\begin{definition}[Definition:Complex Point at Infinity]
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.
This point can be denoted $\infty$.
\end{definition}