Dividing complex numbers

Formula for division of complex numbers

This article describes dividing complex numbers. In the next example we will divide the number \(3 + i\) by the number \(1 - 2i\). Wanted is so

\(\displaystyle(3+i)\,/\,(1-2i)=\frac{3+i}{1-2i}\)

According to the permanence principle, the calculation rules of the real numbers should be valid here. It bothers us that in the denominator of the fraction the \(i\) occurs. Sharing by a real number would be quite easy.

This is where the complex conjugate comes into play. The fraction is extended by the conjugate complex number \(1 + 2i\) of the denominator. This allows the \(i\) to be truncated in the denominator and the denominator becomes a real number. Only in the numerator remains a complex number, which can be easily multiplied out.

The division looks like this

\(\displaystyle\frac{3+i}{1-2i}=\frac{(3+i)·(1+2i)}{(1-2i)·(1+2i)}=\frac{3+6i+i-2}{1+2i-2i+4}=\frac{1+7i}{5}=\frac{1}{5}+\frac{7}{5}i\)

The result is \(\displaystyle\frac{1}{5}+\frac{7}{5}i\)

This article described the division of complex numbers in normal form. Easier to calculate is the division of complex numbers in polar form.