Complex Absolute Value¶

Definition¶

Given a complex number $z=a+\mathrm{i}b$ where $a,b\in\mathbb{R}$ $R$ , the absolute value or modulus of $z$ , denoted $|z|$ , is the real number $|z| = \sqrt{a^2 + b^2}.$

Interpretation¶

The modulus $|z|$ measures the length of $z$ : its distance from $0$ in the complex plane. The square root in the definition is the nonnegative root, so $|z|\ge0$ $0$ , with $|z|=0$ only for $z=0$ $0$ .

A complex number z = a + bi drawn as an arrow from 0, forming a right triangle with sides a and b, whose hypotenuse is the modulus.

The arrow from $0$ to $z=a+b\mathrm{i}$ is the hypotenuse of a right triangle with a horizontal side of length $a$ and a vertical side of length $b$ , so by Pythagoras its length is $\sqrt{a^2+b^2}$ . In another quadrant the sides have lengths $|a|$ and $|b|$ , which gives the same formula, since $|a|^2=a^2$ $a^{2}$ .

For a real number $z=a$ the formula gives $\sqrt{a^2}=|a|$ $∣ a ∣$ , so the modulus extends the absolute value of real numbers, and the same notation is used. As for real numbers, $|z-w|$ is the distance between $z$ and $w$ .

A common slip is to compute $|z|^2$ as $z^2$ . The correct identity uses the conjugate, $|z|^2=z\overline{z}$ $z \overline{z}$ : for $z=\mathrm{i}$ one has $z^2=-1$ but $|z|^2=1$ $1$ .

The modulus is multiplicative, $|zw|=|z|\,|w|$ $∣ z ∣ ∣ w ∣$ , so multiplying by $w$ stretches lengths by the factor $|w|$ . Together with the argument, the modulus gives the polar form of $z$ . The same Pythagorean length later measures vectors in $\mathbb{R}^n$ , as the Euclidean norm.

Key Results¶

While it is true that $|z| = \sqrt{z^2}$ when $z$ is a real number, it does not hold whenever $z$ is a non-real complex number, i.e., $|z| \neq \sqrt{z^2}$ for $z\in (\mathbb{C}\setminus \mathbb{R})$ $R)$ .
The modulus is multiplicative, $|zw|=|z|\,|w|$ $∣ z ∣ ∣ w ∣$ , and $|z|^2=z\bar z$ $z \overset{z}{ˉ}$ .

Examples¶

$|1+\mathrm{i}|=\sqrt{2}$ $2$ .
$|z|=1$ represents a unit circle in the complex plane.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 1

Aliases: complex modulus, modulus of a complex number, magnitude of a complex number Symbols: complex numbers, element of, real numbers, square root

Last updated: 20 October 2026 03:02