Definition:Complex Point at Infinity

Definition
The zero in the set of complex numbers $\C$ has no inverse for multiplication.

That is, the expression:
 * $\dfrac 1 0$

has no meaning.

The point at infinity is the element added to $\C$ in order to allow $\C$ to be closed under division:
 * $\forall x, y \in \C: \dfrac x y \in \C$

The set $\C$ with that point added is known as the extended complex plane.

Conceptually, it can be imagined as a point which is at infinity in all directions.

It can also be considered as the $N$ point on the Riemann sphere which does not map to the complex plane.

Also see

 * Definition:Spherical Representation of Complex Number‎
 * Definition:Stereographic Projection


 * Definition:Extended Complex Plane