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
\[ z_1(z_2z_3) = (z_1z_2)z_3. \]
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,1 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 $w$.
- Future editions will include an account ↩︎