# 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 $\mathbb S$.

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

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Spherical Representation of Complex Numbers. Stereographic Projection