If z is a complex number and z = x + yi, the modulus of z, denoted by |z| (read as ‘mod z’), is equal to $\sqrt{x^2+y^2}$

Modulus is multiplicative, which means |zw| = |z||w| for complex numbers z,w, and also satisfies the triangle inequality |z + w| ≤ |z| + |w|. If z is represented by the point P in the complex plane, the modulus of z equals the distance |OP|. Thus |z|= r, where (r, θ‎) are the polar coordinates of P. If z is real, the modulus of z equals the absolute value of the real number, so the two uses of the same notation are consistent, and the term ‘modulus’ may be used for ‘absolute value’.

The real modulus function |x| is an example of a function which is not differentiable. Its left derivative at 0 is –1, but its right derivative is 1.