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 real subtraction is indeed extended.

Caution
While it is tempting to think of extended real subtraction as simply subtraction, there are some intricacies:
 * It is not the case that $- \left({+\infty}\right) = -\infty$ 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.

Also see

 * Definition:Subtraction
 * Definition:Extended Real Addition
 * Definition:Extended Real Multiplication