Complex number multiplication

Formula for multiplication of complex numbers

This paragraph describes how to multiply two complex numbers. As an example we use the two numbers \(3 + i\) and \(1 - 2i\). So it should be calculated

\((3+i)·(1-2i)\)

According to the permanence principle, the calculation rules of real numbers should continue to apply. Therefore, we will first multiply the parenthesis as normal. So we write

\((3+i)·(1-2i)=\)
\((3·1)+(3·(-2i))+i+(i·(-2i))=\)
\(3-6i+i-2i^2\)

Besides expressions with \(i\) the formula also contains \(i^2\). We can easily replace this \(i^2\). By the definition of \(i\) we have \(i^2 = -1\). So we replace \(i^2\) by the number \(-1\) and continue to calculate with the result from above as usual.

\(3-6i+i-2i^2=\)
\(3-6i+i-2·(-1)=\)
\(3-5i+2=5-5i\)

The result of the calculation is \(5 - 5i\).

This article describes the multiplication of complex numbers in normal form. Easier to calculate is the multiplication of complex numbers in polar form.