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:

* z∞=sgn⁡(z)∞ if z≠0
* 0∞ is undefined, as is z∞w∞
* az∞={sgn⁡(z)∞if a>0,−sgn⁡(z)∞if a<0.
* w∞z∞=sgn⁡(wz)∞

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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