\begin{definition}[Definition:Infinity]
Informally, the term '''infinity''' is used to mean '''some infinite number''', but this concept falls very far short of a usable definition.
The symbol $\infty$ (supposedly invented by {{AuthorRef|John Wallis}}) is often used in this context to mean '''an infinite number'''.
However, outside of its formal use in the definition of limits its use is strongly discouraged until you know what you're talking about.
It is defined as having the following properties:
:$\forall n \in \Z: n < \infty$
:$\forall n \in \Z: n + \infty = \infty$
:$\forall n \in \Z: n \times \infty = \infty$
:$\infty^2 = \infty$
Similarly, the quantity written as $-\infty$ is defined as having the following properties:
:$\forall n \in \Z: -\infty< n$
:$\forall n \in \Z: -\infty + n = -\infty$
:$\forall n \in \Z: -\infty \times n = -\infty$
:$\paren {-\infty}^2 = -\infty$
The latter result seems wrong when you think of the rule that a negative number square equals a positive one, but remember that infinity is not exactly a number as such.
\end{definition}