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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 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:

\(\ds \forall n \in \Z: \, \) \(\ds n\) \(<\) \(\ds \infty\)
\(\ds \forall n \in \Z: \, \) \(\ds n + \infty\) \(=\) \(\ds \infty\)
\(\ds \forall n \in \Z: \, \) \(\ds n \times \infty\) \(=\) \(\ds \infty\)
\(\ds \infty^2\) \(=\) \(\ds \infty\)

Similarly, the quantity written as $-\infty$ is defined as having the following properties:

\(\ds \forall n \in \Z: \, \) \(\ds -\infty\) \(<\) \(\ds n\)
\(\ds \forall n \in \Z: \, \) \(\ds -\infty + n\) \(=\) \(\ds -\infty\)
\(\ds \forall n \in \Z: \, \) \(\ds -\infty \times n\) \(=\) \(\ds -\infty\)
\(\ds \paren {-\infty}^2\) \(=\) \(\ds -\infty\)

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.

Ad Infinitum

The term ad infinitum means:

repeating indefinitely
generating an infinite sequence of terms

and so on.

Sometimes it has the suggestion of an endless loop.

Also, ad infinitum can be used when an infinite sequence is defined merely by showing some initial terms but without explicitly specifying either with a formula or recursively.

Also see

  • Results about infinity can be found here.

Historical Note

The concept of infinity has bothered scientists, mathematicians and philosophers since the time of Plato (who accepted the concept as realisable) Aristotle (who did not).

The symbol $\infty$ for infinity was introduced by John Wallis in the $17$th century.

It was Georg Cantor in the $1870$s who finally made the bold step of positing the actual existence of infinite sets as mathematical objects which paved the way towards a proper understanding of infinity.