Definition:Extended Real Addition

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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.


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 is not defined on all of $\overline \R \times \overline \R$; however, it is a partial mapping

Also see