# Modulus

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 $\sqrt{1^2+2^2}=\sqrt{5}$.