# Definition:Extended Real Subtraction

## Definition

Let $\overline \R$ denote the extended real numbers.

Define extended real subtraction or subtraction 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 subtraction
$\forall x \in \R: x -_{\overline \R} \paren {+\infty} = \paren {-\infty} -_{\overline \R} x := -\infty$
$\forall x \in \R: x -_{\overline \R} \paren {-\infty} = \paren {+\infty} -_{\overline \R} x := +\infty$
$\paren {-\infty} -_{\overline \R} \paren {+\infty} := -\infty$
$\paren {+\infty} -_{\overline \R} \paren {-\infty} := +\infty$

In particular, the expressions:

$\paren {+\infty} -_{\overline \R} \paren {+\infty}$
$\paren {-\infty} -_{\overline \R} \paren {-\infty}$

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 subtraction is appropriate: the operation of real subtraction is indeed extended.

## Caution

While it is tempting to think of extended real subtraction as simply real subtraction, there are some intricacies:

• It is not the case that $-\paren {+\infty} = -\infty$ in the sense of additive inverse, because $\paren {+\infty}+_{\overline \R} \paren {-\infty}$ is not defined, and in particular, not equal to $0$.
• $-_{\overline \R}$ is not a mapping as it isn't defined on all of $\overline \R \times \overline \R$; however, it is a partial mapping.