\begin{definition}[Definition:Multiplication of Cuts]
Let $0^*$ denote the rational cut associated with the (rational) number $0$.
Let $\alpha$ and $\beta$ be cuts.
The operation of '''multiplication''' is defined on $\alpha$ and $\beta$ as:
:$\alpha \beta := \begin {cases}
\size \alpha \, \size \beta & : \alpha \ge 0^*, \beta \ge 0^* \\
-\paren {\size \alpha \, \size \beta} & : \alpha < 0^*, \beta \ge 0^* \\
-\paren {\size \alpha \, \size \beta} & : \alpha \ge 0^*, \beta < 0^* \\
\size \alpha \, \size \beta & : \alpha < 0^*, \beta < 0^*
\end {cases}$
where:
:$\size \alpha$ denotes the absolute value of $\alpha$
:$\size \alpha \, \size \beta$ is defined as in Multiplication of Positive Cuts
:$\ge$ denotes the ordering on cuts.
In this context, $\alpha \beta$ is known as the '''product of $\alpha$ and $\beta$'''.
\end{definition}