# Definition:Complex Number as Vector

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## Definition

Let $z = x + i y$ be a complex number.

Then $z$ can be considered as a vector $OP$ in the complex plane such that:

- its initial point is the origin
- its terminal point $P$ is the point $\tuple {x, y}$.

Two vectors which have the same magnitude and direction, but different initial points, are considered **equal**.

## Also known as

The vector $OP$ is also known as the **position vector** of $P$.

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 2$. Geometrical Representations - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Vector Interpretation of Complex Numbers