Definition:Extended Real Multiplication

Definition
Let $\overline{\R}$ denote the extended real numbers.

Define extended real multiplication or multiplication on $\overline{\R}$, denoted $\cdot_{\overline{\R}}: \overline{\R} \times \overline{\R} \to \overline{\R}$, by:


 * $\forall x \in \overline{\R}: x \cdot_{\overline{\R}} 0 = 0 \cdot_{\overline{\R}} x = 0$
 * $\forall x,y \in \R: x \cdot_{\overline{\R}} y := x \cdot_{\R} y$ where $\cdot_\R$ denotes real multiplication
 * $\forall x \in \R_{>0}: x \cdot_{\overline{\R}} \left({+\infty}\right) = \left({+\infty}\right) \cdot_{\overline{\R}} x := +\infty$
 * $\forall x \in \R_{<0}: x \cdot_{\overline{\R}} \left({+\infty}\right) = \left({+\infty}\right) \cdot_{\overline{\R}} x := -\infty$
 * $\forall x \in \R_{>0}: x \cdot_{\overline{\R}} \left({-\infty}\right) = \left({-\infty}\right) \cdot_{\overline{\R}} x := -\infty$
 * $\forall x \in \R_{<0}: x \cdot_{\overline{\R}} \left({-\infty}\right) = \left({-\infty}\right) \cdot_{\overline{\R}} x := +\infty$
 * $\left({+\infty}\right) \cdot_{\overline{\R}} \left({+\infty}\right) := +\infty$
 * $\left({-\infty}\right) \cdot_{\overline{\R}} \left({-\infty}\right) := +\infty$
 * $\left({+\infty}\right) \cdot_{\overline{\R}} \left({-\infty}\right) := -\infty$
 * $\left({-\infty}\right) \cdot_{\overline{\R}} \left({+\infty}\right) := -\infty$

When no danger of confusion arises, $\cdot_{\overline{\R}}$ is usually replaced with the more familiar $\cdot$, or even suppressed.

From the definition of $\cdot_{\overline{\R}}$ on bona fide real numbers, the name extended real multiplication is appropriate: the real multiplication is indeed extended.