# Complex Products

# Definition

Let $z_1, z_2 \in \C $ with $z_1 = (x_1,
y_1)$ and $z_2 = (x_2, y_2)$.
The complex product of
$z_1$ and $z_2$ is the complex number

\[
(x_1x_2 - y_1y_2, x_1y_2 + y_1x_2).
\]

## Notation

We denote the complex product of $z_1$ and
$z_2$ by $z_1 \cdot z_2$ or $z_1z_2$.

The notation overloads that used for real
numbers.
This overloading is justified by the fact that
the complex product of two purely real complex
numbers $z_1$ and $z_2$ the purely real complex
number whose real part is the product of the
real parts of $z_1$ and $z_2$.

Recall that we denote $z_1 = x_1 + iy_1$ and
$z_2 = x_2 + iy_2$.
This notation is a mnemonic for the definition
of a complex product if we treat $i^2 = -1$.

\[
\begin{aligned}
z_1z_2 &= (x_1 + iy_1)(x_2 + iy_2) \\
&= x_1x_2 + ix_1y_2 + iy_1x_2 + i^2 y_1y_2 \\
&= (x_1x_2 - y_1y_2) + i(x_1y_2 + y_1x_2).
\end{aligned}
\]

# Properties

For all $z_1, z_2 \in \C $, we have $z_1z_2 =
z_2z_1$.

For all $z_1, z_2, z_3 \in \C $, we have
\[
z_1(z_2z_3) = (z_1z_2)z_3.
\]

# Complex multiplication

We call the operation that associates a pair
of complex numbers with their product
complex multiplication.
The operation is symmetric (commutative).

# Multiplicative identity and inverse

Notice that the complex number $(1, 0)$ is the
multiplicative identity.
It is unique,
and so we call it the complex
multiplicative identity.

We call the operation $(z, w) \mapsto z/w$
complex division and we
call $z/w$ the (complex)
quotient of $z$ with $w$.