Definition:Modulus

Complex Number
Let $$z = a + \imath b$$ be a complex number.

Then the (complex) modulus of $$z$$ is written $$\left|{z}\right|$$ and is defined as:

$$\left|{z}\right| \ \stackrel {\mathbf {def}} {=\!=} \ \sqrt {a^2 + b^2}$$.

Real Number
Note that when $$y = 0$$, i.e. when $$z$$ is wholly real, this becomes $$\left|{z}\right| = \sqrt{x^2} = \left|{x}\right|$$, which is consistent with the definition of the absolute value of $\left|{x}\right|$.

Complex-Valued Function
Let $$f: S \to \C$$ be a complex-valued function.

Then the (complex) modulus of $$f$$ is written $$\left|{f}\right|: S \to \R$$ and is the real-valued function defined as:

$$\forall z \in S: \left|{f}\right| \left({z}\right) = \left|{f \left({z}\right)}\right|$$.

Geometric Function Theory
In geometric function theory, the term modulus is used to denote certain conformal invariants of configurations or curve families.

More precisely, the modulus of a curve family $$\Gamma$$ is the reciprocal of its extremal length:
 * $$\operatorname{mod}(\Gamma) := \frac{1}{\lambda(\Gamma)}.$$

Modulus of a Quadrilateral
Consider a quadrilateral; that is, a Jordan domain $$Q$$ in the complex plane (or some other Riemann surface), together with two disjoint closed boundary arcs $$\alpha$$ and $$\alpha'$$.

Then the modulus of the quadrilateral $$Q(\alpha,\alpha')$$ is the extremal length of the family of curves in $$Q$$ that connect $$\alpha$$ and $$\alpha'$$.

Equivalently, there exists a rectangle $$R=\{x+iy: |x|<a, |y|<b\}$$ and a conformal isomorphism between $$Q$$ and $$R$$ under which $$\alpha$$ and $$\alpha'$$ correspond to the vertical sides of $$R$$.

Then the modulus of $$Q(\alpha,\alpha')$$ is equal to the ratio $$a/b$$.

See Modulus of a Quadrilateral.

Modulus of an Annulus
Consider an annulus $$A$$; that is, a domain whose boundary consists of two Jordan curves.

Then the modulus $$\operatorname{mod}(A)$$ is the extremal length of the family of curves in $$A$$ that connect the two boundary components of $$A$$.

Equivalently, there is a round annulus $$\tilde{A}=\{z\in\C: r<|z|<R\}$$ that is conformally equivalent to $$A$$.

Then $$\operatorname{mod}\left ({A}\right) = \frac 1 {2\pi} \log \left({\frac R r}\right)$$.

See Modulus of an Annulus.