Unique Representation of Complex Number in Spherical Form

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Theorem

Let $\mathcal P$ be the complex plane.

Let $\mathbb S$ be the unit sphere which is tangent to $\mathcal P$ at $\tuple {0, 0}$ (that is, where $z = 0$).

Let the diameter of $\mathbb S$ perpendicular to $\mathcal P$ through $\tuple {0, 0}$ be $NS$ where $S$ is the point $\tuple {0, 0}$.

Let the point $N$ be referred to as the north pole of $\mathbb S$ and $S$ be referred to as the south pole of $\mathbb S$.

Spherical-Represenation-of-Complex-Number.png


Let $A$ be a point on $P$.

Let the line $NA$ be constructed.

Then $NA$ passes through exactly one point $A'$ on the surface of $\mathbb S$ apart from $N$.


Similarly, let $A'$ be a point on the surface of $\mathbb S$ apart from $N$.

Let the line $NA'$ be constructed.

Then $NA'$ passes through exactly one point $A$ on $P$.


Proof


Sources