Distance between Points in Complex Plane

Theorem

Let $A$ and $B$ be points in the complex plane such that:

$A = \tuple {x_1, y_1}$

$B = \tuple {x_2, y_2}$

Then the distance between $A$ and $B$ is given by:

\(\ds \size {AB}\)

\(=\)

\(\ds \sqrt {\paren {x_2 - x_1}^2 + \paren {y_2 - y_1}^2}\)

\(\ds \)

\(=\)

\(\ds \cmod {z_1 - z_2}\)

where $z_1$ and $z_2$ are represented by the complex numbers $z_1$ and $z_2$ respectively.

Proof

We have:

\(\ds AB\)

\(=\)

\(\ds z_2 - z_1\)

Geometrical Interpretation of Complex Subtraction

\(\ds \)

\(=\)

\(\ds \paren {x_2 + i y_2} - \paren {x_1 + i y_1}\)

\(\ds \)

\(=\)

\(\ds \paren {x_2 - x_1} + \paren {y_2 - y_1} i\)

Definition of Complex Subtraction

\(\ds \leadsto \ \ \)

\(\ds \size {AB}\)

\(=\)

\(\ds \cmod {\paren {x_2 - x_1} + \paren {y_2 - y_1} i}\)

\(\ds \)

\(=\)

\(\ds \sqrt {\paren {x_2 - x_1}^2 + \paren {y_2 - y_1}^2}\)

Definition of Complex Modulus

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $8 \ \text {(b)}$
• 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.5$ Subsets of the complex plane