The modulus of an object is its size.

For real numbers it is the same as the absolute value.

For complex numbers, the modulus of \(z=x+iy\) is given by \(|z|=\sqrt{x^{2}+y^{2}}\), which is the distance of \(z\) from the origin in the complex plane. It is sometimes convenient to calculate \(|z|\) using the complex conjugate \(z^{*}= x - iy\) since \(|z|^2 = zz^*\). If \(z\) is given in the polar form \(re^{i\theta}\), where \(r\ge0\), then \(|z|=r\).

For vectors, the modulus of a vector \(\mathbf{v}\) is its magnitude (length), written \(|\mathbf{v}|\). It is calculated using Pythagoras’ Theorem. For example, the modulus of \(\begin{pmatrix}1\\2\end{pmatrix}\) is \(\sqrt{1^2+2^2}=\sqrt{5}\).