Directed infinity

A directed infinity is a type of infinity in the complex plane that has a defined complex argument θ but an infinite absolute value r.[1] For example, the limit of 1/x where x is a positive real number approaching zero is a directed infinity with argument 0; however, 1/0 is not a directed infinity, but a complex infinity. Some rules for manipulation of directed infinities (with all variables finite) are:

* [math]\displaystyle{ z\infty = \sgn(z)\infty \text{ if } z\ne 0 }[/math]
* [math]\displaystyle{ 0\infty\text{ is undefined, as is }\frac{z\infty}{w\infty} }[/math]
* [math]\displaystyle{ a z\infty = \begin{cases} \sgn(z)\infty & \text{if }a \gt 0, \\\ -\sgn(z)\infty & \text{if }a \lt 0. \end{cases} }[/math]
* [math]\displaystyle{ w\infty z\infty = \sgn(w z)\infty }[/math]

Here, sgn(z) = z/|z| is the complex signum function.

## See also

* Point at infinity

## References

1. ↑ Weisstein, Eric W.. "Directed Infinity". http://mathworld.wolfram.com/DirectedInfinity.html.

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Infinity (∞)

History|

* Controversy over Cantor's theory

Branches of mathematics|

* Internal set theory
* Non-standard analysis
* Set theory
* Synthetic differential geometry

Formalizations of infinity|

* Cardinal numbers
* Hyperreal numbers
* Infinity + 1
* Ordinal numbers
* Surreal numbers
* Transfinite numbers
* Infinitesimal

Mathematicians|

* Georg Cantor
* Gottfried Wilhelm Leibniz
* Abraham Robinson

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