Complex Numbers in Polar Form

$Z_1 \times Z_2 = \rho \; \; \angle \; \theta$ where
$\rho = \rho_1 \times \rho_2$
and
$\theta = \theta_1 + \theta_2$

### Dividing Complex Numbers in Polar Form

$\dfrac{Z_1}{Z_2} = \rho \; \; \angle \; \theta$ where
$\rho = \dfrac{\rho_1}{\rho_2}$
and
$\theta = \theta_1 - \theta_2$

## Use of Complex Numbers in Polar Form Calculator

1 - Enter the magnitude and argument $\rho_1$ and $\theta_1$ of the complex number $Z_1$ and the magnitude and argument $\rho_2$ and $\theta_2$ of the complex number $Z_2$ as real numbers with the arguments $\theta_1$ and $\theta_2$ in either radians or degrees and then press "Calculate".
The outputs are:
$Z_1$ and $Z_2$ in complex standard form $a + i b$.
and
$Z_1+Z_2$ , $Z_1-Z_2$ , $Z_1 \times Z_2$ and $\dfrac{Z_1}{Z_2}$ in polar form with argument in degrees.

 $\rho_1 =$ 10 $\theta_1 =$ -30degrees radians $\rho_2 =$ 4 $\theta_2 =$ 45degrees radians Number of Decimal Places = 3

## More References and links

Convert a Complex Number to Polar and Exponential Forms
Math Calculators and Solvers .
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