Polar form of complex number

Enter a complex number to calculate the polar shape. The result can be displayed in degrees or radians.

Description of the polar form of a complex number

Every complex number \(z\) can be represented as a vector in the Gaussian number plane. This vector is uniquely defined by the real part and the imaginary part of the complex number \(z\).

A vector emanating from the zero point can also be used as a pointer. This pointer is uniquely defined by its length and the angle \(φ\) to the real axis.

Positive angles are measured counterclockwise, negative angles are clockwise.

A complex number can thus be uniquely defined in the polar form by the pair \((|z|, φ)\). \(φ\) is the angle belonging to the vector. The length of the vector \(r\) equals the magnitude or absolute value \(|z|\) of the complex number.

The general notation \(z = a + bi\) is called normal form (in contrast to the polar form described above).

More information and examples of the polar form can be found in the tutorial