Definition:Extended Real Addition

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

Define extended real addition or addition on $\overline{\R}$, denoted $+_{\overline{\R}}: \overline{\R} \times \overline{\R} \to \overline{\R}$, by:


 * $\forall x,y \in \R: x +_{\overline{\R}} y := x +_{\R} y$ where $+_\R$ denotes real addition
 * $\forall x \in \R: x +_{\overline{\R}} \left({+\infty}\right) = \left({+\infty}\right) +_{\overline{\R}} x := +\infty$
 * $\forall x \in \R: x +_{\overline{\R}} \left({-\infty}\right) = \left({-\infty}\right) +_{\overline{\R}} x := -\infty$
 * $\left({+\infty}\right) +_{\overline{\R}} \left({+\infty}\right) := +\infty$
 * $\left({-\infty}\right) +_{\overline{\R}} \left({-\infty}\right) := -\infty$

In particular, the expressions:


 * $\left({+\infty}\right) +_{\overline{\R}} \left({-\infty}\right)$
 * $\left({-\infty}\right) +_{\overline{\R}} \left({+\infty}\right)$

are considered void and should be avoided.

When no danger of confusion arises, $+_{\overline{\R}}$ is usually replaced with the more familiar $+$.

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

Caution
While it is tempting to think of extended real addition as simply addition, there are some intricacies:
 * It is not the case that $\left({+\infty}\right) +_{\overline{\R}} \left({-\infty}\right) = 0$; this expression is not defined.
 * $+_{\overline{\R}}$ is not a mapping as it isn't defined on all of $\overline{\R} \times \overline{\R}$; however, it is a partial mapping