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.


\( \rho_1 = \)
\( \theta_1 = \)

\( \rho_2 = \)
\( \theta_2 = \)
Number of Decimal Places =

Results of Calculations


    


    

    


    

    



More References and links

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

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