\begin{definition}[Definition:Extended Real Multiplication]
Let $\overline \R$ denote the extended real numbers.
Define '''extended real multiplication''' or '''multiplication on $\overline \R$''', denoted $\times_{\overline \R}: \overline \R \times \overline \R \to \overline \R$, by:
:$\forall x \in \overline \R: x \times_{\overline \R} 0 = 0 \times_{\overline \R} x = 0$
:$\forall x, y \in \R: x \times_{\overline \R} y := x \times_\R y$ where $\times_\R$ denotes real multiplication
:$\forall x \in \R_{>0}: x \times_{\overline \R} \paren {+\infty} = \paren {+\infty} \times_{\overline \R} x := +\infty$
:$\forall x \in \R_{<0}: x \times_{\overline \R} \paren {+\infty} = \paren {+\infty} \times_{\overline \R} x := -\infty$
:$\forall x \in \R_{>0}: x \times_{\overline \R} \paren {-\infty} = \paren {-\infty} \times_{\overline \R} x := -\infty$
:$\forall x \in \R_{<0}: x \times_{\overline \R} \paren {-\infty} = \paren {-\infty} \times_{\overline \R} x := +\infty$
:$\paren {+\infty} \times_{\overline \R} \paren {+\infty} := +\infty$
:$\paren {-\infty} \times_{\overline \R} \paren {-\infty} := +\infty$
:$\paren {+\infty} \times_{\overline \R} \paren {-\infty} := -\infty$
:$\paren {-\infty} \times_{\overline \R} \paren {+\infty} := -\infty$
When no danger of confusion arises, $\times_{\overline \R}$ is usually replaced with the more familiar $\times$, or even suppressed.
From the definition of $\times_{\overline \R}$ on bona fide real numbers, the name '''extended real multiplication''' is appropriate: the real multiplication is indeed extended.
\end{definition}