Complex Magnitude

📐 Definition

For a complex number $z = x + iy \in \mathbb{C}$ , the complex magnitude (or modulus) is defined by

\lvert z \rvert = \sqrt{x^2 + y^2}

Equivalently, using the complex conjugate $\overline{z} = x - iy$ ,

\lvert z \rvert = \sqrt{z\,\overline{z}}

The magnitude measures the Euclidean distance of $z$ from the origin in the complex plane.

|z| \ge 0, \qquad |z| = 0 \iff z = 0

The magnitude is multiplicative with respect to complex multiplication:

|zw| = |z|\,|w|

for all $z,w \in \mathbb{C}$ .

The magnitude satisfies the triangle inequality,

|z + w| \le |z| + |w|

which defines a norm on $\mathbb{C}$ .

|z| = |\overline{z}|

Complex conjugation preserves magnitude while reversing orientation in the complex plane.

If $z = r e^{i\theta}$ with $r \ge 0$ , then

|z| = r

Thus, the magnitude extracts the radial component of a complex number in polar form.

Complex Phase Argument extraction and phase information

Complex Multiplication & Division Products and quotients in polar form

📜 Classical Origins

The geometric interpretation of complex numbers as points in the plane was developed in the late 18th and early 19th centuries by Caspar Wessel, Jean-Robert Argand, and Carl Friedrich Gauss. This viewpoint naturally led to the definition of magnitude as Euclidean distance.
🔬 Formalization in Analysis

In the 19th century, the modulus became central to complex analysis through the work of Cauchy and Weierstrass, providing the foundation for convergence, continuity, and analyticity in the complex plane.
🌍 Modern Perspective

Today, the complex magnitude is understood as the norm induced by the standard inner product on $\mathbb{C} \cong \mathbb{R}^2$ , serving as a prototype for norms in functional analysis and Hilbert spaces.