Multiplying complex numbers

Level: A Level Maths

Multiplying two complex numbers is similar to multiplying brackets in an algebraic equation.

Example

Suppose we have two complex numbers:

${\displaystyle z_{1}=4+3i}$

${\displaystyle z_{2}=2+3i}$

To multiply them, we multiply out the brackets:

${\displaystyle z_{1}.z_{2}=(4+3i)(2+3i)=8+6i+12i+9i^{2}}$

We can simplify this by combining the terms in ${\displaystyle i}$:

${\displaystyle z_{1}.z_{2}=8+18i+9i^{2}}$

We can further simplify the expression using the fact that ${\displaystyle i^{2}=-1}$:

${\displaystyle z_{1}.z_{2}=8+18i-9=-1+18i}$

General case

To multiply two complex numbers ${\displaystyle (a+bi)}$ and ${\displaystyle (c+di)}$:

${\displaystyle (a+bi)(c+di)=(ac-bd)+(ad+bc)i}$