Convert complex number to polar coordinates

This article describes the determination of the polar coordinates of a complex number by calculating the angle \(φ\) and the length of the vector \(z\).

The radius r of the polar form is identical to the magnitude \(|z|\) of the complex number. The formula for calculating the radius is thus the same as that described in the article of the absolute value of a complex number.

For the length \(r\) of the vector results

If the vector is in the 1. or 2. quadrant, the angle \(φ\) applies

or

When calculating the angle, it must be taken into account in which quadrant the vector is located. Consider the following figure:

For the complex number \(3 + 4i\) in the picture above, the ,agnitude is

The angle is

For the complex numbe \(3 - 4i\) the magnitude is also

The calculation of the angle also gives \(53.1°\). In this case \(180°\) must be added to the calculated angle to get into the right quadrant.

After calculating the angle \(φ\) with the aid of the arc sine, a test of the quadrant must always be carried out. For a negative imaginary part, the angle must be corrected.

For a complex number \(a + bi\) applies

If

If

                   or when calculated in radians.

In the calculations above, the angle between \(0°\) and \(360°\) is given as the angle \(φ\) to the real axis. The angle can also be specified between \(0°\) and \(± 180°\).