Multiplication of Complex Numbers

Now that we can add complex numbers, let's look at how we multiply them.

Multiplying Complex Numbers

Let $z_1 = a + bi$ and $z_2 = c + di$. Then the product is given by:

$(a + bi)(c + di)$

$= a(c + di) + bi(c + di)$

$= ac + adi+ bci + bdi^2$

$= ac + bdi^2 + adi + bci$

$= ac + bd \cdot(-1) + i(ad + bc) \quad \quad$ where $(i^2 = -1)$

$= (ac - bd) + (ad + bc)i$

Example

Multiply $(3 + 2i)(1 - 4i)$.

$(3 + 2i)(1 + 4i)$

$= 3(1 + 4i) + b(1 + 4i)$

$= 3 + 12i+ i + 4i^2$

$= 3 + 4i^2 + 13i$

$= 3 + 4\cdot(-1) + 13i$

$= 3 - 4 + 13i$

$= -1 + 13i$

Geometric Interpretation

Multiplying complex numbers has a geometric meaning too: it rotates and scales a number on the complex plane. We'll explore this in more detail later.

Summary

To multiply two complex numbers $(a + bi)(c + di)$, expand the product like you would binomials and simplify using $i^2 = -1$.