\begin{definition}[Definition:Extended Real Addition]
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.
\end{definition}