Multiplication in polar form

Multiplication of complex numbers in polar form

The representation of complex numbers in polar form also simplifies the multiplication of complex numbers. In multiplication, the angles are added and the length of the vectors multiplied.

The figure below shows the geometric multiplication of the complex numbers \(2+2i\) and \(3+1i\)

For multiplication in polar form the following applies

\(z_1·z_2=|z_1·|z_2|\)   und   \(Arg(z_1)+Arg(z_2)\)

The division of complex numbers in polar form

From the handling of multiplication, the division of two complex numbers in polar form can be derived. Complex numbers are divided by dividing their absolute values and subtracting their angles. The following applies

\(\displaystyle \frac{z_1}{z_2}=\frac{|z_1|}{z_2}\)   und   \(Arg(z_1)- Arg(z_2)\)