Definition:Complex Point at Infinity
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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 (complex) 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 $\infty$.
Also see
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Spherical Representation of Complex Numbers. Stereographic Projection