# Definition:Infinity

## Definition

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:

**endlessly****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

- Definition:Extended Real Number Line
- Definition:Extended Natural Number
- Definition:Positive Infinity
- Definition:Negative Infinity
- Definition:Projective Line
- Definition:New Element

- 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.

## Sources

- 1973: G. Stephenson:
*Mathematical Methods for Science Students*(2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.1$ Real Numbers - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.3$: Arithmetic - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**infinity** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**infinity** - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (next): Chapter $1$: General Background: $\S 1$ What is infinity? - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**infinity**