Polar form of complex numbers

Description of the polar form of complex numbers with examples


The polar form of a complex number


In the article on the geometric representation of complex numbers, it has been described that every complex number \(z\) in the Gaussian plane of numbers can be represented as a vector. This vector is uniquely determined 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 clearly defined by its length and the angle \(φ\) to the real axis.

The following figure shows the vector with the length \(r = 2\) and the angle \(φ = 45°\)

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 corresponds to the absolute value \(|z|\) the complex number.

The general spelling is called normal form \(z = a + bi\)

We write in polar form \(r = |z|\) und \(φ = arg(z)\)


Convert Polar form to Normal form


This article describes the conversion from the polar form to the normal form of a complex number.

If the magnitude and angle of a complex number are known, the real and imaginary values can be calculated.

The representation by means of vectors always results in a right triangle, which consists of the two catheters \(a\) and \(b\) and the hypotenuse \(z\). The conversion can therefore be performed using trigonometric functions. With reference to the figure below.

\(Re=r·cos(φ)\)   \(Im=r·sin(φ)\)

To convert a complex number from polar to normal, the following applies

\(z=r·cos(φ)+ir·sin(φ)=a+bi\)


Conversion from coordinates to polar coordinates


This article describes the determination of the polar coordinates of a complex number by calculating the angle \(&phi\) 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

\(r=|z|=\sqrt{a^2+b^2}=\sqrt{Re^2+Im^2}\)

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

\(\displaystyle φ=arccos\left(\frac{a}{r}\right)=arccos\left(\frac{Re}{|z|}\right)\)

or

\(\displaystyle φ=arctan\left(\frac{b}{a}\right)=arctan\left(\frac{Im}{Re}\right)\)

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

\(|z|=\sqrt{3^2+4^2}=5\)

The angle is

\(\displaystyle φ=arccos\left(\frac{Re}{|z|}\right)=arccos\left(\frac{3}{5}\right)=53.1°\)

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

\(|z|=\sqrt{3^2-4^2}=5\)

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 \(b ≥ 0\) is   \(\displaystyle φ=arccos\left(\frac{a}{|z|}\right)\)


If \(b < 0\) is   \(\displaystyle φ= 360 - arccos\left(\frac{a}{|z|}\right)\)


                   or \(\displaystyle φ= 2π - arccos\left(\frac{a}{|z|}\right)\) 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 \(± 180°\).

\(Arg (3 + 4i) = 53.1\)

\(Arg (3 − 4i) = −53.1\)

\(Arg (−3 + 4i)=127\)

\(Arg (−3 − 4i)=−127\)